Abstract We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of t terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes. Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target $$k &lt; t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> , and requires that only k of the terminals be informed in the minimum number of rounds. For this problem, we improve the implications of the previous results and obtain a multiplicative approximation factor of $$\tilde{O}(t^{1/3})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . For the directed version, we obtain an additive $$\tilde{O}(k^{1/2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> approximation algorithm (with a polylogarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding k -trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints. We also study the problem of bounded degree Directed Steiner Tree, for which we obtain improved polylogarithmic approximations for the special case of bounded treewidth graphs. This extends prior work on the Group Steiner Tree problem.

In the Steiner Path Aggregation Problem , our goal is to aggregate paths in a directed network into a single arborescence without significantly disrupting the paths. In particular, we are given a directed multigraph with colored arcs, a root, and k terminals, each of which has a monochromatic path to the root. Our goal is to find an arborescence in which every terminal has a path to the root, and its path does not switch colors too many times. We give an efficient algorithm that finds such a solution with at most 2 log 4 3 ⁡ k color switches. Up to constant factors this is the best possible universal bound, as there are graphs requiring at least log 2 ⁡ k color switches.

Flows and colorings are disparate concepts in graph algorithms -- the former is tractable while the latter is intractable. Tutte introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger showed that nowhere-zero flows are equivalent to cut-balanced orientations. Motivated by connections between nowhere-zero flows, cut-balanced orientations, Nash-Williams' well-balanced orientations, and postman problems, we study optimization versions of nowhere-zero flows and cut-balanced orientations. Given a bidirected graph with asymmetric costs on two orientations of each edge, we study the min cost nowhere-zero $k$-flow problem and min cost $k$-cut-balanced orientation problem. We show that both problems are NP-hard to approximate within any finite factor. Given the strong inapproximability result, we design bicriteria approximations for both problems: we obtain a $(6,6)$-approximation to the min cost nowhere-zero $k$-flow and a $(k,6)$-approximation to the min cost $k$-cut-balanced orientation. For the case of symmetric costs (where the costs of both orientations are the same for every edge), we show that the nowhere-zero $k$-flow problem remains NP-hard and admits a $3$-approximation.

We study an online generalization of the classic Joint Replenishment Problem (JRP) that models the trade-off between ordering costs, holding costs, and backlog costs in supply chain planning systems. A retailer places orders to a supplier for multiple items over time: each request is for some item that the retailer needs in the future, and has an arrival time and a soft deadline. If a request is served before its deadline, the retailer pays a holding cost per unit of the item until the deadline. However, if a request is served after its deadline, the retailer pays a backlog cost per unit. Each service incurs a fixed joint service cost and a fixed item-dependent cost for every item included in a service. These fixed costs are the same irrespective of the units of each item ordered. The goal is to schedule services to satisfy all the online requests while minimizing the sum of the service costs, the holding costs, and the backlog costs.

We analyze a general problem in a crowd-sourced setting where one user asks a question (also called item) and other users return answers (also called labels) for this question. Different from existing crowd sourcing work which focuses on finding the most appropriate label for the question (the “truth”), our problem is to determine a ranking of the users based on their ability to answer questions. We call this problem “ability discovery” to emphasize the connection to and duality with the more well-studied problem of “truth discovery”. To model items and their labels in a principled way, we draw upon Item Response Theory (IRT) which is the widely accepted theory behind standardized tests such as SAT and GRE. We start from an idealized setting where the relative performance of users is consistent across items and better users choose better fitting labels for each item. We posit that a principled algorithmic solution to our more general problem should solve this ideal setting correctly and observe that the response matrices in this setting obey the Consecutive Ones Property (C1P). While C1P is well understood algorithmically with various discrete algorithms, we devise a novel variant of the HITS algorithm which we call “HITSnDIFFs” (or HnD), and prove that it can recover the ideal C1P-permutation in case it exists. Unlike fast combinatorial algorithms for finding the consecutive ones permutation (if it exists), HnD also returns an ordering when such a permutation does not exist. Thus it provides a principled heuristic for our problem that is guaranteed to return the correct answer in the ideal setting. Our experiments show that HnD produces user rankings with robustly high accuracy compared to state-of-the-art truth discovery methods. We also show that our novel variant of HITS scales better in the number of users than ABH, the only prior spectral C1P reconstruction algorithm.

In pool-based active learning, the learner is given an unlabeled data set and aims to efficiently learn the unknown hypothesis by querying the labels of the data points. This can be formulated as the classical Optimal Decision Tree (ODT) problem: Given a set of tests, a set of hypotheses, and an outcome for each pair of test and hypothesis, our objective is to find a low-cost testing procedure (i.e., decision tree) that identifies the true hypothesis. This optimization problem has been extensively studied under the assumption that each test generates a deterministic outcome. However, in numerous applications, for example, clinical trials, the outcomes may be uncertain, which renders the ideas in the deterministic setting invalid. In this work, we study a fundamental variant of the ODT problem in which some test outcomes are noisy, even in the more general case where the noise is persistent, i.e., repeating a test gives the same noisy output. Our approximation algorithms provide guarantees that are nearly best possible and hold for the general case of a large number of noisy outcomes per test or per hypothesis where the performance degrades continuously with this number. Furthermore, most of our results hold for a more general problem called Adaptive Submodular Ranking with Noise (ASRN). We numerically evaluated our algorithms for identifying toxic chemicals and learning linear classifiers and observed that our algorithms have costs very close to the information-theoretic minimum.

We interleave sampling based motion planning methods with pruning ideas from minimum spanning tree algorithms to develop a new approach for solving a Multi-Goal Path Finding (MGPF) problem in high dimensional spaces. The approach alternates between sampling points from selected regions in the search space and de-emphasizing regions that may not lead to good solutions for MGPF. Our approach provides an asymptotic, 2-approximation guarantee for MGPF. We also present extensive numerical results to illustrate the advantages of our proposed approach over uniform sampling in terms of the quality of the solutions found and computation speed. Note to Practitioners—MGPF is concerned with finding a collision-free, near-optimal path for a robot visiting a set of target configurations. This problem arises in applications that use robotic manipulators such as advanced manufacturing, surface inspection, package sorting, and in other logistical applications where the cost of the traveling between any two configurations of a robot cannot be readily determined a-priori. As robots are expected to perform a large number of tasks, the sequencing of these tasks become important specifically when the travel costs are challenging to estimate. This paper provides an approach to handle this problem in higher dimensions with theoretical guarantees as well as provides simulation results on a broad class of environments to corroborate its performance with respect to the state of the art.

Modern platforms leverage randomized experiments to make informed decisions from a given set of alternatives. As a particularly challenging scenario, these alternatives can potentially have (i) high volume, with thousands of new items being released each hour, and (ii) short lifetime, either due to the contents' transient nature, or some underlying non-stationarity that impels the learner to treat the same item as non-identical copies across time. We consider a multiplay bandits model. In each round a set of $k=n^\rho$ actions that will be available for $w$ rounds arrives, each of whose mean reward is drawn from a fixed known distribution. The learner selects a multiset of $n$ actions at a time. We propose an $\ell$-Layered Sieve Policy that recursively refines the action space for $\ell\leq w$ times. We show that for any given $\rho>0$, with suitable $\ell$, the policy achieves $\tilde O (n^{-\min \{\rho, \frac 12 (1+\frac 1w)^{-1}\}})$ regret. We also complement this result with an $\Omega (n^{-\min \{\rho, \frac 12\}})$ lower bound. We further validate the effectiveness of our Sieve Policy via numerical simulations and a field experiment in a large content card serving platform.

In the Steiner Tree Augmentation Problem (STAP), we are given a graph G = (V,E), a set of terminals R ⊆ V, and a Steiner tree T spanning R. The edges L: = E\E(T) are called links and have non-negative costs. The goal is to augment T by adding a minimum cost set of links, so that there are 2 edge-disjoint paths between each pair of vertices in R. This problem is a special case of the Survivable Network Design Problem, which can be approximated to within a factor of 2 using iterative rounding [13]. We give the first polynomial time algorithm for STAP with approximation ratio better than 2. In particular, we achieve an approximation ratio of (1.5 + ε). To do this, we employ the Local Search approach of [24] for the Tree Augmentation Problem and generalize their main decomposition theorem from links (of size two) to hyper-links. We also consider the Node-Weighted Steiner Tree Augmentation Problem (NW-STAP) in which the non-terminal nodes have non-negative costs. We seek a cheapest subset S ⊆ V\R so that G[R ∪ S] is 2-edge-connected. Using a result of Nutov [18], there exists an O(log |R|)-approximation for this problem. We provide an O(log2(|R|))-approximation algorithm for NW-STAP using a greedy algorithm leveraging the spider decomposition of optimal solutions.

We consider an information dissemination problem where the root node in an undirected graph constantly updates its information. The goal is to keep every other node in the graph as freshly informed about the root as possible. Our synchronous information spreading model uses telephone calls at each time step, in which any node can communicate with at most one neighbor, thus forming a matching over which information is transmitted at each step. We introduce two problems in minimizing two natural objectives (Maximum and Average) of the latency of the root's information at all nodes in the network. After deriving a simple reduction from the maximum rooted latency problem to the well-studied minimum broadcast time problem, we focus on the average rooted latency version. We introduce a natural problem of finding a finite schedule that minimizes the average broadcast time from a root. We show that any average rooted latency scheme induces a solution to this average broadcast problem within a constant factor and conversely, this average broadcast time is within a logarithmic factor of the average rooted latency. Then, we derive a log-squared approximation algorithm for the average broadcast time problem via rounding a time-indexed linear programming relaxation, resulting in a log-cubed approximation for the average latency problem. Surprisingly, we show that using the average broadcast time for average rooted latency introduces a necessary logarithmic factor overhead even in trees. We overcome this hurdle and give a 40-approximation for trees. For this, we design an algorithm to find near-optimal locally-periodic schedules in trees where each vertex receives information from its parent in regular intervals. On the other side, we show how such well-behaved schedules approximate the optimal schedule within a constant factor. * This material is based upon work supported in part by the U. S. Office of Naval Research under award number N00014-21-1-2243 and the Air Force Office of Scientific Research under award number FA9550-20-1-0080.

Many applications such as hiring and university admissions involve evaluation and selection of applicants. These tasks are fundamentally difficult, and require combining evidence from multiple different aspects (what we term "attributes"). In these applications, the number of applicants is often large, and a common practice is to assign the task to multiple evaluators in a distributed fashion. Specifically, in the often-used holistic allocation, each evaluator is assigned a subset of the applicants, and is asked to assess all relevant information for their assigned applicants. However, such an evaluation process is subject to issues such as miscalibration (evaluators see only a small fraction of the applicants and may not get a good sense of relative quality), and discrimination (evaluators are influenced by irrelevant information about the applicants). We identify that such attribute-based evaluation allows alternative allocation schemes. Specifically, we consider assigning each evaluator more applicants but fewer attributes per applicant, termed segmented allocation. We compare segmented allocation to holistic allocation on several dimensions via theoretical and experimental methods. We establish various tradeoffs between these two approaches, and identify conditions under which one approach results in more accurate evaluation than the other.

We consider the Continuum Bandit problem where the goal is to find the optimal action under an unknown reward function, with an additional monotonicity constraint (or, "markdown" constraint) that requires that the action sequence be non-increasing. This problem faithfully models a natural single-product dynamic pricing problem, called "markdown pricing", where the objective is to adaptively reduce the price over a finite sales horizon to maximize expected revenues. Jia et al '21 and Chen '21 independently showed a tight $T^{3/4}$ regret bound over $T$ rounds under *minimal* assumptions of unimodality and Lipschitzness in the reward (or, "revenue") function. This bound shows that the demand learning in markdown pricing is harder than unconstrained (i.e., without the monotonicity constraint) pricing under unknown demand which suffers regret only of the order of $T^{2/3}$ under the same assumptions (Kleinberg '04). However, in practice the demand functions are usually assumed to have certain functional forms (e.g. linear or exponential), rendering the demand-learning easier and suggesting lower regret bounds. We investigate two fundamental questions, assuming the underlying demand curve comes from a given parametric family: (1) Can we improve the $T^{3/4}$ regret bound for markdown pricing, under extra assumptions on the functional forms of the demand functions? (2) Is markdown pricing still harder than unconstrained pricing, under these additional assumptions? To answer these, we introduce a concept called markdown dimension that measures the complexity of the parametric family and present tight regret bounds under this framework, thereby completely settling the aforementioned questions.

We interleave sampling based motion planning methods with pruning ideas from minimum spanning tree algorithms to develop a new approach for solving a Multi-Goal Path Finding (MGPF) problem in high dimensional spaces. The approach alternates between sampling points from selected regions in the search space and de-emphasizing regions that may not lead to good solutions for MGPF. Our approach provides an asymptotic, 2-approximation guarantee for MGPF. We also present extensive numerical results to illustrate the advantages of our proposed approach over uniform sampling in terms of the quality of the solutions found and computation speed.

Abstract We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives: min – avg minimizes the average tour cost and min – max minimizes the maximum tour cost over all days. For min – max we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. For min – avg we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.

We consider the deterministic inventory routing problem over a discrete finite time horizon. Given clients on a metric, each with daily demands that must be delivered from a depot and holding costs over the planning horizon, an optimal solution selects a set of daily tours through a subset of clients to deliver all demands before they are due and minimizes the total holding and tour routing costs over the horizon. In the capacitated case, a limited number of vehicles are available, where each vehicle makes at most one trip per day. Each trip from the depot is allowed to carry a limited amount of supply to deliver. We develop fast heuristics for both cases by solving a family of prize-collecting Steiner tree instances. Computational experiments show our heuristics can find near-optimal solutions for both cases and substantially reduce the runtime compared with a pure mixed integer programming formulation approach.

We study a fundamental class of two-layer network design problems. A hub layer is configured by establishing hubs at selected nodes at considerable cost so that the routes between hubs can be operated cheaply. The remaining edges in the network are operated at regular cost. The resulting problem is to determine the set of nodes to open hubs and the set of edges to establish in order to find a network of minimum total cost. We consider the case where the network is required to form a Steiner tree spanning a given set of terminal vertices. When edge costs are non-metric, we show logarithmic approximation hardness even for the special case of spanning trees. On the other hand, we show a polynomial-time reduction for Steiner trees to its corresponding node-weighted version thus proving a logarithmic approximation factor. When edge costs are metric, we show the problem is only a constant factor harder to approximate than its original version (with no hub installation) using a similar reduction.

Consider an agent exploring an unknown graph in search of some goal state. As it walks around the graph, it learns the nodes and their neighbors. The agent only knows where the goal state is when it reaches it. How do we reach this goal while moving only a small distance? This problem seems hopeless, even on trees of bounded degree, unless we give the agent some help. This setting with "help" often arises in exploring large search spaces (e.g., huge game trees) where we assume access to some score/quality function for each node, which we use to guide us towards the goal. In our case, we assume the help comes in the form of distance predictions: each node v provides a prediction f(v) of its distance to the goal vertex. Naturally if these predictions are correct, we can reach the goal along a shortest path. What if the predictions are unreliable and some of them are erroneous? Can we get an algorithm whose performance relates to the error of the predictions? In this work, we consider the problem on trees and give deterministic algorithms whose total movement cost is only O(OPT + Δ ⋅ ERR), where OPT is the distance from the start to the goal vertex, Δ the maximum degree, and the ERR is the total number of vertices whose predictions are erroneous. We show this guarantee is optimal. We then consider a "planning" version of the problem where the graph and predictions are known at the beginning, so the agent can use this global information to devise a search strategy of low cost. For this planning version, we go beyond trees and give an algorithms which gets good performance on (weighted) graphs with bounded doubling dimension.

We study the statistical seriation problem, where the goal is to estimate a matrix whose rows satisfy the same shape constraint after a permutation of the columns. This is a important classical problem, with close connections to statistical literature in permutation-based models and also has wide applications ranging from archaeology to biology. Specifically, we consider the case where the rows are monotonically increasing after an unknown permutation of the columns. Past work has shown that the least-squares estimator is optimal up to logarithmic factors, but efficient algorithms for computing the least-squares estimator remain unknown to date. We approach this important problem from a heuristic perspective. Specifically, we replace the combinatorial permutation constraint by a continuous regularization term, and then use projected gradient descent to obtain a local minimum of the non-convex objective. We show that the attained local minimum is the global minimum in certain special cases under the noiseless setting, and preserves desirable properties under the noisy setting. Simulation results reveal that our proposed algorithm outperforms prior algorithms when (1) the underlying model is more complex than simplistic parametric assumptions such as low-rankedness, or (2) the signal-to-noise ratio is high. Under partial observations, the proposed algorithm requires an initialization, and different initializations may lead to different local minima. We empirically observe that the proposed algorithm yields consistent improvement over the initialization, even though different initializations start with different levels of quality.

In the weighted minimum strongly connected spanning subgraph (WMSCSS ) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative 1.5 cost. This rounding matches a known 1.5 integrality gap lower bound for a half-integral instance. More generally, we show that LP solutions whose non-zero entries are at least a value f>0 can be rounded at a multiplicative cost of 2−f.

A number of applications (e.g., AI bot tournaments, sports, peer grading, crowdsourcing) use pairwise comparison data and the Bradley-Terry-Luce (BTL) model to evaluate a given collection of items (e.g., bots, teams, students, search results). Past work has shown that under the BTL model, the widely-used maximum-likelihood estimator (MLE) is minimax-optimal in estimating the item parameters, in terms of the mean squared error. However, another important desideratum for designing estimators is fairness. In this work, we consider one specific type of fairness, which is the notion of bias in statistics. We show that the MLE incurs a suboptimal rate in terms of bias. We then propose a simple modification to the MLE, which "stretches" the bounding box of the maximum-likelihood optimizer by a small constant factor from the underlying ground truth domain. We show that this simple modification leads to an improved rate in bias, while maintaining minimax-optimality in the mean squared error. In this manner, our proposed class of estimators provably improves fairness in the sense of bias without loss in accuracy.

In this paper we study how to optimally balance cheap inflexible resources with more expensive, reconfigurable resources despite uncertainty in the input problem. Specifically, we introduce the MinEMax model to study "build versus rent" problems. In our model different scenarios appear independently. Before knowing which scenarios appear, we may build rigid resources that cannot be changed for different scenarios. Once we know which scenarios appear, we are allowed to rent reconfigurable but expensive resources to use across scenarios. Although computing the objective in our model might seem to require enumerating exponentially-many possibilities, we show it is well-estimated by a surrogate objective which is representable by a polynomial-size LP. In this surrogate objective we pay for each scenario only to the extent that it exceeds a certain threshold. Using this objective we design algorithms that approximately-optimally balance inflexible and reconfigurable resources for several NP-hard covering problems. For example, we study minimum spanning and Steiner trees, minimum cuts and facility location variants. Up to constants our approximation guarantees match those of previous algorithms for the previously-studied demand-robust and stochastic two-stage models. Lastly, we demonstrate that our problem is sufficiently general to smoothly interpolate between previous demand-robust and stochastic two-stage problems.

We study rumor spreading in graphs, specifically multicommodity multicast problem under the wireless model: given source-destination pairs in the graph, one needs to find the fastest schedule to transfer information from each source to the corresponding destination. Under the wireless model, nodes can transmit to any subset of their neighbors in synchronous time steps, as long as they either transmit or receive from at most one transmitter during the same time step. We improve approximation ratio for this problem from O~(n^(2/3)) to O~(n^((1/2) + epsilon)) on n-node graphs. We also design an algorithm that satisfies p given demand pairs in O(OPT + p) steps, where OPT is the length of an optimal schedule, by reducing it to the well-studied packet routing problem. In the case where underlying graph is an n-node tree, we improve the previously best-known approximation ratio of O((log n)/(log log n)) to 3. One consequence of our proof is a simple constructive rule for optimal broadcasting in a tree under a widely studied telephone model.

A fundamental task in active learning involves performing a sequence of tests to identify an unknown hypothesis that is drawn from a known distribution. This problem, known as optimal decision tree induction, has been widely studied for decades and the asymptotically best-possible approximation algorithm has been devised for it. We study a generalization where certain test outcomes are noisy, even in the more general case when the noise is persistent, i.e., repeating the test on the scenario gives the same noisy output, disallowing simple repetition as a way to gain confidence. We design new approximation algorithms for both the non-adaptive setting, where the test sequence must be fixed a-priori, and the adaptive setting where the test sequence depends on the outcomes of prior tests. Previous work in the area assumed at most a constant number of noisy outcomes per test and per scenario and provided approximation ratios that were problem dependent (such as the minimum probability of a hypothesis). Our new approximation algorithms provide guarantees that are nearly best-possible and work for the general case of a large number of noisy outcomes per test or per hypothesis where the performance degrades smoothly with this number. Our results adapt and generalize methods used for submodular ranking and stochastic set cover. We evaluate the performance of our algorithms on two natural applications with noise: toxic chemical identification and active learning of linear classifiers. Despite our logarithmic theoretical approximation guarantees, our methods give solutions with cost very close to the information theoretic minimum, demonstrating the effectiveness of our methods.

We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an $O(\sqrt{n})$-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. We give an improved $\tilde{O}(n^{1/3})$ approximation for the case of metrics induced by unweighted trees.

In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G=(V,E), nonnegative edge costs {c_e} for e in E, terminal pairs {(s_i,t_i)} for i=1,...,k, and penalties {pi_i} for i=1,...,k for each terminal pair; the goal is to find a forest F to minimize c(F) + sum{ pi_i: (s_i,t_i) is not connected in F }. The Steiner forest problem can be viewed as the special case where pi_i are infinite for all i. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF (PCSF-LP) is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least 9/4 even if the input instance is planar. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than 4. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP-approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most 1/3 and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than 3 using a direct iterative rounding method.

We show that Karger's randomized contraction method (SODA 93) can be adapted to multiobjective global minimum cut problems with a constant number of edge or node budget constraints to give efficient algorithms. For global minimum cuts with a single edge-budget constraint, our extension of the randomized contraction method has running time tilde{O}(n^3) in an n-node graph improving upon the best-known randomized algorithm with running time tilde{O}(n^4) due to Armon and Zwick (Algorithmica 2006). Our analysis also gives a new upper bound of O(n^3) for the number of optimal solutions for a single edge-budget min cut problem. For the case of (k-1) edge-budget constraints, the extension of our algorithm saves a logarithmic factor from the best-known randomized running time of O(n^{2k} log^3 n). A main feature of our algorithms is to adaptively choose, at each step, the appropriate cost function used in the random selection of edges to be contracted. For the global min cut problem with a constant number of node budgets, we give a randomized algorithm with running time tilde{O}(n^2), improving the current best determinisitic running time of O(n^3) due to Goemans and Soto (SIAM Journal on Discrete Mathematics 2013). Our method also shows that the total number of distinct optimal solutions is bounded by O(n^2) as in the case of global min-cuts. Our algorithm extends to the node-budget constrained global min cut problem excluding a given sink with the same running time and bound on number of optimal solutions, again improving upon the best-known running time by a factor of O(n). For node-budget constrained problems, our improvements arise from incorporating the idea of merging any infeasible super-nodes that arise during the random contraction process. In contrast to cuts excluding a sink, we note that the node-cardinality constrained min-cut problem containing a given source is strongly NP-hard using a reduction from graph bisection.

We study a generalization of the Steiner tree problem, where we are given a weighted network G together with a collection of k subsets of its vertices and a root r. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for k=2, and give a 3/2-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known O(log n)-approximation. Despite these obstacles, we conjecture that this problem should have an O(1)-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.

Collaborative filtering is a broad and powerful framework for building recommendation systems that has seen widespread adoption. Over the past decade, the propensity of such systems for favoring popular products and thus creating echo chambers have been observed. This has given rise to an active area of research that seeks to diversify recommendations generated by such algorithms. We address the problem of increasing diversity in recom- mendation systems that are based on collaborative filtering that use past ratings to predict a rating quality for potential recommendations. Following our earlier work, we formulate recommendation system design as a subgraph selection problem from a candidate super-graph of potential recommendations where both diversity and rating quality are explicitly optimized: (1) On the modeling side, we define a new flexible notion of diversity that allows a system designer to prescribe the number of recommendations each item should receive, and smoothly penalizes deviations from this distribution. (2) On the algorithmic side, we show that minimum-cost network flow methods yield fast algorithms in theory and practice for designing recommendation subgraphs that optimize this notion of diversity. (3) On the empirical side, we show the effectiveness of our new model and method to increase diversity while maintaining high rating quality in standard rating data sets from Netflix and MovieLens.

The online (uniform) buy-at-bulk network design problem asks us to design a network, where the edge-costs exhibit economy-of-scale. Previous approaches to this problem used tree-embeddings, giving us randomized algorithms. Moreover, the optimal results with a logarithmic competitive ratio requires the metric on which the network is being built to be known up-front; the competitive ratios then depend on the size of this metric (which could be much larger than the number of terminals that arrive). We consider the buy-at-bulk problem in the least restrictive model where the metric is not known in advance, but revealed in parts along with the demand points seeking connectivity arriving online. For the single sink buy-at-bulk problem, we give a deterministic online algorithm with competitive ratio that is logarithmic in k, the number of terminals that have arrived, matching the lower bound known even for the online Steiner tree problem. In the oblivious case when the buy-at-bulk function used to compute the edge-costs of the network is not known in advance (but is the same across all edges), we give a deterministic algorithm with competitive ratio polylogarithmic in k, the number of terminals. At the heart of our algorithms are optimal constructions for online Light Approximate Shortest-path Trees (LASTs) and spanners, and their variants. We give constructions that have optimal trade-offs in terms of cost and stretch. We also define and give constructions for a new notion of LASTs where the set of roots (in addition to the points) expands over time. We expect these techniques will find applications in other online network-design problems.

Consider the following problem: given a set system ( U , Ω) and an edge-weighted graph G = ( U , E ) on the same universe U , find the set A ∈ Ω such that the Steiner tree cost with terminals A is as large as possible—“which set in Ω is the most difficult to connect up?” This is an example of a max-min problem : find the set A ∈ Ω such that the value of some minimization (covering) problem is as large as possible. In this article, we show that for certain covering problems that admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system Ω is given by a p -system or knapsack constraints or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p -systems and knapsack constraints.

From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks. In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars.

We study the problem of computing a minimum time schedule to spread rumors in a given graph under several models: In the radio model, all neighbors of a transmitting node listen to the messages and are able to record it only when no other neighbor is transmitting; In the wireless model (also called the edge-star model), each transmitter is at a different frequency to which any neighbor can tune to, but only one neighboring transmission can be accessed in this way; In the telephone model, the set of transmitter-receiver pairs form a matching in the graph. The rumor spreading problems assume a message at one or several nodes of the graph that must reach a target node or set of nodes. The transmission proceeds in synchronous rounds under the rules of the corresponding model. The goal is to compute a schedule that completes in the minimum number of rounds. We present a comprehensive study of approximation algorithms for these problems, and show several reductions from the harder to the easier models for special demands. We show a new hardness of approximation of Omega(n^1/2 - epsilon) for the minimum radio gossip time by a connection to maximum induced matchings. We give the first sublinear approximation algorithms for the most general case of the problem under the wireless model; we also consider various special cases such as instances with symmetric demands and give better approximation algorithms. Our work exposes the relationships across the models and opens up several new avenues for further study.

Recommendations are central to the utility of many popular e-commerce websites. Such sites typically contain a set of recommendations on every product page that enables visitors and crawlers to easily navigate the website. These recommendations are essentially universally present on all e-commerce websites. Choosing an appropriate set of recommendations at each page is a critical task performed by dedicated backend software systems. We formalize the concept of recommendations used for discovery as a natural graph optimization problem on a bipartite graph and propose three methods for solving the problem in increasing order of sophistication: a local random sampling algorithm, a greedy algorithm and a more involved partitioning based algorithm. We first theoretically analyze the performance of these three methods on random graph models and characterize when each method will yield a solution of sufficient quality and the parameter ranges when more sophistication is needed. We complement this by roviding an empirical analysis of these algorithms on simulated and real-world production data from a retail website. Our results confirm that it is not always necessary to implement complicated algorithms in the real-world, and demonstrate that very good practical results can be obtained by using simple heuristics that are backed by the confidence of concrete theoretical guarantees.

We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset linear programming relaxation for this problem. For directed graphs and $k$-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node $v$ in the solution is at most $2b(v) + O(k)$, where $b(v)$ is the degree upper bound on $v$. For undirected graphs and element-connectivity requirements with maximum connectivity requirement $k$, our algorithm computes a solution that is a $4$-approximation on the cost, and the degree of each node $v$ in the solution is at most $4b(v)+O(k)$. These ratios improve the previous $O(\log k)$-approximation on the cost and $O(2^k b(v))$-approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k$-out-connectivity, directed and undirected $k$-connectivity, and undirected rooted $k$-connectivity and subset $k$-connectivity.

Dial-a-Ride problems consist of a setVofnvertices in a metric space (denoting travel time between vertices) and a set ofmobjects represented as source-destination pairs {(si,ti)}mi=1, where each object requires to be moved from its source to destination vertex. In themulti-vehicle Dial-a-Rideproblem, there areqvehicles, each having capacitykand where each vehiclej∈ [q] has its own depot-vertexrj∈ V. A feasible schedule consists of a capacitated route for each vehicle (where vehiclejoriginates and ends at its depotrj) that together move all objects from their sources to destinations. The objective is to find a feasible schedule that minimizes the maximum completion time (i.e.,makespan) of vehicles, where the completion time of vehiclejis the time when it returns to its depotrjat the end of its route. We study thepreemptiveversion of multi-vehicle Dial-a-Ride, in which an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are anO(log3n)-approximation algorithm forpreemptive multi-vehicle Dial-a-Ride, and an improvedO(logt)-approximation for its special case when there is no capacity constraint (heret≤nis the number of distinct depot-vertices). There is an Ω (log1/4-ϵn) hardness of approximation known even for single vehicle capacitated Dial-a-Ride [Gørtz 2006]. For uncapacitated multi-vehicle Dial-a-Ride, we show that there are instances when natural lower bounds (used in our algorithm) are ˜Ω(logt) factor away from the optimum. We also consider the special class of metrics induced by graphs excluding any fixed minor (e.g., planar metrics). In this case, we obtain improved guarantees ofO(log2n) for capacitated multi-vehicle Dial-a-Ride, andO(1) for the uncapacitated problem.

The inventory routing problem involves trading off inventory holding costs at client locations with vehicle routing costs to deliver frequently from a single central depot to meet deterministic client demands over a finite planing horizon. In this paper, we consider periodic solutions that visit clients in one of several specified frequencies, and focus on the case when the frequencies of visiting nodes are nested. We give the first constant-factor approximation algorithms for designing optimum nested periodic schedules for the problem with no limit on vehicle capacities by simple reductions to prize-collecting network design problems. For instance, we present a 2.55-approximation algorithm for the minimum-cost nested periodic schedule where the vehicle routes are modeled as minimum Steiner trees. We also show a general reduction from the capacitated problem where all vehicles have the same capacity to the uncapacitated version with a slight loss in performance. This reduction gives a 4.55-approximation for the capacitated problem. In addition, we prove several structural results relating the values of optimal policies of various types.

We prove new results for approximating Graphic TSP. Specifically, we provide a polynomial-time 9/7-approximation algorithm for cubic bipartite graphs and a (9/7+1/(21(k-2)))-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with our knowledge of the structure of cubic graphs.

Detecting and quantifying the timing and the genetic contributions of parental populations to a hybrid population is an important but challenging problem in reconstructing evolutionary histories from genetic variation data. With the advent of high throughput genotyping technologies, new methods suitable for large-scale data are especially needed. Furthermore, existing methods typically assume the assignment of individuals into subpopulations is known, when that itself is a difficult problem often unresolved for real data. Here, we propose a novel method that combines prior work for inferring non reticulate population structures with an MCMC scheme for sampling over admixture scenarios to both identify population assignments and learn divergence times and admixture proportions for those populations using genome-scale admixed genetic variation data. We validated our method using coalescent simulations and a collection of real bovine and human variation data. On simulated sequences, our methods show better accuracy and faster run time than leading competitive methods in estimating admixture fractions and divergence times. Analysis on the real data further shows our methods to be effective at matching our best current knowledge about the relevant populations.

We consider the problem of finding a minimum edge cost subgraph of an undirected or a directed graph satisfying given connectivity requirements and degree bounds b(·) on nodes. We present an iterative rounding algorithm for this problem. When the graph is undirected and the connectivity requirements are on the element-connectivity with maximum value k, our algorithm computes a solution that is an O(k)-approximation for the edge cost in which the degree of each node v is at most O(k) · b(v). We also consider the no edge cost case where the objective is to find a subgraph satisfying connectivity requirements and degree bounds. Our algorithm for this case outputs a solution in which the degree of each node v is at most 6·b(v)+O(k2). These algorithms can be extended to other well-studied undirected node-connectivity requirements such as uniform, subset and rooted connectivity. When the graph is directed and the connectivity requirement is k-out-connectivity from a root, our algorithm computes a solution that is a 2-approximation for the edge cost in which the degree of each node v is at most 2 · b(v) + O(k).

In the Stochastic Orienteering problem, we are given a metric, where each node also has a job located there with some deterministic reward and a random size. (Think of the jobs as being chores one needs to run, and the sizes as the amount of time it takes to do the chore.) The goal is to adaptively decide which nodes to visit to maximize total expected reward, subject to the constraint that the total distance traveled plus the total size of jobs processed is at most a given budget of B. (I.e., we get reward for all those chores we finish by the end of the day). The (random) size of a job is not known until it is completely processed. Hence the problem combines aspects of both the stochastic knapsack problem with uncertain item sizes and the deterministic orienteering problem of using a limited travel time to maximize gathered rewards located at nodes. In this paper, we present a constant-factor approximation algorithm for the best non-adaptive policy for the Stochastic Orienteering problem. We also show a small adaptivity gap—i.e., the existence of a non-adaptive policy whose reward is at least an Ω(1/ log log B) fraction of the optimal expected reward—and hence we also get an O(log log B)-approximation algorithm for the adaptive problem. Finally we address the case when the node rewards are also random and could be correlated with the waiting time, and give a non-adaptive policy which is an O(log n log B)-approximation to the best adaptive policy on n-node metrics with budget B.

In these lectures, I will describe a simple iterative method that supplies new proofs of integrality of linear characterizations of various basic problems in combinatorial optimization, and also allows adaptations to design approximation algorithms for NP-hard variants of these problems involving extra "degree-like" budget constraints. It is inspired by Jain's iterative rounding method for designing approximation algorithms for survivable network design problems, and augmented with a relaxation idea in the work of Lau, Naor, Salvatipour and Singh in their work on designing the approximation algorithm for its degree bounded version. Its application was further refined in recent work of Bansal, Khandekar and Nagarajan on degree-bounded directed network design. I will begin by reviewing the background material on LP relaxations and their solvability and properties of extreme point or vertex solutions to such problems. I will then introduce the basic framework of the method using the assignment problem, and show its application by re-deriving the approximation results of Shmoys and Tardos for the generalized assignment problem. I will then discuss linear characterizations for the spanning tree polyhedron in undirected graphs and give a new proof of integrality using an iterative method. I will then illustrate an application to approximating the degree-bounded version of the undirected problem, by proving the results of Goemans and Lau & Singh. I will continue with showing how these methods for spanning trees simplify and generalize to showing linear descriptions of maximum weight matroid bases and also maximum weight sets that are independent in two different matroids. This also leads to good additive approximation algorithms for a bounded degree version of the matroid basis problem. I will close with applications of the iterative method by revisiting Jain's original proof for the SNDP and giving a new proof that unifies its treatment with that for the Symmetric TSP polyhedron (describing joint work with Nagarajan and Singh). I will also outline the versatility of the method by pointing out the other problems for which the method has been applied, summarizing the discussion in a recent monograph I have co-authored on this topic with Lap Chi Lau and Mohit Singh (published by Cambridge University Press, 2011).

Abstract We study the distance constrained vehicle routing problem (DVRP) (Laporte et al., Networks 14 (1984), 47–61, Li et al., Oper Res 40 (1992), 790–799): given a set of vertices in a metric space, a specified depot, and a distance bound D, find a minimum cardinality set of tours originating at the depot that covers all vertices, such that each tour has length at most D. This problem is NP‐complete, even when the underlying metric is induced by a weighted star. Our main result is a 2‐approximation algorithm for DVRP on tree metrics; we also show that no approximation factor better than 1.5 is possible unless P = NP. For the problem on general metrics, we present a $(O(\log {1 \over \varepsilon }),1 + \varepsilon )$ ‐bicriteria approximation algorithm: i.e., for any ε > 0, it obtains a solution violating the length bound by a 1 + ε factor while using at most $O(\log {1 \over \varepsilon })$ times the optimal number of vehicles. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Consider the edge-connectivity survivable network design problem (SNDP): given a graph $G = (V,E)$ with edge-costs, and edge-connectivity requirements $r_{ij} \in \mathbb{Z}_{\geq 0}$ for every pair of vertices $i, j \in V$, find an (approximately) minimum-cost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized $\tilde{O}(r_{\max} \log^3 n)$-competitive online algorithm for this edge-connectivity network design problem that runs in time $O(m^{r_{\max}})$, where $r_{\max} = \max_{ij} r_{ij}$. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. As a consequence of using these random embeddings, our algorithms are competitive only against oblivious adversaries. Our results for the online problem give us approximation algorithms that admit strict cost-shares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rent-or-buy version and (b) the (two-stage) stochastic version of the edge-connected network design problem with independent arrivals. If we are in the case when the underlying graph is complete and the edge-costs are metric (i.e., the triangle inequality is satisfied), we improve on our results to give an $O(\log n)$-competitive deterministic online algorithm for the rooted version of the problem, and constant-factor approximation algorithms for the rent-or-buy and stochastic variants of SNDP.

In the stochastic knapsack problem, we are given a knapsack of size B, and a set of items whose sizes and rewards are drawn from a known probability distribution. To know the actual size and reward we have to schedule the item-when it completes, we get to know these values. The goal is to schedule the items (possibly making adaptive decisions based on the sizes seen so far) to maximize the expected total reward of items which successfully pack into the knapsack. We know constant-factor approximations when (i) the rewards and sizes are independent, and (ii) we cannot prematurely cancel items after we schedule them. What if either or both assumptions are relaxed? Related stochastic packing problems are the multi-armed bandit (and budgeted learning) problems, here one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to (adaptively) decide which arms to pull, in order to maximize the expected reward obtained after B pulls in total. Much recent work on this problem focuses on the case when the evolution of each arm follows a martingale, i.e., when the expected reward from one pull of an arm is the same as the reward at the current state. What if the rewards do not form a martingale? In this paper, we give O(1)-approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations. Extending the ideas developed here, we give O(1)-approximations for MAB problems without the martingale assumption. Indeed, we can show that previously proposed linear programming relaxations for these problems have large integrality gaps. So we propose new time-indexed LP relaxations, using a decomposition and "gap-filling" approach, we convert these fractional solutions to distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise randomized adaptive scheduling algorithms.

Understanding a network's temporal evolution appears to require multiple observations of the graph over time. These often expensive repeated crawls are only able to answer questions about what happened from observation to observation, and not what happened before or between network snapshots. Contrary to this picture, we propose a method for Twitter's social network that takes a single static snapshot of network edges and user account creation times to accurately infer when these edges were formed. This method can be exact in theory, and we demonstrate empirically for a large subset of Twitter relationships that it is accurate to within a few hours in practice.

This special section comprises eight fully refereed papers whose extended abstracts were presented at the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008) in Philadelphia, Pennsylvania, October 26–28, 2008. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2008 proceedings. The regular conference program consisted of 79 papers chosen from among 276 submissions. These were selected by a program committee consisting of Scott Aaronson, Yossi Azar, Avrim Blum, Harry Buhrman, Artur Czumaj, Yevgeniy Dodis, David Eppstein, Jeff Erickson, Naveen Garg, Tom Hayes, Sampath Kannan, Jonathan Katz, Valerie King, Mohammad Mahdian, Yury Makarychev, Yishay Mansour, Rafail Ostrovsky, Toniann Pitassi, Harald Raecke, R. Ravi (chair), Madhu Sudan, and Emanuele Viola. The papers invited to this special section were also selected with the input of the program committee. The eight papers in this section span a broad range of topics, including algorithmic game theory, computational complexity, hardness of approximation, learning theory, pseudorandomness, and quantum algorithms. Each paper underwent an extensive refereeing process; we thank both the authors and the anonymous referees for their efforts. In addition, we would like to thank Eva Tardos, who was SICOMP's editor-in-chief as this project began, and SIAM staff member Cherie Trebisky for their help in preparing this special section.

We consider two- and multistage versions of stochastic combinatorial optimization problems with recourse: in this framework, the instance for the combinatorial optimization problem is drawn from a known probability distribution $\pi$ and is only revealed to the algorithm over two (or multiple) stages. At each stage, on receiving some more information about the instance, the algorithm is allowed to build some partial solution. Since the costs of elements increase with each passing stage, there is a natural tension between waiting for later stages, to gain more information about the instance, and purchasing elements in earlier stages, to take advantages of lower costs. We provide approximation algorithms for stochastic combinatorial optimization problems (such as the Steiner tree problem, the Steiner network problem, and the vertex cover problem) by means of a simple sampling-based algorithm. In every stage, our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample. We show that if one can construct cost-sharing functions associated with the algorithms used to construct these partial solutions, then this strategy results in provable approximation guarantees for the overall stochastic optimization problem. We also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem, a problem that does not fit into the simpler framework of our main model.

The random accumulation of variations in the human genome over time implicitly encodes a history of how human populations have arisen, dispersed, and intermixed since we emerged as a species. Reconstructing that history is a challenging computational and statistical problem but has important applications both to basic research and to the discovery of genotype-phenotype correlations. We present a novel approach to inferring human evolutionary history from genetic variation data. We use the idea of consensus trees, a technique generally used to reconcile species trees from divergent gene trees, adapting it to the problem of finding robust relationships within a set of intraspecies phylogenies derived from local regions of the genome. Validation on both simulated and real data shows the method to be effective in recapitulating known true structure of the data closely matching our best current understanding of human evolutionary history. Additional comparison with results of leading methods for the problem of population substructure assignment verifies that our method provides comparable accuracy in identifying meaningful population subgroups in addition to inferring relationships among them. The consensus tree approach thus provides a promising new model for the robust inference of substructure and ancestry from large-scale genetic variation data.

The group Steiner problem is a classical network design problem where we are given a graph and a collection of groups of vertices, and want to build a min-cost subgraph that connects the root vertex to at least one vertex from each group. What if we wanted to build a subgraph that two-edge-connects the root to each group—that is, for every group g ⊆ V, the subgraph should contain two edge-disjoint paths from the root to some vertex in g? What if we wanted the two edge-disjoint paths to end up at distinct vertices in the group, so that the loss of a single member of the group would not destroy connectivity? In this paper, we investigate tree-embedding techniques that can be used to solve these and other 2-edge-connected network design problems. We illustrate the potential of these techniques by giving poly-logarithmic approximation algorithms for two-edge-connected versions of the group Steiner, connected facility location, buy-at-bulk, and the k-MST problems.

Multicommodity facility location refers to the extension of facility location to allow for different clients' demands for different goods from among a finite set of goods. This leads to several optimization problems, depending on the cost of opening a facility (now a function of the commodities it serves). In this paper, we introduce and provide approximation algorithms for two fairly general versions of multicommodity facility location. We formulate integer programming models for these problems, and use them to provide approximation algorithms for the problems that are close to the inapproximability thresholds.

The k-forest problem is a common generalization of both the k-MST and the dense-k-subgraph problems. Formally, given a metric space on n vertices V , with m demand pairs ⊆ V × V and a “target” k ≤ m , the goal is to find a minimum cost subgraph that connects at least k pairs. In this paper, we give an O (min{√ n ⋅log k ,√ k })-approximation algorithm for k -forest, improving on the previous best ratio of O (min { n 2/3 ,√ m }log n ) by Segev and Segev. We then apply our algsorithm for k -forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an n point metric space with m objects each with its own source and destination, and a vehicle capable of carrying at most k objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We want that the tour be non-preemptive : that is, each object, once picked up at its source, is dropped only at its destination. We prove that an α-approximation algorithm for the k -forest problem implies an O (α⋅log 2 n )-approximation algorithm for Dial-a-Ride. Using our results for k -forest, we get an O (min{√ n ,√ k }⋅log 2 n )-approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an O (√ k log n )-approximation by Charikar and Raghavachari; our results give a different proof of a similar approximation guarantee—in fact, when the vehicle capacity k is large, we give a slight improvement on their results. The reduction from Dial-a-Ride to the k -forest problem is fairly robust, and allows us to obtain approximation algorithms (with the same guarantee) for some interesting generalizations of Dial-a-Ride.

We describe a simple iterative method for proving a variety of results in combinatorial optimization. It is inspired by Jain's iterative rounding method (FOCS 1998) for designing approximation algorithms for survivable network design problems, and augmented with a relaxation idea in the work of Lau, Naor, Salvatipour and Singh (STOC 2007) on designing an approximation algorithm for its degree bounded version. At the heart of the method is a counting argument that redistributes tokens from the columns to the rows of an LP extreme point. This token argument was further refined to fractional assignment and redistribution in work of Bansal, Khandekar and Nagarajan on degree-bounded directed network design (STOC 2008). In this presentation, we introduce the method using the assignment problem, describe its application to showing the integrality of Edmond's characterization (1971) of the spanning tree polyhedron, and then extend the argument to show a simple proof of the Singh and Lau's approximation algorithm (STOC 2007) for its degree constrained version, due to Bansal, Khandekar and Nagarajan. We conclude by showing how Jain's original proof can also be simplified by using a fractional token argument (joint work with Nagarajan and Singh). This presentation is extracted from an upcoming monograph on this topic co-authored with Lau and Singh.

Consider the edge-connectivity survivable network design problem: given a graph G = (V,E) with edge-costs, and edge-connectivity requirements rij for every pair of vertices i,j, find an (approximately) minimum-cost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting.

We propose an exact solution method for a routing and capacity installation problem in networks. Given an input graph, the problem is to route traffic from a set of source nodes to a sink node and to install transmission facilities on the edges of the graph to accommodate the flow at minimum cost. We give a branch-and-bound algorithm that solves relaxations obtained by approximating the noncontinuous cost function by its lower convex envelope. The approximations are refined by branching on the flow ranges on selected edges. Our computational experiments indicate that this method is effective in solving moderate-size problems and provides very good candidate solutions early in the branch-and-bound tree.

We consider a combinatorial problem derived from haplotyping a population with respect to a genetic disease, either recessive or dominant. Given a set of individuals, partitioned into healthy and diseased, and the corresponding sets of genotypes, we want to infer "bad'' and "good'' haplotypes to account for these genotypes and for the disease. Assume e.g. the disease is recessive. Then, the resolving haplotypes must consist of bad and good haplotypes, so that (i) each genotype belonging to a diseased individual is explained by a pair of bad haplotypes and (ii) each genotype belonging to a healthy individual is explained by a pair of haplotypes of which at least one is good. We prove that the associated decision problem is NP-complete. However, we also prove that there is a simple solution, provided the data satisfy a very weak requirement.

Reconstruction of phylogenetic trees is a fundamental problem in computational biology. While excellent heuristic methods are available for many variants of this problem, new advances in phylogeny inference will be required if we are to be able to continue to make effective use of the rapidly growing stores of variation data now being gathered. In this paper, we present two integer linear programming (ILP) formulations to find the most parsimonious phylogenetic tree from a set of binary variation data. One method uses a flow-based formulation that can produce exponential numbers of variables and constraints in the worst case. The method has, however, proven extremely efficient in practice on datasets that are well beyond the reach of the available provably efficient methods, solving several large mtDNA and Y-chromosome instances within a few seconds and giving provably optimal results in times competitive with fast heuristics than cannot guarantee optimality. An alternative formulation establishes that the problem can be solved with a polynomial-sized ILP. We further present a web server developed based on the exponential-sized ILP that performs fast maximum parsimony inferences and serves as a front end to a database of precomputed phylogenies spanning the human genome.

BACKGROUND: Maximum parsimony phylogenetic tree reconstruction from genetic variation data is a fundamental problem in computational genetics with many practical applications in population genetics, whole genome analysis, and the search for genetic predictors of disease. Efficient methods are available for reconstruction of maximum parsimony trees from haplotype data, but such data are difficult to determine directly for autosomal DNA. Data more commonly is available in the form of genotypes, which consist of conflated combinations of pairs of haplotypes from homologous chromosomes. Currently, there are no general algorithms for the direct reconstruction of maximum parsimony phylogenies from genotype data. Hence phylogenetic applications for autosomal data must therefore rely on other methods for first computationally inferring haplotypes from genotypes. RESULTS: In this work, we develop the first practical method for computing maximum parsimony phylogenies directly from genotype data. We show that the standard practice of first inferring haplotypes from genotypes and then reconstructing a phylogeny on the haplotypes often substantially overestimates phylogeny size. As an immediate application, our method can be used to determine the minimum number of mutations required to explain a given set of observed genotypes. CONCLUSION: Phylogeny reconstruction directly from unphased data is computationally feasible for moderate-sized problem instances and can lead to substantially more accurate tree size inferences than the standard practice of treating phasing and phylogeny construction as two separate analysis stages. The difference between the approaches is particularly important for downstream applications that require a lower-bound on the number of mutations that the genetic region has undergone.

We consider the problem of reconstructing near-perfect phylogenetic trees using binary character states (referred to as BNPP). A perfect phylogeny assumes that every character mutates at most once in the evolutionary tree, yielding an algorithm for binary character states that is computationally efficient but not robust to imperfections in real data. A near-perfect phylogeny relaxes the perfect phylogeny assumption by allowing at most a constant number of additional mutations. We develop two algorithms for constructing optimal near-perfect phylogenies and provide empirical evidence of their performance. The first simple algorithm is fixed parameter tractable when the number of additional mutations and the number of characters that share four gametes with some other character are constants. The second, more involved algorithm for the problem is fixed parameter tractable when only the number of additional mutations is fixed. We have implemented both algorithms and shown them to be extremely efficient in practice on biologically significant data sets. This work proves the BNPP problem fixed parameter tractable and provides the first practical phylogenetic tree reconstruction algorithms that find guaranteed optimal solutions while being easily implemented and computationally feasible for data sets of biologically meaningful size and complexity.

Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worst-case among the various uncertain scenarios in the model. While these approaches may be thought of defining data- or cost-robust problems, we formulate a new "demand-robust" model motivated by recent work on two-stage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of first-stage solutions for such problems: there exists a first-stage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multi-cut, shortest paths, Steiner trees, vertex cover and un-capacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demand-robust setting.

In 1995, Gremban, Miller, and Zagha introduced support-tree preconditioners and a parallel algorithm called support-tree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the off-diagonal entries are non-positive, and the row and column sums are zero. A Laplacian A with 2m non-zeros can be interpreted as an undirected positively-weighted graph G with n vertices and m edges, where there is an edge between two nodes i and j with weight c((i, j)) = −Ai,j = −Aj,i if Ai,j = Aj,i < 0. Gremban et al. showed experimentally that STCG performs well on several classes of graphs commonly used in scientific computations. In his thesis, Gremban also proved upper bounds on the number of iterations required for STCG to converge for certain classes of graphs. In this paper, we present an algorithm for finding a preconditioner for an arbitrary graph G = (V, E) with n nodes, m edges, and a weight function c> 0 on the edges, where w.l.o.g., mine∈E c(e) = 1. Equipped with this preconditioner, STCG requires O(log 4 n · � ∆/α) iterations, where α = min U⊂V,|U|≤|V |/2 c(U, V \\U)/|U | is the minimum edge expansion of the graph, and ∆ = maxv∈V c(v) is the maximum incident weight on any vertex. Each iteration requires O(m) work and can be implemented in O(log n) steps in parallel, using only O(m) space. Our results generalize to matrices that are symmetric and diagonally-dominant (SDD). 1

We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree (MST) problem: Given an undirected graph with nonnegative edge weights and a degree bound B, find a spanning tree of maximum node-degree B and minimum total edge-cost. Our algorithm outputs a tree of maximum degree at most a constant times B and total edge-cost at most a constant times that of a minimum-cost degree-B-bounded spanning tree. While our new algorithm is based on ideas from Lagrangian relaxation, as is our previous work [SIAM J. Comput., 31 (2002), pp. 1783--1793], it does not rely on computing a solution to a linear program. Instead, it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangian node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee. A second useful feature of our algorithm is that it can handle nonuniform degree bounds on the nodes: Given distinct bounds B v for every node $v \in V$, the output tree has degree at most O(B v + log|V|) for every $v \in V$. As before, the cost of the tree is at most a constant times that of a minimum-cost tree obeying all degree bounds.

Real-world networks often need to be designed under uncertainty, with only partial information and predictions of demand available at the outset of the design process. The field of stochastic optimization deals with such problems where the forecasts are specified in terms of probability distributions of future data. In this paper, we broaden the set of models as well as the techniques being considered for approximating stochastic optimization problems. For example, we look at stochastic models where the cost of the elements is correlated to the set of realized demands, and risk-averse models where upper bounds are placed on the amount spent in each of the stages. These generalized models require new techniques, and our solutions are based on a novel combination of the primal-dual method truncated based on optimal LP relaxation values, followed by a tree-rounding stage. We use these to give constant-factor approximation algorithms for the stochastic Steiner tree and single sink network design problems in these generalized models.

No abstract available.

Several combinatorial optimization problems choose elements to minimize the total cost of constructing a feasible solution that satisfies requirements of clients. In the Steiner Tree problem, for example, edges must be chosen to connect terminals (clients); in Vertex Cover, vertices must be chosen to cover edges (clients); in Facility Location, facilities must be chosen and demand vertices (clients) connected to these chosen facilities. We consider a stochastic version of such a problem where the solution is constructed in two stages: Before the actual requirements materialize, we can choose elements in a first stage. The actual requirements are then revealed, drawn from a pre-specified probability distribution π thereupon, some more elements may be chosen to obtain a feasible solution for the actual requirements. However, in this second (recourse) stage, choosing an element is costlier by a factor of σ> 1. The goal is to minimize the first stage cost plus the expected second stage cost.We give a general yet simple technique to adapt approximation algorithms for several deterministic problems to their stochastic versions via the following method. First stage: Draw σ independent sets of clients from the distribution π and apply the approximation algorithm to construct a feasible solution for the union of these sets. Second stage: Since the actual requirements have now been revealed, augment the first-stage solution to be feasible for these requirements. We use this framework to derive constant factor approximations for stochastic versions of Vertex Cover, Steiner Tree and Uncapacitated Facility Location for arbitrary distributions π in one fell swoop. For special (product) distributions, we obtain additional and improved results. Our techniques adapt and use the notion of strict cost-shares introduced in [5].

Abstract We give quasipolynomial‐time approximation algorithms for designing networks with a minimum degree. Using our methods, one can design networks whose connectivity is specified by “proper” functions, a class of 0–1 functions indicating the number of edges crossing each cut. We also provide quasipolynomial‐time approximation algorithms for finding two‐edge‐connected spanning subgraphs of approximately minimum degree of a given two‐edge‐connected graph, and a spanning tree (branching) of approximately minimum degree of a directed graph. The degree of the output network in all cases is guaranteed to be at most (1 + ϵ) times the optimal degree, plus an additive O(log1+ϵn) for any ϵ > 0. Our analysis indicates that the degree of an optimal subgraph for each of the problems above is well estimated by certain polynomially solvable linear programs. This suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch and bound. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 203–215 2004

No abstract available.

We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree problem: Given an undirected graph with nonnegative edge weights and degree bounds Bv > 1 for all vertices v, find a spanning tree T of minimum total edge-cost such that the maximum degree of each node v in T is at most Bv. Our algorithm finds a tree in which the degree of each node v is O(Bv + log n) and the total edge-cost is at most a constant times the cost of any tree that obeys all degree constraints.Our previous algorithm[9] with similar guarantees worked only in the case of uniform degree bounds (i.e. Bv=B for all vertices v). While the new algorithm is based on ideas from Lagrangean relaxation as is our previous work, it does not rely on computing a solution to a linear program. Instead it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangean node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee.

In this paper, we present a new bicriteria approximation algorithm for the degree-bounded minimum spanning tree problem. In this problem, we are given an undirected graph, a nonnegative cost function on the edges, and a positive integer B * , and the goal is to find a minimum-cost spanning tree T with maximum degree at most B * . In an n-node graph, our algorithm finds a spanning tree with maximum degree O(B * +logn) and cost O(opt B * ), where opt B * is the minimum cost of any spanning tree whose maximum degree is at most B * . Our algorithm uses ideas from Lagrangean duality. We show how a set of optimum Lagrangean multipliers yields bounds on both the degree and the cost of the computed solution.

In this paper, we present a new bicriteria approximation algorithm for the degree- bounded minimum spanning tree problem. In this problem, we are given an undirected graph, a nonnegative cost function on the edges, and a positive integer B ∗ , and the goal is to find a minimum- cost spanning tree T with maximum degree at most B ∗ .I n ann-node graph, our algorithm finds a spanning tree with maximum degree O(B∗ + log n) and cost O( opt B∗ ), where opt B∗ is the minimum cost of any spanning tree whose maximum degree is at most B ∗ . Our algorithm uses ideas from Lagrangean duality. We show how a set of optimum Lagrangean multipliers yields bounds on both the degree and the cost of the computed solution.

Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NP-hard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomial-time approximation scheme valid for both versions of the problem. In particular, we show how to build a spanning tree of an n-vertex weighted graph with routing cost at most $(1+\epsilon)$ of the minimum in time $O(n^{O({\frac{1}{\epsilon}}% )})$. Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a generalization of the minimum routing cost tree problem where the routing costs of different pairs are weighted by different requirement amounts. We observe that a randomized O(log n log log n)-approximation for this problem follows directly from a recent result of Bartal, where n is the number of nodes in a metric graph. This also yields the same approximation for the generalized sum-of-pairs alignment problem in computational biology.

We present simple semidefinite programming relaxations for the m-hard minimum bandwidth and minimum length linear ordering problems.We then show how these relaxations can be rounded in a natural way (via random projection) to obtain new approximation guarantees for both of these vertex-ordering problems.

We present new design and analysis techniques for the synthesis of parallel multiplier circuits that have smaller predicted delay than the best current multipliers. V.G. Oklobdzija et al. (1996) suggested a new approach, the Three-Dimensional Method (TDM), for Partial Product Reduction Tree (PPRT) design that produces multipliers that outperform the current best designs. The goal of TDM is to produce a minimum delay PPRT using full adders. This is done by carefully modeling the relationship of the output delays to the input delays in an adder and, then, interconnecting the adders in a globally optimal way. Oklobdzija et al. suggested a good heuristic for finding the optimal PPRT, but no proofs about the performance of this heuristic were given. We provide a formal characterization of optimal PPRT circuits and prove a number of properties about them. For the problem of summing a set of input bits within the minimum delay, we present an algorithm that produces a minimum delay circuit in time linear in the size of the inputs. Our techniques allow us to prove tight lower bounds on multiplier circuit delays. These results are combined to create a program that finds optimal TDM multiplier designs. Using this program, we can show that, while the heuristic used by Oklobdzija et al. does not always find the optimal TDM circuit, it performs very well in terms of overall PPRT circuit delay. However, our search algorithms find better PPRT circuits for reducing the delay of the entire multiplier.

This paper presents an algorithm for finding parallel elimination orders for Gaussian elimination. Viewing a system of equations as a graph, the algorithm can be applied directly to interval graphs and chordal graphs. For general graphs, the algorithm can be used to parallelize the order produced by some other heuristic such as minimum degree. In this case, the algorithm is applied to the chordal completion that the heuristic generates from the input graph. In general, the input to the algorithm is a chordal graph G with n nodes and m edges. The algorithm produces an order with height at most O(log/sup 3/ n) times optimal, fill at most O(m), and work at most O(W*(G)), where W*(G) is the minimum possible work over all elimination orders for G. Experimental results show that when applied after some other heuristic, the increase in work and fill is usually small. In some instances the algorithm obtains an order that is actually better, in terms of work and fill, than the original one. We also present an algorithm that produces an order with a factor of log n less height, but with a factor of O(/spl radic/log n) more fill.

Given an undirected graph with non-negative edge costs and an integer k, the k-MST problem is that of finding a tree of minimum cost on k nodes.This problem is known to be NP-hard.We present a simple approximation algorithm that finds a solution whose cost is less than 17 times the cost of the optimum.This improves upon previous performance ratios for this problem -O(w) due to Ravi et al., 0(log2 k) due to Awerbuch et al, and the previous best bound of O(log k) due to Rajagopalan and Vazirani.Given any O < cr < 1, we first present a bicriteria approximation algorithm that ~o~tputs a tree on p z cYk vertices of total cost at most ~1~, where L is the cost of the optimal k-

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k} $ for the general edge-weighted case and $O(k^{1/4} )$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded graphs, which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees and, more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding k-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

We present new design and analysis techniques for the synthesis of fast parallel multiplier circuits. V.G. Oklobdzija, D. Villeger, and S.S. Lui (1995) suggested a new approach, the three dimensional method (TDM), for partial product reduction tree (PPRT) design that produces multipliers which outperform the current best designs. The goal of TDM is to produce a minimum delay PPRT using full adders. This is done by carefully modelling the relationship of the output delays to the input delays an an adder, and then interconnecting the adders in a globally optimal way. Oklobdzija, et. al. suggested a good heuristic for finding the optimal PPRT, but no proofs about the performance of this heuristic were given. We provide a formal characterization of optimal PPRT circuits and prove a number of properties about them. For the problem of summing a set of input bits within the minimum delay, we present an algorithm that produces a minimum delay circuit in time linear in the size of the inputs. Our techniques allow us to prove tight lower bounds on multiplier circuit delays. These results are combined to create a program which finds optimal TDM multiplier designs.>

We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair $\{i, j\}$ of nodes, an edgeconnectivity requirement $r_{ij}$. The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within $2 \lceil {\log_{2}(r + 1)} \rceil $ of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

Given an undirected graph representing a network of processors, and a source node containing a message that must be broadcast to all the nodes, find a scheme that accomplishes the broadcast in the minimum number of time steps. At each time step, any processor that has received the message is allowed to communicate the message to at most one of its neighbors in the network, i.e. can communicate via a telephone call to a neighbor. This has been termed the minimum broadcast time problem under the telephone model and is known to be NP-complete. The minimum broadcast time in a graph is closely related to the poise of the graph. The poise of a tree is defined to be the quantity (maximum degree of any node in the tree+diameter of the tree). The poise of a graph is the minimum poise of any of its spanning trees. Computing the poise of a graph is shown to be NP-hard and an algorithm for computing a spanning tree of approximately minimum poise is derived. This algorithm is then used to derive an O(log/sup 2/n/log log n)-approximation for the minimum broadcast time problem on an n-node graph. Our algorithm extends to many generalizations of the problem such as the multicast problem, a telephone model allowing conference calls, and to the closely related minimum gossip time problem.>

We study network-design problems with multiple design objectives.In particular, we look at two cost NY 12222.

Abstract We present an O(n2) time‐optimal algorithm for solving the unweighted all‐pair shortest path problem on interval graphs, an important subclass of perfect graphs. An interesting structure called the neighborhood tree is studied and used in the algorithm. This tree is formed by identifying the successive neighborhoods of the vertex labeled last in the graph according to the IG‐ordering.

Article When trees collide: an approximation algorithm for the generalized Steiner problem on networks Share on Authors: Ajit Agrawal Brown Univ., Providence, RI Brown Univ., Providence, RIView Profile , Philip Klein Brown Univ., Providence, RI Brown Univ., Providence, RIView Profile , R. Ravi Brown Univ., Providence, RI Brown Univ., Providence, RIView Profile Authors Info & Claims STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of ComputingJanuary 1991 Pages 134–144https://doi.org/10.1145/103418.103437Online:03 January 1991Publication History 47citation861DownloadsMetricsTotal Citations47Total Downloads861Last 12 Months16Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

The first approximate max-flow-min-cut theorem for general multicommodity flow is proved. It is used to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF identical to formula, via minimization problems, and other problems. Also presented are approximation algorithms for chordalization of a graph and for register sufficiency that are based on undirected and directed node separators.>