In this paper we consider a classic scheduling problem known as Machine Minimization (MM). The input to MM is a set J of n jobs, where each job j has processing time p_j, release time r_j, and deadline d_j. The goal is to schedule the jobs to run non-preemptively on minimum number of machines such that each job is fully scheduled within its [r_j,d_j] interval and each machine runs at most one job at a time. This problem generalizes several NP-hard problems (e.g. the case of identical release time and deadlines reduces to the bin packing problem). Using Randomized Rounding [Prabhakar Raghavan and Clark D. Thompson, 1987] one can get an O((log n)/(log log n))-approximation. Chuzhoy et al. [Julia Chuzhoy and Paolo Codenotti, 2009] presented an algorithm that uses O(opt²) machines (i.e. O(opt)-approximation). Combined with the earlier work this yields an O(√{(log n)/(log log n)})-approximation and this remains the best known result for over 20 years. Even for when the ratio of largest to smallest processing time p_{max}/p_{min} is bounded, or the number of distinct processing times are bounded, there is no (universal) constant approximation. In this paper, we present a number of results. When p_max/p_min = ρ we present an algorithm that yields an asymptotic (2+ε)-approximation in time n^{O(ρ⁴/ε⁴)}. When the number of distinct processing times is c, we present a 2-approximation with run time n^{O(c²log³ n)}. If we have c distinct processing times and p_max/p_min = ρ we present an asymptotic (1+ε)-approximation that runs in time n^O(c²⋅ρ²⋅log³n/ε²).

In this paper we look at various extensions of the classic Traveling Salesman Problem (TSP) on graphs with bounded doubling dimension and bounded treewidth and present approximation schemes for them. Suppose we are given a weighted graph G = (V,E) with a start node s ∈ V, distances on the edges d:E → ℚ^+ and integer k. In k-stroll problem the goal is to find a path from s of minimum length that visits at least k vertices. In k-path we are given an additional end node t ∈ V and the path is supposed to go from s to t. The dual problem to k-stroll is the rooted orienteering in which instead of k we are given a budget B and the goal is to find a walk of length at most B starting at s that visits as many vertices as possible. In the point-to-point orienteering (P2P orienteering) we are given start and end nodes s,t and the walk is supposed to start at s and end at t. In the deadline TSP (which generalizes P2P orienteering) we are given a deadline D(v) for each v ∈ V and the goal is to find a walk starting at s that visits as many vertices as possible before their deadline (where the visit time of a node is the distance travelled from s to that node). The best approximation for rooted orienteering (or P2P orienteering) is (2+ε)-approximation [Chekuri et al., 2012] and O(log n)-approximation for deadline TSP [Nikhil Bansal et al., 2004]. For Euclidean metrics of fixed dimension, Chen and Har-Peled present [Chen and Har-Peled, 2008] a PTAS for rooted orienteering. There is no known approximation scheme for deadline TSP for any metric (not even trees). Our main result is the first approximation scheme for deadline TSP on metrics with bounded doubling dimension (which includes Euclidean metrics). To do so we first we present a quasi-polynomial time approximation scheme for k-path and P2P orienteering on such metrics. More specifically, if G is a metric with doubling dimension κ and aspect ratio Δ, we present a (1+ε)-approximation that runs in time n^{O((logΔ/ε) ^{2κ+1})}. Building upon these, we obtain an approximation scheme for deadline TSP when the distances and deadlines are integer which runs in time n^{O((log Δ/ε) ^{2κ+2})}. The same approach also implies a bicriteria (1+ε,1+ε)-approximation for deadline TSP for when distances and deadlines are in ℚ^+. For graphs with bounded treewidth ω we show how to solve k-path and P2P orienteering exactly in polynomial time and a (1+ε)-approximation for deadline TSP in time n^O((ωlogΔ/ε)²).

We consider the Travelling Salesman Problem with Neighbourhoods (TSPN) on the Euclidean plane (ℝ²) and present a Polynomial-Time Approximation Scheme (PTAS) when the neighbourhoods are parallel line segments with lengths between [1, λ] for any constant value λ ≥ 1. In TSPN (which generalizes classic TSP), each client represents a set (or neighbourhood) of points in a metric and the goal is to find a minimum cost TSP tour that visits at least one point from each client set. In the Euclidean setting, each neighbourhood is a region on the plane. TSPN is significantly more difficult than classic TSP even in the Euclidean setting, as it captures group TSP. A notable case of TSPN is when each neighbourhood is a line segment. Although there are PTASs for when neighbourhoods are fat objects (with limited overlap), TSPN over line segments is APX-hard even if all the line segments have unit length. For parallel (unit) line segments, the best approximation factor is 3√2 from more than two decades ago. The PTAS we present in this paper settles the approximability of this case of the problem. Our algorithm finds a (1 + ε)-factor approximation for an instance of the problem for n segments with lengths in [1,λ] in time n^O(λ/ε³).

We study the classic (Uncapacitated) Facility Location problem on Unit Disk Graphs (UDGs). For a given point set P in the plane, the unit disk graph UDG(P) on P has vertex set P and an edge between two distinct points p, q ∈ P if and only if their Euclidean distance |pq| is at most 1. The weight of the edge pq is equal to their distance |pq|. An instance of {Facility Location} on UDG(P) consists of a set C ⊆ P of clients and a set F ⊆ P of facilities, each having an opening cost f_i. The goal is to pick a subset F' ⊆ F to open while minimizing ∑_{i ∈ F'} f_i + ∑_{v ∈ C} d(v,F'), where d(v,F') is the distance of v to nearest facility in F' through UDG(P). In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.

Abstract. We investigate the complexity of solving stable or perturbation-resilient instances of [Formula: see text]-means and [Formula: see text]-median clustering in fixed-dimensional Euclidean metrics (or more generally doubling metrics). The notion of stable or perturbation-resilient instances was introduced by Bilu and Linial [ Are stable instances easy?, 2010] and Awasthi, Blum, and Sheffet [ Stability yields a PTAS for k-median and k-means clustering, IEEE Computer Society, Washington, DC, 2010]. In our context, we say a [Formula: see text]-means instance is [Formula: see text]-stable if there is a unique optimum solution which remains unchanged if distances are (nonuniformly) stretched by a factor of at most [Formula: see text]. Stable clustering instances have been studied to explain why heuristics such as Lloyd’s algorithm perform well in practice. In this work we show that for any fixed [Formula: see text], [Formula: see text]-stable instances of [Formula: see text]-means in doubling metrics, which include fixed-dimensional Euclidean metrics, can be solved in polynomial time. More precisely, we show a natural multiswap local-search algorithm in fact finds the optimum solution for [Formula: see text]-stable instances of [Formula: see text]-means and [Formula: see text]-median in a polynomial number of iterations. We complement this result by showing that it is essentially tight: when the dimension [Formula: see text] is part of the input there is a fixed [Formula: see text] such that there is not even a PTAS for [Formula: see text]-stable [Formula: see text]-means in [Formula: see text] with [Formula: see text] unless NP = RP. To do this, we consider a robust property of CSPs: call an instance stable if there is a unique optimum solution [Formula: see text] and for any other solution [Formula: see text], the number of unsatisfied clauses is proportional to the Hamming distance between [Formula: see text] and [Formula: see text]. Dinur, Goldreich, and Gur have already shown stable QSAT is hard to approximate for some constant [Formula: see text] [ 20 ]. Recently, Paradise [ Comput. Complexity, 30 (2021), 1] extended this to the setting with bounded variable occurrence. More specifically, this implies that stable QSAT with bounded variable occurrence is APX-hard. Given this, we consider “stability-preserving” reductions to prove our hardness for stable [Formula: see text]-means. Such reductions seem to be more fragile and intricate than standard [Formula: see text]-reductions and may be of further use to demonstrate other stable optimization problems are hard to solve.

In this paper, we present approximation algorithms for the airport and railway problem (AR) on several classes of graphs. The AR problem, introduced by [Anna Adamaszek et al., 2016], is a combination of the Capacitated Facility Location problem (CFL) and the network design problem. An AR instance consists of a set of points (cities) V in a metric d(.,.), each of which is associated with a non-negative cost f_v and a number k, which represent respectively the cost of establishing an airport (facility) in the corresponding point, and the universal airport capacity. A feasible solution is a network of airports and railways providing services to all cities without violating any capacity, where railways are edges connecting pairs of points, with their costs equivalent to the distance between the respective points. The objective is to find such a network with the least cost. In other words, find a forest, each component having at most k points and one open facility, minimizing the total cost of edges and airport opening costs. Adamaszek et al. [Anna Adamaszek et al., 2016] presented a PTAS for AR in the two-dimensional Euclidean metric ℝ² with a uniform opening cost. In subsequent work [Anna Adamaszek et al., 2018] presented a bicriteria 4/3 (2+1/α)-approximation algorithm for AR with non-uniform opening costs but violating the airport capacity by a factor of 1+α, i.e. (1+α)k capacity where 0 < α ≤ 1, a (2+k/(k-1)+ε)-approximation algorithm and a bicriteria Quasi-Polynomial Time Approximation Scheme (QPTAS) for the same problem in the Euclidean plane ℝ². In this work, we give a 2-approximation for AR with a uniform opening cost for general metrics and an O(log n)-approximation for non-uniform opening costs. We also give a QPTAS for AR with a uniform opening cost in graphs of bounded treewidth and a QPTAS for a slightly relaxed version in the non-uniform setting. The latter implies O(1)-approximation on graphs of bounded doubling dimensions, graphs of bounded highway dimensions and planar graphs in quasi-polynomial time.

In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [ 14 ], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V , and a vehicle of bounded capacity Q . The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand d v and the total demand of each tour must be no more than Q . Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed \(ε \gt \frac{1}{3000}\) , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [ 5 ] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [ 15 ] presented an approximation scheme with time n log O(1/ε) n for Euclidean plane ℝ 2 . No other approximation scheme is known for any other class of metrics (without further restrictions on Q ). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time n O(log 6 n/ε 5 ) .

In this paper, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [13], we are given a graph G = (V, E) with metric edges costs, a depot r ∊ V, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand dv and the total demand of each tour must be no more than Q. Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best known approximation algorithm for general graphs has ratio α + 2(1–∊) (for the unsplittable) and α + 1–∊ (for the splittable) for some fixed is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [5], and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [14] presented an approximation scheme with time for Euclidean plane ℝ2. No other approximation scheme is known for any other class of metrics (without further restrictions on Q). In this paper, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for Euclidean plane with run time .

Resource Minimization Fire Containment (RMFC) is a natural model for optimal inhibition of harmful spreading phenomena on a graph. In the RMFC problem on trees, we are given an undirected tree G, and a vertex r where the fire starts at, called root. At each time step, the firefighters can protect up to B vertices of the graph while the fire spreads from burning vertices to all their neighbors that have not been protected so far. The task is to find the smallest B that allows for saving all the leaves of the tree. The problem is hard to approximate up to any factor better than 2 even on trees unless P = NP (King and MacGillivray in Discret Math 310(3):614–621, 2010). Chalermsook and Chuzhoy (In: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, Texas, USA, 17–19 Jan 2010, SIAM, pp 1334–1349, 2010) presented a Linear Programming (LP) based $$O(\log ^* n)$$ approximation for RMFC on trees that matches the integrality gap of the natural Linear Programming relaxation. This was recently improved by Adjiashvili et al. (ACM Trans Algorithms 15(2):20:1–20:33, 2019) to a 12-approximation through a combination of LP rounding along with several new techniques. In this paper we present an asymptotic QPTAS for RMFC on trees. More specifically, let $$\epsilon >0$$ , and $$\mathcal {I}$$ be an instance of RMFC where the optimum number of firefighters to save all the leaves is $$OPT(\mathcal {I})$$ . We present an algorithm which uses at most $$\lceil (1+\epsilon )OPT(\mathcal {I})\rceil $$ many firefighters at each time step and runs in time $$n^{O(\log \log n/\epsilon )}$$ . This suggests that the existence of an asymptotic PTAS is plausible especially since the exponent is $$O(\log \log n)$$ , not $$O(\log n)$$ . Our result combines a more refined height reduction lemma than the one in Adjiashvili et al. (2019) with LP rounding and dynamic programming to find the solution. We also apply our height reduction lemma to the algorithm provided in Adjiashvili et al. (2019) plus a more careful analysis to improve their 12-approximation and provide a polynomial time ( $$5+\epsilon $$ )-approximation.

In this paper we study the classical problem of throughput maximization. In this problem we have a collection J of n jobs, each having a release time r_j, deadline d_j, and processing time p_j. They have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their [r_j,d_j] window. This problem has been studied extensively (even for the case of m = 1). Several special cases of the problem remain open. Bar-Noy et al. [STOC1999] presented an algorithm with ratio 1-1/(1+1/m)^m for m machines, which approaches 1-1/e as m increases. For m = 1, Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation with ratio 1-1/e-ε (for any ε > 0). Recently Im-Li-Moseley [IPCO2017] presented an algorithm with ratio 1-1/e+ε₀ for some absolute constant ε₀ > 0 for any fixed m. They also presented an algorithm with ratio 1-O(√(log m/m))-ε for general m which approaches 1 as m grows. The approximability of the problem for m = O(1) remains a major open question. Even for the case of m = 1 and c = O(1) distinct processing times the problem is open (Sgall [ESA2012]). In this paper we study the case of m = O(1) and show that if there are c distinct processing times, i.e. p_j’s come from a set of size c, then there is a randomized (1-ε)-approximation that runs in time O(n^{mc⁷ε^(-6)}log T), where T is the largest deadline. Therefore, for constant m and constant c this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning.

Scheduling problems where the machines can be represented as the edges of a network and each job needs to be processed by a sequence of machines that form a path in this network have been the subject of many research articles (e.g. flow shop is the special case where the network as well as the sequence of machines for each job is a simple path). In this paper we consider one such problem, called Generalized Path Scheduling (GPS) problem, which can be defined as follows. Given a set of non-preemptive jobs J and identical machines M ( |J| = n and |M| = m ). The machines are ordered on a path. Each job j = {P_j = {l_j, r_j}, p_j} is defined by its processing time p_j and a sub-path P_j from machine with index l_j to r_j (l_j, r_j ∈ M, and l_j ≤ r_j) specifying the order of machines it must go through. We assume each machine has a queue of infinite size where jobs can sit in the queue to resolve conflicts. Two objective functions, makespan and total completion time, are considered. Machines can be identical or unrelated. In the latter case, this problem generalizes the classical Flow shop problem (in which all jobs have to go through all machines from 1 to m in that order). Generalized Path Scheduling has been studied (e.g. see [Ronald Koch et al., 2009; Zachary Friggstad et al., 2019]). In this paper, we present several improved approximation algorithms for both objectives. For the case of number of machines being sub-logarithmic in the number of jobs we present a PTAS for both makespan and total completion time. The PTAS holds even on unrelated machines setting and therefore, generalizes the result of Hall [Leslie A. Hall, 1998] for the classic problem of Flow shop. For the case of identical machines, we present an O((log m)/(log log m))-approximation algorithms for both objectives, which improve the previous best result of [Zachary Friggstad et al., 2019]. We also show that the GPS problem is NP-complete for both makespan and total completion time objectives.

We investigate the complexity of solving stable or perturbation-resilient instances of k-means and k-median clustering in fixed dimension Euclidean metrics (or more generally doubling metrics). The notion of stable or perturbation resilient instances was introduced by Bilu and Linial [2010] and Awasthi, Blum, and Sheffet [2012]. In our context, we say a k-MEANS instance is α-stable if there is a unique optimum solution which remains unchanged if distances are (non-uniformly) stretched by a factor of at most α. Stable clustering instances have been studied to explain why heuristics such as Lloyd's algorithm perform well in practice. In this work we show that for any fixed ∊ > 0, (1 + ∊)-stable instances of k-MEANS in doubling metrics, which include fixed-dimensional Euclidean metrics, can be solved in polynomial time. More precisely, we show a natural multi-swap local-search algorithm in fact finds the optimum solution for (1 + ∊)-stable instances of k-MEANS and k-median in a polynomial number of iterations. We complement this result by showing that under a plausible PCP hypothesis this is essentially tight: that when the dimension d is part of the input, there is a fixed ∊0 > 0 such there is not even a PTAS for (1 + ∊0)-stable k-MEANS in ℝd unless NP=RP. To do this, we consider a robust property of CSPs; call an instance stable if there is a unique optimum solution x* and for any other solution x’, the number of unsatisfied clauses is proportional to the Hamming distance between x* and x’. Dinur, Goldreich, and Gur have already shown stable QSAT is hard to approximation for some constant Q [16], our hypothesis is simply that stable QSAT with bounded variable occurrence is also hard (there is in fact work in progress to prove this hypothesis). Given this hypothesis, we consider “stability-preserving” reductions to prove our hardness for stable k-MEANS. Such reductions seem to be more fragile and intricate than standard L-reductions and may be of further use to demonstrate other stable optimization problems are hard to solve.

Clustering problems are well studied in a variety of fields, such as data science, operations research, and computer science. Such problems include variants of center location problems, k -median and k -means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. For instance, some points may arise from noise in a dataset or one might be willing to discard a certain fraction of the points to avoid incurring unnecessary overhead in the cost of a clustering solution. We study clustering problems with outliers. More specifically, we look at uncapacitated facility location (UFL), k - median , and k - means . In these problems, we are given a set X of data points in a metric space δ(., .), a set C of possible centers (each maybe with an opening cost), maybe an integer parameter k , plus an additional parameter z as the number of outliers. In uncapacitated facility location with outliers, we have to open some centers, discard up to z points of X , and assign every other point to the nearest open center, minimizing the total assignment cost plus center opening costs. In k - median and k - means , we have to open up to k centers, but there are no opening costs. In k - means , the cost of assigning j to i is δ 2 ( j , i ). We present several results. Our main focus is on cases where δ is a doubling metric (this includes fixed dimensional Euclidean metrics as a special case) or is the shortest path metrics of graphs from a minor-closed family of graphs. For uniform-cost UFL with outliers on such metrics, we show that a multiswap simple local search heuristic yields a PTAS. With a bit more work, we extend this to bicriteria approximations for the k - median and k - means problems in the same metrics where, for any constant ϵ > 0, we can find a solution using (1 + ϵ) k centers whose cost is at most a (1 + ϵ)-factor of the optimum and uses at most z outliers. Our algorithms are all based on natural multiswap local search heuristics. We also show that natural local search heuristics that do not violate the number of clusters and outliers for k - median (or k - means ) will have unbounded gap even in Euclidean metrics. Furthermore, we show how our analysis can be extended to general metrics for k - means with outliers to obtain a (25 + ϵ, 1 + ϵ)-approximation: an algorithm that uses at most (1 + ϵ) k clusters and whose cost is at most 25 + ϵ of optimum and uses no more than z outliers.

Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, k-median, and k-means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. For instance, some points may arise from noise in a data set or one might be willing to discard a certain fraction of the points to avoid incurring unnecessary overhead in the cost of a clustering solution. We study clustering problems with outliers. More specifically, we look at UNCAPACITATED FACILITY LOCATION (UFL), k-MEDIAN, and k-MEANS. In these problems, we are given a set χ of data points in a metric space δ(.,.), a set C of possible centres (each maybe with an opening cost), maybe an integer parameter k, plus an additional parameter z as the number of outliers. In UNCAPACITATED FACILITY LOCATION with outliers, we have to open some centres, discard up to z points of χ and assign every other point to the nearest open centre, minimizing the total assignment cost plus centre opening costs. In k-MEDIAN and k-MEANS, we have to open up to k centres but there are no opening costs. In k-MEANS, the cost of assigning j to i is δ2(j, i). We present several results. Our main focus is on cases where δ is a doubling metric (this includes fixed dimensional Euclidean metrics as a special case) or is the shortest path metrics of graphs from a minor-closed family of graphs. For UNIFORM-COST UFL with outliers on such metrics we show that a multiswap simple local search heuristic yields a PTAS. With a bit more work, we extend this to bicriteria approximations for the k-MEDIAN and k-MEANS problems in the same metrics where, for any constant ε > 0, we can find a solution using (1 + ε)k centres whose cost is at most a (1 + ε)-factor of the optimum and uses at most z outliers. Our algorithms are all based on natural multiswap local search heuristics. We also show that natural local search heuristics that do not violate the number of clusters and outliers for k-MEDIAN (or k-MEANS) will have unbounded gap even in Euclidean metrics. Furthermore, we show how our analysis can be extended to general metrics for k-MEANS with outliers to obtain a (25 + ε, 1 + ε)-approximation: an algorithm that uses at most (1 + ε)k clusters and whose cost is at most 25 + ε of optimum and uses no more than z outliers.

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (ML k FL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k ≥ 0. Assigning client j to a facility f incurs a connection cost d ( f , j ). The goal is to open a set F ⊆ F of k facilities and assign each client j to a facility f ( j )∈ F so as to minimize max f ∈ F ∑ j ∈ C : f ( j )= f d ( f , j ); we call ∑ j ∈ C : f ( j )= f d ( f , j ) the load of facility f . This problem was studied under the name of min-max star cover in References [3, 7], who (among other results) gave bicriteria approximation algorithms for ML k FL for when F = C . ML k FL is rather poorly understood, and only an O ( k )-approximation is currently known for ML k FL, even for line metrics . Our main result is the first polytime approximation scheme (PTAS) for ML k FL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that ML k FL is strongly NP -hard on line metrics. We also devise a quasi-PTAS for ML k FL on tree metrics. ML k FL turns out to be surprisingly challenging even on line metrics and resilient to attack by a variety of techniques that have been successfully applied to facility-location problems. For instance, we show that (a) even a configuration-style LP-relaxation has a bad integrality gap and (b) a multi-swap k -median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack ML k FL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding of the problem and argue that a MILP-solution minimizing a certain potential function possesses the desired structure and then use a rounding algorithm for the generalized-assignment problem to “transfer” this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.

We consider scheduling problems in which jobs need to be processed through a (shared) network of machines. The network is given in the form of a graph the edges of which represent the machines. We are also given a set of jobs, each specified by its processing time and a path in the graph. Every job needs to be processed in the order of edges specified by its path. We assume that jobs can wait between machines and preemption is not allowed; that is, once a job is started being processed on a machine, it must be completed without interruption. Every machine can only process one job at a time. The makespan of a schedule is the earliest time by which all the jobs have finished processing. The flow time (a.k.a. the completion time) of a job in a schedule is the difference in time between when it finishes processing on its last machine and when the it begins processing on its first machine. The total flow time (or the sum of completion times) is the sum of flow times (or completion times) of all jobs. Our focus is on finding schedules with the minimum sum of completion times or minimum makespan. In this paper, we develop several algorithms (both approximate and exact) for the problem both on general graphs and when the underlying graph of machines is a tree. Even in the very special case when the underlying network is a simple star, the problem is very interesting as it models a biprocessor scheduling with applications to data migration.

The most well-known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centers and partition the data points into $k$ clusters around these centers so that the sum of squares of distances of the points to their cluster center is minimized. Typically these data points lie in Euclidean space $\mathbb{R}^d$ for some $d\geq 2$. $k$-means and the first algorithms for it were introduced in the 1950s. Over the last six decades, hundreds of papers have studied this problem and different algorithms have been proposed for it. The most commonly used algorithm in practice is known as Lloyd--Forgy, which is also referred to as “the” $k$-means algorithm, and various extensions of it often work very well in practice. However, they may produce solutions whose cost is arbitrarily large compared to the optimum solution. Kanungo et al. [ Comput. Geom., 28 (2004), pp. 89--112] analyzed a very simple local search heuristic to get a polynomial-time algorithm with approximation ratio $9+\epsilon$ for any fixed $\epsilon>0$ for $k$-means in Euclidean space. Finding an algorithm with a better worst-case approximation guarantee has remained one of the biggest open questions in this area, in particular, whether one can get a true polynomial-time approximation scheme (PTAS) for fixed dimension Euclidean space. We settle this problem by showing that a simple local search algorithm provides a PTAS for $k$-means for $\mathbb{R}^d$ for any fixed $d$. More precisely, for any error parameter $\epsilon>0$, the local search algorithm that considers swaps of up to $\rho=d^{O(d)}\cdot{\epsilon}^{-O(d/\epsilon)}$ centers at a time will produce a solution using exactly $k$ centers whose cost is at most a $(1+\epsilon)$-factor greater than the optimum solution. Although the algorithm is not practical due to the large polynomial running time, it settles the approximability of this important problem. Our analysis extends very easily to the more general settings where we want to minimize the sum of $q$th powers of the distances between data points and their cluster centers (instead of sum of squares of distances as in $k$-means) for any fixed $q\geq 1$ and where the metric may not be Euclidean but still has fixed doubling dimension. Finally, our techniques also extend to other classic clustering problems. We provide the first demonstration that local search yields a PTAS for uncapacitated facility location and the generalization of $k$-median to the setting with nonuniform opening costs in doubling metrics.

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem.

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F' of k facilities, and assign each client j to a facility f(j) in F' so as to minimize maximum, over all facilities in F', of the sum of distances of clients j assigned to F' to F'. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.

We consider the unsplittable flow problem on a line. In this problem, we are given a set of n tasks, each specified by a start time s i , an end time t i , a demand d i > 0, and a profit p i > 0. A task, if accepted, requires d i units of “bandwidth” from time s i to t i and accrues a profit of p i . For every time t , we are also specified the available bandwidth c t , and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints. We present the first polynomial time O (log n ) approximation algorithm for this problem. This significantly advances the state of the art, as no polynomial time o ( n ) approximation was known previously. Previous results for this problem were known only in more restrictive settings; in particular, either the instance satisfies the so-called “no-bottleneck” assumption: max i d i ≤ min t c t , or the ratio of both maximum to minimum demands and maximum to minimum capacities are polynomially (or quasi-polynomially) bounded in n . Our result, on the other hand, does not require these assumptions. Our algorithm is based on a combination of dynamic programming and rounding a natural linear programming relaxation for the problem. While there is an Ω( n ) integrality gap known for this LP relaxation, our key idea is to exploit certain structural properties of the problem to show that instances that are bad for the LP can in fact be handled using dynamic programming.

We study integrality gaps and approximability of three closely related problems on directed graphs with edge lengths that satisfy the triangle inequality. Given two specified vertices $s$ and $t$, two of these problems ask to find an $s$-$t$ path in the graph visiting all other vertices. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total length of this path. In the directed latency problem, the objective is to minimize the sum of the latencies of the vertices, where the latency of a vertex $v$ is the distance from $s$ to $v$ along the path. The third problem that we study is the $k$-person ATSPP, in which the goal is to find $k$ paths from $s$ to $t$, of minimum total length, such that every vertex is on at least one of these paths. All of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio $O(\log n)$ [C. Chekuri and M. Pal, Theory Comput., 3 (2007), pp. 197--209], [U. Feige and M. Singh, “Improved approximation ratios for traveling salesperson tours and paths in directed graphs,” in Proceedings of the 10th APPROX, 2007, pp. 107--118] until the recent result that improves it to $O(\log n/\log \log n)$ [A. Asadpour et al., “An $O(\log n/\log \log n)$-approximation algorithm for the asymmetric traveling salesman problem,” in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, 2010, pp. 379--389], [U. Feige and M. Singh, “Improved approximation ratios for traveling salesperson tours and paths in directed graphs,” in Proceedings of the 10th APPROX, 2007, pp. 107--118]. However, the best known bound on the integrality gap of any linear programming relaxation for ATSPP is only $O(\sqrt{n})$. For directed latency, the best previously known approximation algorithm has a guarantee of $O(n^{1/2+\epsilon})$ for any constant $\epsilon>0$ [V. Nagarajan and R. Ravi, “The directed minimum latency problem,” in Proceedings of the 11th APPROX, 2008, pp. 193--206]. We present a new algorithm for the ATSPP problem that has an approximation ratio of $O(\log n)$, but whose analysis also upper bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed in [C. Chekuri and M. Pal, Theory Comput., 3 (2007), pp. 197--209]. We then pursue a deeper study of this linear program and its variations, which leads to an $O(\log n)$-approximation for the directed latency problem, a significant improvement over previously known results. Our result for $k$-person ATSPP is an $O(k^2 \log n)$-approximation that bounds the integrality gap of a linear programming relaxation by the same factor. We are not aware of any previous work on this problem.

We describe a O(log n)-approximation algorithm for computing the homotopic Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms where known only for curves on the Euclidean plane with polygonal obstacles.

In the [Formula: see text]-subgraph problem, we are given an undirected graph [Formula: see text] with edge costs and two positive integers [Formula: see text] and [Formula: see text], and the goal is to find a minimum cost simple [Formula: see text]-edge-connected subgraph of [Formula: see text] with at least [Formula: see text] nodes. This generalizes several classical problems, such as the minimum cost [Formula: see text]-spanning tree problem, or [Formula: see text]-MST (which is a [Formula: see text]-subgraph), and the minimum cost [Formula: see text]-edge-connected spanning subgraph (which is a [Formula: see text]-subgraph). The only previously known results on this problem [L. C. Lau, J. S. Naor, M. R. Salavatipour and M. Singh, SIAM J. Comput., 39 (2009), pp. 1062–1087], [C. Chekuri and N. Korula, in Proceedings of the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Bangalore, India, LIPIcs 2, Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2008, pp. 119–130] show that the [Formula: see text]-subgraph problem has an [Formula: see text]-approximation (even for 2-node-connectivity) and that the [Formula: see text]-subgraph problem in general is almost as hard as the densest [Formula: see text]-subgraph problem. In this paper we show that if the edge costs are metric (i.e., satisfy the triangle inequality), like in the [Formula: see text]-MST problem, then there is an [Formula: see text]-approximation algorithm for the [Formula: see text]-subgraph problem. This essentially generalizes the [Formula: see text]-MST constant factor approximability to higher connectivity.

In the mobile facility location problem, which is a variant of the classical facility location, each facility and client is assigned to a start location in a metric graph and our goal is to find a destination node for each client and facility such that every client is sent to a node which is the destination of some facility. The quality of a solution can be measured either by the total distance clients and facilities travel or by the maximum distance traveled by any client or facility. As we show in this article (by an approximation-preserving reduction), the problem of minimizing the total movement of facilities and clients generalizes the classical k -median problem. The class of movement problems was introduced by Demaine et al. [2007] where a simple 2-approximation was proposed for the minimum maximum movement mobile facility location problem while an approximation for the minimum total movement variant and hardness results for both were left as open problems. Our main result here is an 8-approximation algorithm for the minimum total movement mobile facility location problem. Our algorithm is obtained by rounding an LP relaxation in five phases. For the minimum maximum movement mobile facility location problem, we show that we cannot have a better than a 2-approximation for the problem, unless P = NP so the simple algorithm proposed by Demaine et al. [2007] is essentially best possible.

We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find an s-t path visiting all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio O(log n) [7, 9] until the very recent result that improves it to O(log n/ log log n) [3,9]. However, only abound of for the integrality gap of its linear programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n1/2+ε), for any constant ε > 0 [23]. We present a new algorithm for the ATSPP problem that has approximation ratio of O(log n), but whose analysis also bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed in [7]. We then pursue a deeper study of this LP and its variations and their use in approximating directed latency. Our second major result is an O(log n)-approximation to the directed latency problem. This also places an O(log n) bound on the integrality gap of a new LP relaxation of the latency problem that we introduce.

Buy-at-bulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buy-at-bulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first poly-logarithmic approximation ratio for the non-uniform problem that allows different cost functions on each edge and node; the ratio we achieve is $O(\log^4 h)$, where h is the number of demand pairs. In addition we present an $O(\log h)$ approximation for the single sink problem. Poly-logarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems.

We consider the unsplittable flow problem on a line. In this problem, we are given a set of n tasks, each specified by a start time si, an end time ti, a demand di > 0, and a profit pi > 0. A task, if accepted, requires di units of “bandwidth” from time si to ti and accrues a profit of pi. For every time t, we are also specified the available bandwidth ct, and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints. In this paper, we present the first polynomial-time O(log n)-approximation algorithm for this problem. No polynomial-time o(n)-approximation was known prior to this work. Previous results for this problem were known only in more restrictive settings, in particular, either if the given instance satisfies the so-called “no-bottleneck” assumption: maxi di ≤ mint ct, or else if the ratio of the maximum to the minimum demands and ratio of the maximum to the minimum capacities are polynomially (or quasi-polynomially) bounded in n. Our result, on the other hand, does not require any of these assumptions. Our algorithm is based on a combination of dynamic programming and rounding a natural linear programming relaxation for the problem. While there is an Ω(n) integrality gap known for this LP relaxation, our key idea is to exploit certain structural properties of the problem to show that instances that are bad for the LP can in fact be handled using dynamic programming.

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds $B_v$ on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a $(2,2B_v+3)$-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most $2B_v+3$. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost $\lambda$-edge-connected subgraph with at least k vertices, which we call the $(k,\lambda)$-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost $\lambda$-edge-connected subgraph problems. We give a polylogarithmic approximation for the $(k,2)$-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the $(k,\lambda)$-subgraph problem might be hard to approximate for arbitrary $\lambda$.

In the mobile facility location problem, which is a variant of the classical uncapacitated facility location and k-median problems, each facility and client is assigned to a start location in a metric graph and our goal is to find a destination node for each client and facility such that every client is sent to a node which is the destination of some facility. The quality of a solution can be measured either by the total distance clients and facilities travel or by the maximum distance traveled by any client or facility. As we show in this paper (by an approximation preserving reduction), the problem of minimizing the total movement of facilities and clients generalizes the classical k-median problem. The class of movement problems was introduced by Demaine et al. in SODA 2007, where it was observed a simple 2-approximation for the minimum maximum movement mobile facility location while an approximation for the minimum total movement variant and hardness results for both were left as open problems. Our main result here is an 8-approximation algorithm for the minimum total movement mobile facility location problem. Our algorithm is obtained by rounding an LP relaxation in five phases. We also show that this problem generalizes the classical k-median problem using an approximation preserving reduction. For the minimum maximum movement mobile facility location problem, we show that we cannot have a better than a 2-approximation for the problem, unless P = NP; so the simple algorithm observed in is essentially best possible.

We prove semilogarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that $\mathrm{NP}\not\subseteq\operatorname{BPTIME}(2^{n^\varepsilon})$ for an arbitrary $\varepsilon>0$, we prove $O(1/\log^{\sigma}n)$ inapproximability for some constant $\sigma=\sigma(\varepsilon)$. We also prove $O(1/\log^{1/3-\varepsilon}n)$ inapproximability for any $\varepsilon>0$, assuming that refuting random instances of 3SAT is hard on average; and we prove $O(1/\log n)$ inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an $\Omega(1/\log n)$-approximation algorithm, even for a more general (budgeted) setting, and obtain an $\Omega(1/\log B)$-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems, including max cut, maximum coverage, and radio broadcasting.

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv +3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv + 3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic

BACKGROUND: Gene expression microarray is a powerful technology for genetic profiling diseases and their associated treatments. Such a process involves a key step of biomarker identification, which are expected to be closely related to the disease. A most important task of these identified genes is that they can be used to construct a classifier which can effectively diagnose disease and even recognize the disease subtypes. Binary classification, for example, diseased or healthy, in microarray data analysis has been successful, while multi-class classification, such as cancer subtyping, remains challenging. RESULTS: We target on the challenging multi-class classification in microarray data analysis, especially on the cancer subtyping using gene expression microarray. We present a novel class discrimination strength vector to represent individual genes and introduce a new measurement to quantify the class discrimination strength difference between two genes. Such a new distance measure is employed in gene clustering, and subsequently the gene cluster information is exploited to select a set of genes which can be used to construct a sample classifier. We tested our method on four real cancer microarray datasets each contains multiple subtypes of cancer patients. The experimental results show that the constructed classifiers all achieved a higher classification accuracy than the previously best classification results obtained on these four datasets. Additional tests show that the selected genes by our method are less correlated and they all contribute statistically significantly to the more accurate cancer subtyping. CONCLUSION: The proposed novel class discrimination strength vector is a better representation than the gene expression vector, in the sense that it can be used to effectively eliminate highly correlated but redundant genes for classifier construction. Such a method can build a classifier to achieve a higher classification accuracy, which is demonstrated via cancer subtyping.

We consider random instances of constraint satisfaction problems where each variable has domain size d and each constraint contains t restrictions on k variables. For each $(d,k,t)$ we determine whether the resolution complexity is a.s. constant, polynomial, or exponential in the number of variables. For a particular range of $(d,k,t)$, we determine a sharp threshold for resolution complexity where the resolution complexity drops from a.s. exponential to a.s. polynomial when the clause density passes a specific value.

Given an undirected graph G ( V , E ) with terminal set T ⊆ V , the problem of packing element-disjoint Steiner trees is to find the maximum number of Steiner trees that are disjoint on the nonterminal nodes and on the edges. The problem is known to be NP-hard to approximate within a factor of Ω(log n ), where n denotes | V |. We present a randomized O (log n )-approximation algorithm for this problem, thus matching the hardness lower bound. Moreover, we show a tight upper bound of O (log n ) on the integrality ratio of a natural linear programming relaxation.

The cycle packing number ν e ( G ) of a graph G is the maximum number of pairwise edge-disjoint cycles in G . Computing ν e ( G ) is an NP-hard problem. We present approximation algorithms for computing ν e ( G ) in both undirected and directed graphs. In the undirected case we analyze a variant of the modified greedy algorithm suggested by Caprara et al. [2003] and show that it has approximation ratio Θ(√log n ), where n = | V ( G )|. This improves upon the previous O (log n ) upper bound for the approximation ratio of this algorithm. In the directed case we present a √ n -approximation algorithm. Finally, we give an O ( n 2/3 )-approximation algorithm for the problem of finding a maximum number of edge-disjoint cycles that intersect a specified subset S of vertices. We also study generalizations of these problems. Our approximation ratios are the currently best-known ones and, in addition, provide upper bounds on the integrality gap of standard LP-relaxations of these problems. In addition, we give lower bounds for the integrality gap and approximability of ν e ( G ) in directed graphs. Specifically, we prove a lower bound of Ω(log n /loglog n ) for the integrality gap of edge-disjoint cycle packing. We also show that it is quasi-NP-hard to approximate ν e ( G ) within a factor of O (log 1 − ε n ) for any constant ε > 0. This improves upon the previously known APX-hardness result for this problem.

A single DNA microarray measures thousands to tens of thousands of gene expression levels, but experimental datasets normally consist of much fewer such arrays, typically in tens to hundreds, taken over a selection of tissue samples. The biological interpretation of these data relies on identifying subsets of induced or repressed genes that can be used to discriminate various categories of tissue, to provide experimental evidence for connections between a subset of genes and the tissue pathology. A variety of methods can be used to identify discriminatory gene subsets, which can be ranked by classification accuracy. But the high dimensionality of the gene expression space, coupled with relatively fewer tissue samples, creates the dimensionality problem: gene subsets that are too large to provide convincing evidence for any plausible causal connection between that gene subset and the tissue pathology. We propose a new gene selection method, clustered gene selection (CGS) which, when coupled with existing methods, can identify gene subsets that overcome the dimensionality problem and improve classification accuracy. Experiments on eight real datasets showed that CGS can identify many more cancer related genes and clearly improve classification accuracy, compared with three other non-CGS based gene selection methods

We consider approximation algorithms for non-uniform buy-at-bulk network design problems. The first non-trivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC 05); for an instance on h pairs their algorithm has an approximation guarantee of exp(O(radic(log h log log h)))for the uniform-demand case, and log D middot exp(O(radic(log h log log h))) for the general demand case, where D is the total demand. We improve upon this result, by presenting the first poly-logarithmic approximation for this problem. The ratio we obtain is O(log3h middot min{log D, gamma(h2)}) where his the number of pairs and gamma(n) is the worst case distortion in embedding the metric induced by a n vertex graph into a distribution over its spanning trees. Using the best known upper bound on gamma(n) we obtain an O(min{log3h middot log D, log5h log log h}) ratio approximation. We also give poly-logarithmic approximations for some variants of the single-source problem that we need for the multicommodity problem

We consider random instances of constraint satisfaction problems where each variable has domain size d, and each constraint contains t restrictions on k variables. For each (d, k, t) we determine whether the resolution complexity is a.s. constant, polynomial or exponential in the number of variables. For a particular range of (d, k, t) we determine a sharp threshold for resolution complexity where the resolution complexity drops from a.s. exponential to a.s. polynomial when the clause density passes a specific value.