We study a general framework of optimization with the aim to compute fair solutions in settings with a set of agents whose valuations are combined using an aggregation function. The strength of our framework lies (1) in its generality and (2) in the fact that we leverage the power of ex-ante fairness, a concept that has recently gained much attention in the scope of fair allocation and fairness in AI in general. More precisely, in our setting there are n set functions f₁, …, fₙ (e.g., the valuation functions of n agents) that are combined using an aggregation function g (e.g., the minimum, Nash social welfare, p-norm). The power of ex-ante fairness is obtained by allowing as a feasible solution not simply a finite set S, but instead a distribution Π over feasible sets. The goal in our setting is then to find a probability distribution p in Π that maximizes the value resulting from aggregating (using g) the n expected values of the functions f₁, …, fₙ obtained when sampling a set S according to the distribution p. We stress that this is different from maximizing the expected value of g (ex-post fairness) and typically allows for much fairer solutions. We give three different greedy algorithms for three different settings of this framework and prove that they achieve constant approximation guarantees under certain realistic assumptions. For some of the settings, we show that these approximation guarantees are tight. Specific scenarios that can be modelled using our framework include fair information diffusion in social networks, fair submodular matching problems, and ex-ante versions of item assignment problems.

Wheeler automata have recently emerged as a powerful generalization of the Burrows-Wheeler Transform, enabling optimal linear-time pattern matching on compressed labeled graphs - a task that is otherwise computationally hard. Consequently, when an automaton recognizes a Wheeler language (i.e., it is equivalent to some Wheeler automaton), computing its minimum equivalent Wheeler DFA is a powerful indexing strategy. This problem is particularly relevant in computational pangenomics, where pangenome graphs frequently recognize Wheeler languages. However, constructing the minimum Wheeler DFA for a Wheeler language has remained a computational bottleneck. The problem is known to be PSPACE-hard for nondeterministic inputs. When the input is a DFA, state-of-the-art solutions forced a compromise: they were either fast but limited to acyclic DFAs (Alanko et al., SODA 2020) or capable of handling general topologies but prohibitively slow (D'Agostino et al., TCS 2023). In this work, we bridge this gap with the first algorithm solving the problem for general DFAs in near-optimal, linearithmic output-sensitive time. By matching the efficiency of acyclic-only solutions while retaining full generality, our approach improves upon the previous general solution by at least a quadratic factor. We demonstrate the practical impact of our algorithm on real-world pangenome graphs; our tool achieves a processing throughput of over 10⁵ transitions per second on a standard workstation, enabling the construction of a provably optimal pattern matching data structure in such applications.

Co-lex partial orders (Cotumaccio et al., SODA 2021 and Journal of the ACM 2023) are a powerful tool to index finite automata, with applications to regular expression matching, generalizing Wheeler orders (Gagie et al., Theoretical Computer Science 2017). The co-lex width p of an automaton naturally measures how sortable its states are w.r.t. the co-lexicographic order among its accepted strings. Automata of co-lex width p can be compressed to O ( log ⁡ p ) bits per edge and admit regular expression matching in time proportional to p 2 per matched character. The deterministic co-lex width of a regular language L is the smallest width of such a co-lex order, among all DFAs recognizing L . Since languages of small co-lex width admit efficient solutions to hard computational problems on the language, computing the co-lex width of a language is relevant in applications. Previous work shows that the deterministic co-lex width p of a language L can be computed in m O ( p ) , given as input any DFA A with m transitions accepting L . For constant p (in particular Wheeler languages, where p = 1 ), the constant in the exponent is large and the exact complexity remains unknown. In this work, using new techniques, we show that one can decide in O ( m p ) if the deterministic co-lex width of the language recognized by a given minimum DFA is strictly smaller than p ≥ 2 . We complement this with a matching conditional lower bound based on the Strong Exponential Time Hypothesis. Hence, our paper essentially settles the complexity of the problem.

Abstract. A temporal graph is a graph whose edges appear only at certain points in time. Recently, the second and the last three authors proposed a natural temporal analog of the Erdős–Rényi random graph model. The proposed model is obtained by randomly permuting the edges of an Erdős–Rényi random graph and interpreting this permutation as an ordering of presence times. It was shown that the connectivity threshold in the Erdős–Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting. In the present paper, we identify a sharp threshold for the emergence of a giant temporally connected component. We show that at [Formula: see text] the size of the largest temporally connected component increases from [Formula: see text] to [Formula: see text]. This threshold holds for both open and closed connected components, i.e., components that allow (respectively, forbid) their connecting paths to use external nodes.

In this paper, we study the following problem: given n subsets S₁, … , S_n of an integer universe U = {0,… , u-1}, having total cardinality N = ∑_{i = 1}ⁿ |S_i|, find a prefix-free encoding enc : U → {0,1}^+ minimizing the so-called trie measure, i.e., the total number of edges in the n binary tries T₁, … , T_n, where T_i is the trie packing the encoded integers {enc(x):x ∈ S_i}. We first observe that this problem is equivalent to that of merging u sets with the cheapest sequence of binary unions, a problem which in [Ghosh et al., ICDCS 2015] is shown to be NP-hard. Motivated by the hardness of the general problem, we focus on particular families of prefix-free encodings. We start by studying the fixed-length shifted encoding of [Gupta et al., Theoretical Computer Science 2007]. Given a parameter 0 ≤ a < u, this encoding sends each x ∈ U to (x + a) mod u, interpreted as a bit-string of log u bits. We develop the first efficient algorithms that find the value of a minimizing the trie measure when this encoding is used. Our two algorithms run in O(u + Nlog u) and O(Nlog² u) time, respectively. We proceed by studying ordered encodings (a.k.a. monotone or alphabetic), and describe an algorithm finding the optimal such encoding in O(N+u³) time. Within the same running time, we show how to compute the best shifted ordered encoding, provably no worse than both the optimal shifted and optimal ordered encodings. We provide implementations of our algorithms and discuss how these encodings perform in practice.

Wheeler automata were introduced in 2017 as a tool to generalize existing indexing and compression techniques based on the Burrows-Wheeler transform. Intuitively, an automaton is said to be Wheeler if there exists a total order on its states reflecting the co-lexicographic order of the strings labeling the automaton's paths; this property makes it possible to represent the automaton's topology in a constant number of bits per transition, as well as efficiently solving pattern matching queries on its accepted regular language. After their introduction, Wheeler automata have been the subject of a prolific line of research, both from the algorithmic and language-theoretic points of view. A recurring issue faced in these studies is the lack of large datasets of Wheeler automata on which the developed algorithms and theories could be tested. One possible way to overcome this issue is to generate random Wheeler automata. Motivated by this observation, in this paper we initiate the theoretical study of random Wheeler automata, focusing on the deterministic case (Wheeler DFAs -- WDFAs). We start by extending the Erdős-Rényi random graph model to WDFAs, and proceed by providing an algorithm generating uniform WDFAs according to this model. Our algorithm generates a uniform WDFA with $n$ states, $m$ transitions, and alphabet's cardinality $σ$ in $O(m)$ expected time ($O(m\log m)$ worst-case time w.h.p.) and constant working space for all alphabets of size $σ\le m/\ln m$. As a by-product, we also give formulas for the number of distinct WDFAs and obtain that $ nσ+ (n - σ) \log σ$ bits are necessary and sufficient to encode a WDFA with $n$ states and alphabet of size $σ$, up to an additive $Θ(n)$ term. We present an implementation of our algorithm and show that it is extremely fast in practice, with a throughput of over 8 million transitions per second.

We initiate the study of sub-linear sketching and streaming techniques for estimating the output size of common dictionary compressors such as Lempel-Ziv ’77, the run-length Burrows-Wheeler transform, and grammar compression. To this end, we focus on a measure that has recently gained much attention in the information-theoretic community and which approximates up to a polylogarithmic multiplicative factor the output sizes of those compressors: the normalized substring complexity function δ. As a matter of fact, δ itself is a very accurate measure of compressibility: it is monotone under concatenation, invariant under reversals and alphabet permutations, sub-additive, and asymptotically tight (in terms of worst-case entropy) for representing strings, up to polylogarithmic factors.We present a data sketch of O(ε−3log n + ε−1log2n) words that allows computing a multiplicative (1 ± ε)-approximation of δ with high probability, where n is the string length. The sketches of two strings S1,S2can be merged in O(ε−1log2n) time to yield the sketch of {S1,S2}, speeding up the computation of Normalized Compression Distances (NCD). If random access is available on the input, our sketch can be updated in O(ε−1log2n) time for each character right-extension of the string. This yields a polylogarithmic-space algorithm for approximating δ, improving exponentially over the working space of the state-of-the-art algorithms running in nearly-linear time. Motivated by the fact that random access is not always available on the input data, we then present a streaming algorithm computing our sketch in $O(\sqrt n \cdot \log n)$ working space and O(ε−1log2n) worst-case delay per character. We show that an implementation of our streaming algorithm can estimate δ on a dataset of 189GB with a throughput of 203MB per minute while using only 5MB of RAM, and that our sketch speeds up the computation of all-pairs NCD distances by one order of magnitude, with applications to phylogenetic tree reconstruction.

In this paper, we propose a symbolic-numeric algorithm to count the number of solutions of a zero-dimensional square polynomial system within a local region. We show that the algorithm succeeds under the condition that the region is sufficiently small and well-isolating for a k-fold solution z of the system. In our analysis, we derive a bound on the size of the region that guarantees success. We further argue that this size depends on local parameters such as the norm and multiplicity of z as well as the distances between z and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots of the original system to the problem of solving a truncated system of degree k. In particular, if the multiplicity k of z is small compared to the total degrees of the original polynomials, our method considerably improves upon known complete and certified methods. We see a series of applications of our approach. When combined with a numerical solver in the fashion of an a posteriori certification step, we obtain a certified and reliable method for solving polynomial systems while profiting both from the efficiency of the numerical algorithm and the reliability of the symbolic approach. An alternative application results from incorporating our algorithm as inclusion predicate into an elimination method. For the special case of bivariate systems, we experimentally show that this approach leads to a significant improvement over an existing state-of-the-art elimination method.

Abstract. We study the problem of approximating the distances in an undirected weighted graph [Formula: see text] by the distances in trees based on the notion of stretch. Focusing on decentralized models of computation such as the [Formula: see text], [Formula: see text], and semi-streaming models, our main results are as follows: (1) We develop a simple randomized algorithm that constructs a spanning tree such that the expected stretch of every edge is [Formula: see text], where [Formula: see text] is the number of nodes in [Formula: see text]. If [Formula: see text] is unweighted, then this algorithm can be implemented to run in [Formula: see text] rounds in the [Formula: see text] model, where [Formula: see text] is the hop-diameter of [Formula: see text]; thus our algorithm is asymptotically optimal in this case. In the weighted case, the run-time of the algorithm matches the currently best known bound for exact single source shortest path (SSSP) computations, which despite recent progress is still separated from the lower bound of [Formula: see text] by polynomial factors. A naive attempt to replace exact SSSP computations with approximate ones in order to improve the complexity in the weighted case encounters a fundamental challenge, as the underlying decomposition technique fails to work under distance approximation. (2) We overcome this obstacle by developing a technique termed blurry ball growing. This technique, in combination with a clever algorithmic idea of Miller, Peng, and Xu (SPAA 2013), allows us to obtain low diameter graph decompositions with small edge cutting probabilities based solely on approximate SSSP computations. (3) Using these decompositions, we in turn obtain metric tree embedding algorithms in the vein of the celebrated work of Bartal (FOCS 1996), whose computational complexity is optimal up to polylogarithmic factors not only in the [Formula: see text] model but also in the [Formula: see text] and semi-streaming models. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is “used” only logarithmically many times. This property is of interest for capacitated problems and for simulating [Formula: see text] algorithms on the tree into which the graph is embedded.

Modeling and shaping how information spreads through a network is a major research topic in network analysis. While initially the focus has been mostly on efficiency, recently fairness criteria have been taken into account in this setting. Most work has focused on the maximin criteria however, and thus still different groups can receive very different shares of information. In this work we propose to consider fairness as a notion to be guaranteed by an algorithm rather than as a criterion to be maximized. To this end, we propose three optimization problems that aim at maximizing the overall spread while enforcing strict levels of demographic parity fairness via constraints (either ex-post or ex-ante). The level of fairness hence becomes a user choice rather than a property to be observed upon output. We study this setting from various perspectives. First, we prove that the cost of introducing demographic parity can be high in terms of both overall spread and computational complexity, i.e., the price of fairness may be unbounded for all three problems and optimal solutions are hard to compute, in some case even approximately or when fairness constraints may be violated. For one of our problems, we still design an algorithm with both constant approximation factor and fairness violation. We also give two heuristics that allow the user to choose the tolerated fairness violation. By means of an extensive experimental study, we show that our algorithms perform well in practice, that is, they achieve the best demographic parity fairness values. For certain instances we additionally even obtain an overall spread comparable to the most efficient algorithms that come without any fairness guarantee, indicating that the empirical price of fairness may actually be small when using our algorithms.

Fairness in influence maximization has been a very active research topic recently. Most works in this context study the question of how to find seeding strategies (deterministic or probabilistic) such that nodes or communities in the network get their fair share of coverage. Different fairness criteria have been used in this context. All these works assume that the entity that is spreading the information has an inherent interest in spreading the information fairly, otherwise why would they want to use the developed fair algorithms? This assumption may however be flawed in reality -- the spreading entity may be purely efficiency-oriented. In this paper we propose to study two optimization problems with the goal to modify the network structure by adding links in such a way that efficiency-oriented information spreading becomes automatically fair. We study the proposed optimization problems both from a theoretical and experimental perspective, that is, we give several hardness and hardness of approximation results, provide efficient algorithms for some special cases, and more importantly provide heuristics for solving one of the problems in practice. In our experimental study we then first compare the proposed heuristics against each other and establish the most successful one. In a second experiment, we then show that our approach can be very successful in practice. That is, we show that already after adding a few edges to the networks the greedy algorithm that purely maximizes spread surpasses all fairness-tailored algorithms in terms of ex-post fairness. Maybe surprisingly, we even show that our approach achieves ex-post fairness values that are comparable or even better than the ex-ante fairness values of the currently most efficient algorithms that optimize ex-ante fairness.

Wheeler nondeterministic finite automata (WNFAs) were introduced as a generalization of prefix sorting from strings to labeled graphs. WNFAs admit optimal solutions to classic hard problems on labeled graphs and languages. The problem of deciding whether a given NFA is Wheeler is known to be NP-complete. Recently, however, Alanko et al. showed how to side-step this complexity by switching to preorders: letting $Q$ be the set of states, $E$ the set of transitions, $|Q|=n$, and $|E|=m$, they provided a $O(mn^2)$-time algorithm computing a totally-ordered partition of the WNFA's states such that (1) equivalent states recognize the same regular language, and (2) the order of non-equivalent states is consistent with any Wheeler order, when one exists. Then, the output is a preorder of the states as useful for pattern matching as standard Wheeler orders. Further research generalized these concepts to arbitrary NFAs by introducing co-lex partial preorders: any NFA admits a partial preorder of its states reflecting the co-lex order of their accepted strings; the smaller the width of such preorder is, the faster regular expression matching queries can be performed. To date, the fastest algorithm for computing the smallest-width partial preorder on NFAs runs in $O(m^2+n^{5/2})$ time, while on DFAs the same can be done in $O(\min(n^2\log n,mn))$ time. In this paper, we provide much more efficient solutions to the problem above. Our results are achieved by extending a classic algorithm for the relational coarsest partition refinement problem to work with ordered partitions. Specifically, we provide a $O(m\log n)$-time algorithm computing a co-lex total preorder when the input is a WNFA, and an algorithm with the same time complexity computing the smallest-width co-lex partial order of any DFA. Also, we present implementations of our algorithms and show that they are very efficient in practice.

The influence maximization paradigm has been used by researchers in various fields in order to study how information spreads in social networks. While previously the attention was mostly on efficiency, more recently fairness issues have been taken into account in this scope. In the present paper, we propose to use randomization as a mean for achieving fairness. While this general idea is not new, it has not been applied in this area. Similar to previous works like Fish et al. (WWW ’19) and Tsang et al. (IJCAI ’19), we study the maximin criterion for (group) fairness. In contrast to their work however, we model the problem in such a way that, when choosing the seed sets, probabilistic strategies are possible rather than only deterministic ones. We introduce two different variants of this probabilistic problem, one that entails probabilistic strategies over nodes (node-based problem) and a second one that entails probabilistic strategies over sets of nodes (set-based problem). After analyzing the relation between the two probabilistic problems, we show that, while the original deterministic maximin problem was inapproximable, both probabilistic variants permit approximation algorithms that achieve a constant multiplicative factor of 1 − 1/e minus an additive arbitrarily small error that is due to the simulation of the information spread. For the node-based problem, the approximation is achieved by observing that a polynomial-sized linear program approximates the problem well. For the set-based problem, we show that a multiplicative-weight routine can yield the approximation result. For an experimental study, we provide implementations of multiplicative-weight routines for both the set-based and the node-based problems and compare the achieved fairness values to existing methods. Maybe non-surprisingly, we show that the ex-ante values, i.e., minimum expected value of an individual (or group) to obtain the information, of the computed probabilistic strategies are significantly larger than the (ex-post) fairness values of previous methods. This indicates that studying fairness via randomization is a worthwhile path to follow. Interestingly and maybe more surprisingly, we observe that even the ex-post fairness values, i.e., fairness values of sets sampled according to the probabilistic strategies computed by our routines, dominate over the fairness achieved by previous methods on many of the instances tested.

The influence maximization paradigm has been used by researchers in various fields in order to study how information spreads in social networks. While previously the attention was mostly on efficiency, more recently fairness issues have been taken into account in this scope. In the present paper, we propose to use randomization as a mean for achieving fairness. While this general idea is not new, it has not been applied in the area of information spread in networks. Similar to previous works like Fish et al. (WWW '19) and Tsang et al. (IJCAI '19), we study the maximin criterion for (group) fairness. By allowing randomized solutions, we introduce two different variants of this problem. While the original deterministic maximin problem has been shown to be inapproximable, interestingly, we show that both probabilistic variants permit approximation algorithms with a constant multiplicative factor of 1-1/e plus an additive arbitrarily small error that is due to the simulation of the information spread. For an experimental study, we provide implementations of our methods and compare the achieved fairness values to existing methods. Non-surprisingly, the ex-ante values, i.e., minimum expected value of an individual (or group) to obtain the information, of the computed probabilistic strategies are significantly larger than the (ex-post) fairness values of previous methods. This confirms that studying fairness via randomization is a worthwhile direction. More surprisingly, we observe that even the ex-post fairness values, i.e., fairness values of sets sampled according to the probabilistic strategies, computed by our routines dominate over the fairness achieved by previous methods on most of the instances tested.

Centrality measures characterize important nodes in networks. Efficiently computing such nodes has received a lot of attention. When considering the generalization of computing central groups of nodes, challenging optimization problems occur. In this work, we study two such problems, group-harmonic maximization and group-closeness maximization both from a theoretical and from an algorithm engineering perspective. On the theoretical side, we obtain the following results. For group-harmonic maximization, unless P = NP, there is no polynomial-time algorithm that achieves an approximation factor better than (directed) and (undirected), even for unweighted graphs. On the positive side, we show that a greedy algorithm achieves an approximation factor of (directed) and (undirected), where λ is the ratio of minimal and maximal edge weights. For group-closeness maximization, we obtain a strong separation between undirected and directed graphs (that holds even in the unweighted case). The undirected case is NP-hard to be approximated to within a factor better than and a constant approximation factor is achieved by a local-search algorithm. For the directed case, however, we show that, for any , the problem is NP-hard to be approximated within a factor of 4|V|−∊. From the algorithm engineering perspective, we provide efficient implementations of the above greedy and local search algorithms. In our extensive experimental study we show that, on instances small enough so that an optimum solution can be computed in reasonable time, the quality of both the greedy and the local search algorithms come very close to the optimum. On larger instances, our local search algorithms yield results with superior quality compared to existing greedy and local search solutions, at the cost of additional running time. We thus advocate local search for scenarios where solution quality is of highest concern.

In the classical influence maximization problem we aim to select a set of nodes, called seeds, to start an efficient information diffusion process. More precisely, the goal is to select seeds such that the expected number of nodes reached by the diffusion process is maximized. In this work we study a variant of this problem where an unknown (up to a probability distribution) set of nodes, referred to as co-existing seeds, joins in starting the diffusion process even if not selected. This setting allows to model that, in certain situations, some nodes are willing to act as "voluntary seeds'' even if not chosen by the campaign organizer. This may for example be due to the positive nature of the information campaign (e.g., public health awareness programs, HIV prevention, financial aid programs), or due to external social driving effects (e.g., nodes are friends of selected seeds in real life or in other social media).

We present a method for solving the transshipment problem-also known as uncapacitated minimum cost flow-up to a multiplicative error of 1+ε in undirected graphs with nonnegative edge weights using a tailored gradient descent algorithm. Using O(\cdot ) to hide polylogarithmic factors in n (the number of nodes in the graph), our gradient descent algorithm takes O(ε 2) iterations, and in each iteration it solves an instance of the transshipment problem up to a multiplicative error of polylog n. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a randomized rounding scheme, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve upon prior works by obtaining the following results: (1) Broadcast CONGEST model: (1 + ε)-approximate SSSP using O(( n + D)ε 3) rounds, where D is the (hop) diameter of the network. (2) Broadcast Congested Clique model: (1 + ε)-approximate transshipment and SSSP using O (ε 2) rounds. (3) Multipass Streaming model: (1 + ε)-approximate transshipment and SSSP using O(n) space and O(ε 2) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume nonnegative edge weights that are polynomially bounded in n; for general nonnegative weights, there is an additional multiplicative overhead equal to the logarithm of the maximum ratio between nonzero weights. Our algorithms can also handle asymmetric costs for traversing edges in opposite directions. In this case, we obtain an additional multiplicative dependence of the maximum ratio between the two costs on some edge.

The personalization of our news consumption on social media has a tendency to reinforce our pre-existing beliefs instead of balancing our opinions. To tackle this issue, Garimella et al. (NIPS'17) modeled the spread of these viewpoints, also called campaigns, using the independent cascade model introduced by Kempe, Kleinberg and Tardos (KDD'03) and studied an optimization problem that aims to balance information exposure when two opposing campaigns propagate in a network. This paper investigates a natural generalization of this optimization problem in which μ different campaigns propagate in the network and we aim to maximize the expected number of nodes that are reached by at least ν or none of the campaigns, where μ ≥ ν ≥ 2. Following Garimella et al., despite this general setting, we also investigate a simplified one, in which campaigns propagate in a correlated manner. While for the simplified setting, we show that the problem can be approximated within a constant factor for any constant μ and ν, for the general setting, we give reductions leading to several approximation hardness results when ν ≥ 3. For instance, assuming the gap exponential time hypothesis to hold, we obtain that the problem cannot be approximated within a factor of n−g(n) for any g(n) = o(1) where n is the number of nodes in the network. We complement our hardness results with an Ω(n−1/2)-approximation algorithm for the general setting when ν = 3 and μ is arbitrary.

In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of low diameter graph decomposition with small edge cutting probabilities inherently rely on the subtractive form of the triangle inequality, which fails to hold under distance approximation. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 2013), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the CONGEST, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 1996) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logaritmically many times, which is of interest for capacitated problems and simulating CONGEST algorithms on the tree into which the graph is embedded.

Given an undirected graph with integer edge lengths, we study the problem of approximating the distances in the graph by a spanning tree based on the notion of stretch. Our main contribution is a distributed algorithm in the CONGEST model of computation that constructs a random spanning tree with the guarantee that the expected stretch of every edge is O(log^{3} n), where n is the number of nodes in the graph. If the graph is unweighted, then this algorithm can be implemented to run in O(D) rounds, where D is the hop-diameter of the graph, thus being asymptotically optimal. In the weighted case, the run-time of our algorithm matches the currently best known bound for exact distance computations, i.e., O~ (min{sqrt{n D}, sqrt{n} D^{1 / 4} + n^{3 / 5} + D}). We stress that this is the first distributed construction of spanning trees leading to poly-logarithmic expected stretch with non-trivial running time.

We present a method for solving the shortest transshipment problem-also known as uncapacitated minimum cost flow-up to a multiplicative error of 1 + ε in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes ε-3 polylog n iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog n, where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results: 1. Broadcast CONGEST model: (1+")-approximate SSSP using Õ((√ n+D) · ε-O(1)) rounds, 1 where D is the (hop) diameter of the network. 2. Broadcast congested clique model: (1+ε)-approximate shortest transshipment and SSSP using Õ (ε-O(1)) rounds. 3. Multipass streaming model: (1+ε)-approximate shortest transshipment and SSSP using Õ (n) space and Õ(ε-O(1)) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general nonnegative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges.

We present a novel algorithm for the min-cost flow problem that is competitive with recent third-party implementations of well-known algorithms for this problem and even outperforms them on certain realistic instances. We formally prove correctness of our algorithm and show that the worst-case running time is in O(‖b‖1(m + n log n)) where b is the vector of demands. Combined with standard scaling techniques, this pseudo-polynomial bound can be made polynomial in a straightforward way. Furthermore, we evaluate our approach experimentally. Our empirical findings indeed suggest that the running time does not significantly depend on the costs and that a linear dependence on ‖b‖1 is overly pessimistic.

Let F(z) be an arbitrary complex polynomial. We introduce the {local root clustering problem}, to compute a set of natural epsilon-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters.