arXiv subjects

Michal Dory

Publications and source records attributed to Michal Dory.

At least 19 recordsLinked to original sources

arXiv2026

$\tilde{\text{O}}$ptimal Distributed Maximum Flow Approximation in Undirected Planar Graphs

Persistent efforts in recent years have been devoted to devising distributed algorithms for fundamental optimization problems in planar graphs. In particular, for Single-Source Shortest-Paths, there is an $\tilde O(D^2)$-rounds exact algorithm [Li, Parter STOC'19] for directed planar graphs, and an $\tilde {O}(D)$-rounds $(1+o(1))$-approximation algorithm [Rozhon, Grunau, Haeupler, Zuzic, Li STOC'22] for undirected planar graphs (where $D$ is the graph's hop-diameter). Recently [Abd-Elhaleem, Dory, Parter, Weimann PODC'25], a matching bound for the exact case was obtained for the Maximum $st$-Flow problem. Namely, an $\tilde O(D^2)$-rounds exact algorithm for directed planar graphs. However, for the approximate case, they give a $D\cdot n^{o(1)}$-rounds $(1-o(1))$-approximation algorithm for undirected planar graphs that works only for the special case where both $s$ and $t$ lie on the same face. In this paper, we remove the restriction that both $s$ and $t$ must lie on the same face (we also eliminate the $n^{o(1)}$ factor). Namely, we present the first distributed near-optimal $\tilde{O}(D)$-rounds $(1-o(1))$-approximation algorithm for Maximum $st$-Flow in general undirected planar graphs. Our main technical contribution is a distributed implementation of the classical Reif's [SICOMP'83] centralized algorithm. This is achieved by a careful recursive incision procedure on the planar dual $G^*$ of the graph $G$. It is challenging, because we need to simulate dynamic changes (incisions) over the dual graph $G^*$, while we can only communicate over the input graph $G$.

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arXiv2026

Deterministic Distance Approximation in MPC via Improved Hitting Sets

In this paper, we provide the first deterministic algorithms with sublogarithmic round complexity for spanners and approximate shortest paths in various MPC models. Moreover, we significantly improve upon the state of the art in the deterministic Congested Clique. In particular, we obtain the following four results on undirected graphs: 1. In both linear MPC and Congested Clique, we obtain an $O(k)$ stretch-spanner of a weighted graph of size $O(n^{1+1/k})$ in $O(1)$ rounds, for some parameter $k\ge 0$. For $k=O(\log{n})$, this leads to an $O(\log n)$ approximation of APSP in constant rounds in both models. 2. In sublinear MPC, we obtain an $O(k^{1+\varepsilon})$-stretch spanner of a weighted graph of size $O(n^{1+1/k})$ in $O(\log k)$ rounds, for any fixed constant $\varepsilon>0$. 3. In Congested Clique, we obtain $O(1)$-approximate APSP for weighted graphs in $O(\log \log \log n)$ rounds. 4. In near-linear MPC, we obtain $(1+\varepsilon)$-approximate single-source shortest paths and $O(1)$-approximate all-pairs shortest paths for unweighted graphs in $\textsf{poly}\log \log n$ rounds. Our algorithm only requires a single near-linear memory machine, where the rest can have sublinear memory. Our deterministic algorithms obtain similar guarantees to the state of the art randomized algorithms without incurring additional factors in the round complexity. To obtain these results, we inspect the randomized algorithms and isolate a randomized sampling routine. Then we derandomize these sampling routines by using a deterministic hitting set. Hereto, we develop a versatile deterministic hitting set algorithm, which we hope will have further derandomization applications.

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arXiv2026

A Simple Distributed Deterministic Planar Separator

A balanced separator of a graph $G$ is a set of vertices whose removal disconnects the graph into connected components that are a constant factor smaller than $G$. Lipton and Tarjan [FOCS'77] famously proved that every planar graph admits a balanced separator of size $O(\sqrt{n})$, as well as a balanced separator of size $O(D)$ that is a simple path (where $D$ is $G$'s diameter). In the centralized setting, both separators can be found in linear time. In the distributed setting, $D$ is a universal lower bound for the round complexity of solving many optimization problems, so, separators of size $O(D)$ are preferable. It was not until [DISC'17] that a distributed algorithm was devised by Ghaffari and Parter to compute such an $O(D)$-size separator in $\tilde O(D)$ rounds, by adapting the Lipton-Tarjan algorithm to the distributed model. Since then, this algorithm was used in several distributed algorithms for planar graphs, e.g., [GP, DISC'17], [LP, STOC'19], [AEDPW, PODC'25]. However, the algorithm is randomized, deeming the algorithms that use it to be randomized as well. Obtaining a deterministic algorithm remained an interesting open question until [PODC'25], when a (complex) deterministic separator algorithm was given by Jauregui, Montealegre and Rapaport. We present a much simpler deterministic separator algorithm with the same (near-optimal) $\tilde O(D)$-round complexity. While previous works devised either complicated or randomized ways of transferring weights from vertices to faces of $G$, we show that a straightforward way also works: Each vertex simply transfers its weight to one arbitrary face it lies on. That's it! We note that a deterministic separator algorithm directly derandomizes the state-of-the-art distributed algorithms for classical problems on planar graphs such as single-source shortest-paths, maximum-flow, directed global min-cut, and reachability.

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arXiv2024

Massively Parallel Algorithms for Approximate Shortest Paths

We present fast algorithms for approximate shortest paths in the massively parallel computation (MPC) model. We provide randomized algorithms that take $poly(\log{\log{n}})$ rounds in the near-linear memory MPC model. Our results are for unweighted undirected graphs with $n$ vertices and $m$ edges. Our first contribution is a $(1+\epsilon)$-approximation algorithm for Single-Source Shortest Paths (SSSP) that takes $poly(\log{\log{n}})$ rounds in the near-linear MPC model, where the memory per machine is $\tilde{O}(n)$ and the total memory is $\tilde{O}(mn^{\rho})$, where $\rho$ is a small constant. Our second contribution is a distance oracle that allows to approximate the distance between any pair of vertices. The distance oracle is constructed in $poly(\log{\log{n}})$ rounds and allows to query a $(1+\epsilon)(2k-1)$-approximate distance between any pair of vertices $u$ and $v$ in $O(1)$ additional rounds. The algorithm is for the near-linear memory MPC model with total memory of size $\tilde{O}((m+n^{1+\rho})n^{1/k})$, where $\rho$ is a small constant. While our algorithms are for the near-linear MPC model, in fact they only use one machine with $\tilde{O}(n)$ memory, where the rest of machines can have sublinear memory of size $O(n^{\gamma})$ for a small constant $\gamma < 1$. All previous algorithms for approximate shortest paths in the near-linear MPC model either required $\Omega(\log{n})$ rounds or had an $\Omega(\log{n})$ approximation. Our approach is based on fast construction of near-additive emulators, limited-scale hopsets and limited-scale distance sketches that are tailored for the MPC model. While our end-results are for the near-linear MPC model, many of the tools we construct such as hopsets and emulators are constructed in the more restricted sublinear MPC model.

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arXiv2024

Distributed Maximum Flow in Planar Graphs

The dual of a planar graph $G$ is a planar graph $G^*$ that has a vertex for each face of $G$ and an edge for each pair of adjacent faces of $G$. The profound relationship between a planar graph and its dual has been the algorithmic basis for solving numerous (centralized) classical problems on planar graphs. In the distributed setting however, the only use of planar duality is for finding a recursive decomposition of $G$ [DISC 2017, STOC 2019]. We extend the distributed algorithmic toolkit to work on the dual graph $G^*$. These tools can then facilitate various algorithms on $G$ by solving a suitable dual problem on $G^*$. Given a directed planar graph $G$ with positive and negative edge-lengths and hop-diameter $D$, our key result is an $\tilde{O}(D^2)$-round algorithm for Single Source Shortest Paths on $G^*$, which then implies $\tilde{O}(D^2)$-round algorithms for Maximum $st$-Flow and Directed Global Min-Cut on $G$. Prior to our work, no $\tilde{O}(\text{poly}(D))$-round algorithm was known for those problems. We further obtain a $D\cdot n^{o(1)}$-rounds $(1-\epsilon)$-approximation algorithm for Maximum $st$-Flow on $G$ when $G$ is undirected and $st$-planar. Finally, we give a near optimal $\tilde O(D)$-round algorithm for computing the weighted girth of $G$. The main challenges in our work are that $G^*$ is not the communication graph (e.g., a vertex of $G$ is mapped to multiple vertices of $G^*$), and that the diameter of $G^*$ can be much larger than $D$ (i.e., possibly by a linear factor). We overcome these challenges by carefully defining and maintaining subgraphs of the dual graph $G^*$ while applying the recursive decomposition on the primal graph $G$. The main technical difficulty, is that along the recursive decomposition, a face of $G$ gets shattered into (disconnected) components yet we still need to treat it as a dual node.

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arXiv2024

Improved All-Pairs Approximate Shortest Paths in Congested Clique

In this paper, we present a new randomized $O(1)$-approximation algorithm for the All-Pairs Shortest Paths (APSP) problem in weighted undirected graphs that runs in just $O(\log \log \log n)$ rounds in the Congested-Clique model. Before our work, the fastest algorithms achieving an $O(1)$-approximation for APSP in weighted undirected graphs required $\operatorname{poly}(\log n)$ rounds, as shown by Censor-Hillel, Dory, Korhonen, and Leitersdorf (PODC 2019 & Distributed Computing 2021). In the unweighted undirected setting, Dory and Parter (PODC 2020 & Journal of the ACM 2022) obtained $O(1)$-approximation in $\operatorname{poly}(\log \log n)$ rounds. By terminating our algorithm early, for any given parameter $t \geq 1$, we obtain an $O(t)$-round algorithm that guarantees an $O\left(\log^{1/2^t} n\right)$ approximation in weighted undirected graphs. This tradeoff between round complexity and approximation factor offers flexibility, allowing the algorithm to adapt to different requirements. In particular, for any constant $\varepsilon > 0$, an $O\left(\log^\varepsilon n\right)$-approximation can be obtained in $O(1)$ rounds. Previously, $O(1)$-round algorithms were only known for $O(\log n)$-approximation, as shown by Chechik and Zhang (PODC 2022). A key ingredient in our algorithm is a lemma that, under certain conditions, allows us to improve an $a$-approximation for APSP to an $O(\sqrt{a})$-approximation in $O(1)$ rounds. To prove this lemma, we develop several new techniques, including an $O(1)$-round algorithm for computing the $k$-nearest nodes, as well as new types of hopsets and skeleton graphs based on the notion of $k$-nearest nodes.

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arXiv2024

Fast Broadcast in Highly Connected Networks

We revisit the classic broadcast problem, wherein we have $k$ messages, each composed of $O(\log{n})$ bits, distributed arbitrarily across a network. The objective is to broadcast these messages to all nodes in the network. In the distributed CONGEST model, a textbook algorithm solves this problem in $O(D+k)$ rounds, where $D$ is the diameter of the graph. While the $O(D)$ term in the round complexity is unavoidable$\unicode{x2014}$given that $\Omega(D)$ rounds are necessary to solve broadcast in any graph$\unicode{x2014}$it remains unclear whether the $O(k)$ term is needed in all graphs. In cases where the minimum cut size is one, simply transmitting messages from one side of the cut to the other would require $\Omega(k)$ rounds. However, if the size of the minimum cut is larger, it may be possible to develop faster algorithms. This motivates the exploration of the broadcast problem in networks with high edge connectivity. In this work, we present a simple randomized distributed algorithm for performing $k$-message broadcast in $O(((n+k)/\lambda)\log n)$ rounds in any $n$-node simple graph with edge connectivity $\lambda$. When $k = \Omega(n)$, our algorithm is universally optimal, up to an $O(\log n)$ factor, as its complexity nearly matches an information-theoretic $\Omega(k/\lambda)$ lower bound that applies to all graphs, even when the network topology is known to the algorithm. The setting $k = \Omega(n)$ is particularly interesting because several fundamental problems can be reduced to broadcasting $\Omega(n)$ messages. Our broadcast algorithm finds several applications in distributed computing, enabling $O(1)$-approximation for all distances and $(1+\epsilon)$-approximation for all cut sizes in $\tilde{O}(n/\lambda)$ rounds.

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arXiv2023

Fast 2-Approximate All-Pairs Shortest Paths

In this paper, we revisit the classic approximate All-Pairs Shortest Paths (APSP) problem in undirected graphs. For unweighted graphs, we provide an algorithm for $2$-approximate APSP in $\tilde O(n^{2.5-r}+n^{\omega(r)})$ time, for any $r\in[0,1]$. This is $O(n^{2.032})$ time, using known bounds for rectangular matrix multiplication $n^{\omega(r)}$ [Le Gall, Urrutia, SODA 2018]. Our result improves on the $\tilde{O}(n^{2.25})$ bound of [Roditty, STOC 2023], and on the $\tilde{O}(m\sqrt n+n^2)$ bound of [Baswana, Kavitha, SICOMP 2010] for graphs with $m\geq n^{1.532}$ edges. For weighted graphs, we obtain $(2+\epsilon)$-approximate APSP in $\tilde O(n^{3-r}+n^{\omega(r)})$ time, for any $r\in [0,1]$. This is $O(n^{2.214})$ time using known bounds for $\omega(r)$. It improves on the state of the art bound of $O(n^{2.25})$ by [Kavitha, Algorithmica 2012]. Our techniques further lead to improved bounds in a wide range of density for weighted graphs. In particular, for the sparse regime we construct a distance oracle in $\tilde O(mn^{2/3})$ time that supports $2$-approximate queries in constant time. For sparse graphs, the preprocessing time of the algorithm matches conditional lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer, STOC 2023]. To the best of our knowledge, this is the first 2-approximate distance oracle that has subquadratic preprocessing time in sparse graphs. We also obtain new bounds in the near additive regime for unweighted graphs. We give faster algorithms for $(1+\epsilon,k)$-approximate APSP, for $k=2,4,6,8$. We obtain these results by incorporating fast rectangular matrix multiplications into various combinatorial algorithms that carefully balance out distance computation on layers of sparse graphs preserving certain distance information.

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arXiv2022

A Nearly Time-Optimal Distributed Approximation of Minimum Cost $k$-Edge-Connected Spanning Subgraph

The minimum-cost $k$-edge-connected spanning subgraph ($k$-ECSS) problem is a generalization and strengthening of the well-studied minimum-cost spanning tree (MST) problem. While the round complexity of distributedly computing the latter has been well-understood, the former remains mostly open, especially as soon as $k\geq 3$. In this paper, we present the first distributed algorithm that computes an approximation of $k$-ECSS in sublinear time for general $k$. Concretely, we describe a randomized distributed algorithm that, in $\tilde{O}(k(D+k\sqrt{n}))$ rounds, computes a $k$-edge-connected spanning subgraph whose cost is within an $O(\log n\log k)$ factor of optimal. Here, $n$ and $D$ denote the number of vertices and diameter of the graph, respectively. This time complexity is nearly optimal for any $k=poly(\log n)$, almost matching an $\tilde{\Omega}(D+\sqrt{n/k})$ lower bound. Our algorithm is the first to achieve a sublinear round complexity for $k\geq 3$. We note that this case is considerably more challenging than the well-studied and well-understood $k=1$ case -- better known as MST -- and the closely related $k=2$ case. Our algorithm is based on reducing the $k$-ECSS problem to $k$ set cover instances, in which we gradually augment the connectivity of the spanning subgraph. To solve each set cover instance, we combine new structural observations on minimum cuts with graph sketching ideas. One key ingredient in our algorithm is a novel structural lemma that allows us to compress the information about all minimum cuts in a graph into a succinct representation, which is computed in a decentralized fashion. We hope that this succinct representation may find applications in other computational settings or for other problems.

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arXiv2022

New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate decremental APSP algorithms, two for weighted and two for unweighted graphs. Our first result is $(2+ \epsilon)$-APSP with total update time $\tilde{O}(m^{1/2}n^{3/2})$ (when $m= n^{1+c}$ for any constant $0 0$). For comparison, in the special case of $(1+\epsilon, 2)$-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of $\tilde{O}(n^{2.5})$. All of our results are randomized, work against an oblivious adversary, and have constant query time.

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arXiv2022

Near-Optimal Distributed Dominating Set in Bounded Arboricity Graphs

We describe a simple deterministic $O( \varepsilon^{-1} \log \Delta)$ round distributed algorithm for $(2\alpha+1)(1 + \varepsilon)$ approximation of minimum weighted dominating set on graphs with arboricity at most $\alpha$. Here $\Delta$ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation [Kuhn, Moscibroda, and Wattenhofer JACM'16]. Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $O(\alpha^2)$ approximation in $O(\log n)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(\alpha \log \Delta)$ approximation in $O(\log \Delta)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(\alpha)$ approximation in $O(\log^2 \Delta)$ rounds [implicit in Bansal and Umboh IPL'17 and Kuhn, Moscibroda, and Wattenhofer SODA'06], and a randomized $O(\alpha)$ approximation in $O(\alpha\log n)$ rounds [Morgan, Solomon and Wein DISC'21]. We also provide a randomized $O(\alpha \log\Delta)$ round distributed algorithm that sharpens the approximation factor to $\alpha(1+o(1))$. If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $\alpha - 1 - \varepsilon$ approximation [Bansal and Umboh IPL'17].

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arXiv2021

Fault-Tolerant Labeling and Compact Routing Schemes

The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given $n$-vertex graph $G$ and a bound $f$ on the number of faults, an $f$-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of the vertices $s$ and $t$, and at most $f$ failing edges $F$, one can determine if $s$ and $t$ are connected in $G \setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], compact FT labeling schemes have been devised only for a limited collection of graph families. In this work, we fill in this gap by proposing two (independent) FT connectivity labeling schemes for general graphs, with a nearly optimal label length. This serves the basis for providing also FT approximate distance labeling schemes, and ultimately also routing schemes. Our main results for an $n$-vertex graph and a fault bound $f$ are: -- There is a randomized FT connectivity labeling scheme with a label length of $O(f+\log n)$ bits, hence optimal for $f=O(\log n)$. This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG '11]. -- There is a randomized FT connectivity labeling scheme with a label length of $O(\log^3 n)$ bits (independent of the number of faults $f$). This scheme is based on the notion of linear sketches of [Ahn et al., SODA '12]. -- For $k\geq 1$, there is a randomized routing scheme that routes a message from $s$ to $t$ in the presence of a set $F$ of faulty edges, with stretch $O(|F|^2 k)$ and routing tables of size $\tilde{O}(f^3 n^{1/k})$. This significantly improves over the state-of-the-art bounds by [Chechik, ICALP '11], providing the first scheme with sub-linear FT labeling and routing schemes for general graphs.

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arXiv2020

Distributed Weighted Min-Cut in Nearly-Optimal Time

Minimum-weight cut (min-cut) is a basic measure of a network's connectivity strength. While the min-cut can be computed efficiently in the sequential setting [Karger STOC'96], there was no efficient way for a distributed network to compute its own min-cut without limiting the input structure or dropping the output quality: In the standard CONGEST model, existing algorithms with nearly-optimal time (e.g. [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]) can guarantee a solution that is $(1+\epsilon)$-approximation at best while the exact $\tilde O(n^{0.8}D^{0.2} + n^{0.9})$-time algorithm [Ghaffari, Nowicki, Thorup, SODA'20] works only on *simple* networks (no weights and no parallel edges). Here $n$ and $D$ denote the network's number of vertices and hop-diameter, respectively. For the weighted case, the best bound was $\tilde O(n)$ [Daga, Henzinger, Nanongkai, Saranurak, STOC'19]. In this paper, we provide an *exact* $\tilde O(\sqrt n + D)$-time algorithm for computing min-cut on *weighted* networks. Our result improves even the previous algorithm that works only on simple networks. Its time complexity matches the known lower bound up to polylogarithmic factors. At the heart of our algorithm are a clever routing trick and two structural lemmas regarding the structure of a minimum cut of a graph. These two structural lemmas considerably strengthen and generalize the framework of Mukhopadhyay-Nanongkai [STOC'20] and can be of independent interest.

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arXiv2020

Exponentially Faster Shortest Paths in the Congested Clique

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain $poly(\log\log n)$-round algorithms for the following problems in unweighted undirected $n$-vertex graphs: -- $(1+ε)$-approximation of multi-source shortest paths (MSSP) from $O(\sqrt{n})$ sources. -- $(2+ε)$-approximation of all pairs shortest paths (APSP). -- $(1+ε,β)$-approximation of APSP where $β=O(\frac{\log\log n}ε)^{\log\log n}$. These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., PODC19]. It also provides the first nearly-additive bounds for the APSP problem in sub-polynomial time. Our approach is based on distinguishing between short and long distances based on some distance threshold $t = O(\fracβε)$ where $β=O(\frac{\log\log n}ε)^{\log\log n}$. Handling the long distances is done by devising a new algorithm for computing sparse $(1+ε,β)$ emulator with $O(n\log\log n)$ edges. For the short distances, we provide distance-sensitive variants for the distance tool-kit of [Censor-Hillel et al., PODC19]. By exploiting the fact that this tool-kit should be applied only on local balls of radius $t$, their round complexities get improved from $poly(\log n)$ to $poly(\log t)$. Finally, our deterministic solutions for these problems are based on a derandomization scheme of a novel variant of the hitting set problem, which might be of independent interest.

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arXiv2020

Massively Parallel Algorithms for Distance Approximation and Spanners

Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(\log\log n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(\log k) \in poly(\log\log n)$ round MPC algorithms for computing $O(k^{1+{o(1)}})$-spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an $O(\log^2\log n)$-round algorithm for $O(\log^{1+o(1)} n)$ approximation of all pairs shortest paths (APSP) in the near-linear regime of local memory. To the best of our knowledge, this is the first sublogarithmic-time MPC algorithm for distance approximations. -Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sub-logarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model.

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arXiv2019

Fast Approximate Shortest Paths in the Congested Clique

We design fast deterministic algorithms for distance computation in the congested clique model. Our key contributions include: -- A $(2+ε)$-approximation for all-pairs shortest paths in $O(\log^2{n} / ε)$ rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model. -- A $(1+ε)$-approximation for multi-source shortest paths from $O(\sqrt{n})$ sources in $O(\log^2{n} / ε)$ rounds on weighted undirected graphs. This is the first sub-polynomial algorithm obtaining this approximation for a set of sources of polynomial size. Our main techniques are new distance tools that are obtained via improved algorithms for sparse matrix multiplication, which we leverage to construct efficient hopsets and shortest paths. Furthermore, our techniques extend to additional distance problems for which we improve upon the state-of-the-art, including diameter approximation, and an exact single-source shortest paths algorithm for weighted undirected graphs in $\tilde{O}(n^{1/6})$ rounds.

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arXiv2019

Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

The minimum-weight $2$-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of $1$ between each pair of vertices while the former strengthens this edge-connectivity requirement to $2$. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a $2$-approximation. In this paper, we give a deterministic distributed algorithm with round complexity of $\widetilde{O}(D+\sqrt{n})$ that computes a $(5+ε)$-approximation of 2-ECSS, for any constant $ε>0$. Up to logarithmic factors, this complexity matches the $\widetildeΩ(D+\sqrt{n})$ lower bound that can be derived from Das Sarma et al. [STOC'11], as shown by Censor-Hillel and Dory [OPODIS'17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC'18], which achieved an $O(\log n)$-approximation in $\widetilde{O}(D+\sqrt{n})$ rounds. We also present an alternative algorithm for $O(\log n)$-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network, following a framework introduced by Ghaffari and Haeupler [SODA'16]. This algorithm has round complexity $\widetilde{O}(D+\sqrt{n})$ in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in $\widetilde{O}(D)$ time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.

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arXiv2019

Hardness of Distributed Optimization

This paper studies lower bounds for fundamental optimization problems in the CONGEST model. We show that solving problems exactly in this model can be a hard task, by providing $\tildeΩ(n^2)$ lower bounds for cornerstone problems, such as minimum dominating set (MDS), Hamiltonian path, Steiner tree and max-cut. These are almost tight, since all of these problems can be solved optimally in $O(n^2)$ rounds. Moreover, we show that even in bounded-degree graphs and even in simple graphs with maximum degree 5 and logarithmic diameter, it holds that various tasks, such as finding a maximum independent set (MaxIS) or a minimum vertex cover, are still difficult, requiring a near-tight number of $\tildeΩ(n)$ rounds. Furthermore, we show that in some cases even approximations are difficult, by providing an $\tildeΩ(n^2)$ lower bound for a $(7/8+ε)$-approximation for MaxIS, and a nearly-linear lower bound for an $O(\log{n})$-approximation for the $k$-MDS problem for any constant $k \geq 2$, as well as for several variants of the Steiner tree problem. Our lower bounds are based on a rich variety of constructions that leverage novel observations, and reductions among problems that are specialized for the CONGEST model. However, for several additional approximation problems, as well as for exact computation of some central problems in $P$, such as maximum matching and max flow, we show that such constructions cannot be designed, by which we exemplify some limitations of this framework.

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