Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation to the maximum weight matching, running in $\tilde O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$ notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or $\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$ rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly

Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a $(2\Delta-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $\Delta$, in $O(\log^7 \Delta \log n)$ rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrt{\Delta}) + O(\log^* n)$ by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of $(2\Delta-1)$-edge-coloring to poly$(\log\log n)$ rounds. The key technical ingredient is a deterministic distributed algorithm for \emph{hypergraph maximal matching}, which we believe will be of interest beyond this result. In any hypergraph of rank $r$ --- where each hyperedge has at most $r$ vertices --- with $n$ nodes and maximum degree $\Delta$, this algorithm computes a maximal matching in $O(r^5 \log^{6+\log r } \Delta \log n)$ rounds. This hypergraph matching algorithm and its extensions lead to a number of other results. In particular, a polylogarithmic-time deterministic distributed maximal independent set algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a $((\log \Delta/\varepsilon)^{O(\log (1/\varepsilon))})$-round deterministic algorithm for $(1+\varepsilon)$-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting $\lambda$-arboricity graphs with out-degree at most $(1+\varepsilon)\lambda$, for any constant $\varepsilon>0$, hence partially answering Open Problem 10 of Barenboim and Elkin's book

Risk-Averse Matchings over Uncertain Graph Databases

A large number of applications such as querying sensor networks, and analyzing protein-protein interaction (PPI) networks, rely on mining uncertain graph and hypergraph databases. In this work we study the following problem: given an uncertain, weighted (hyper)graph, how can we efficiently find a (hyper)matching with high expected reward, and low risk? This problem naturally arises in the context of several important applications, such as online dating, kidney exchanges, and team formation. We introduce a novel formulation for finding matchings with maximum expected reward and bounded risk under a general model of uncertain weighted (hyper)graphs that we introduce in this work. Our model generalizes probabilistic models used in prior work, and captures both continuous and discrete probability distributions, thus allowing to handle privacy related applications that inject appropriately distributed noise to (hyper)edge weights. Given that our optimization problem is NP-hard, we turn our attention to designing efficient approximation algorithms. For the case of uncertain weighted graphs, we provide a $\frac{1}{3}$-approximation algorithm, and a $\frac{1}{5}$-approximation algorithm with near optimal run time. For the case of uncertain weighted hypergraphs, we provide a $\Omega(\frac{1}{k})$-approximation algorithm, where $k$ is the rank of the hypergraph (i.e., any hyperedge includes at most $k$ nodes), that runs in almost (modulo log factors) linear time. We complement our theoretical results by testing our approximation algorithms on a wide variety of synthetic experiments, where we observe in a controlled setting interesting findings on the trade-off between reward, and risk. We also provide an application of our formulation for providing recommendations of teams that are likely to collaborate, and have high impact.Comment: 25 page

Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O(\log^2 n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an $O(\Delta^{\epsilon}\text{polylog}(n))$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum node degree of the hypergraph and $\epsilon > 0$ is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

Sketching Cuts in Graphs and Hypergraphs

Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that $(1-\epsilon)$-approximation for Max-Cut must use $n^{1-O(\epsilon)}$ space; moreover, beating $4/5$-approximation requires polynomial space. For the sketching model, we show that $r$-uniform hypergraphs admit a $(1+\epsilon)$-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with $O(\epsilon^{-2} n (r+\log n))$ edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems)

Distributed Approximation of Maximum Independent Set and Maximum Matching

We present a simple distributed $\Delta$-approximation algorithm for maximum weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes in $O(\texttt{MIS}(G)\cdot \log W)$ rounds, where $\Delta$ is the maximum degree, $\texttt{MIS}(G)$ is the number of rounds needed to compute a maximal independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the \texttt{MIS} algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in $O(\log n \log W)$ rounds, where $n$ is the number of nodes. We also present a deterministic $O(\Delta +\log^* n)$-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a $2$-approximation for maximum weight matching without incurring any additional round penalty in the $\mathsf{CONGEST}$ model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of \emph{local aggregation algorithms} for which we describe a mechanism that allows the simulation to run in the $\mathsf{CONGEST}$ model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ($2+\varepsilon$) allows us to devise a distributed algorithm requiring $O(\frac{\log \Delta}{\log\log\Delta})$ rounds for any constant $\varepsilon>0$. For the unweighted case, we can even obtain a $(1+\varepsilon)$-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on $\Delta$

On Derandomizing Local Distributed Algorithms

The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing LOCAL algorithms. We also exhibit how this simple recipe leads to significant improvements on a number of problem. Two main results are: - An improved distributed hypergraph maximal matching algorithm, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The first gives an improved algorithm for Open Problem 11.4 of the book of Barenboim and Elkin, and the last gives the first positive resolution of their Open Problem 11.10. - An improved distributed algorithm for the Lov\'{a}sz Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and $k$-SAT.Comment: 37 page

Almost-Smooth Histograms and Sliding-Window Graph Algorithms

We study algorithms for the sliding-window model, an important variant of the data-stream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We extend the smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes all subadditive functions. Specifically, we show that if a subadditive function can be $(1+\epsilon)$-approximated in the insertion-only streaming model, then it can be $(2+\epsilon)$-approximated also in the sliding-window model with space complexity larger by factor $O(\epsilon^{-1}\log w)$, where $w$ is the window size. We demonstrate how our framework yields new approximation algorithms with relatively little effort for a variety of problems that do not admit the smooth-histogram technique. For example, in the frequency-vector model, a symmetric norm is subadditive and thus we obtain a sliding-window $(2+\epsilon)$-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window $(\sqrt{2}+\epsilon)$-approximation algorithm for Schatten $4$-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum $k$-cover, thereby deriving sliding-window $O(1)$-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every $d\in (1,2]$ an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly $d$