Deterministic Massively Parallel Symmetry Breaking for Sparse Graphs

Massively Parallel Computation (MPC) is an emerging model which distills core aspects of distributed and parallel computation. It was developed as a tool to solve (typically graph) problems in systems where input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n, number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. There are, however, no prior corresponding deterministic algorithms. A major challenge in the sublinear space setting is that the local space of each machine may be too small to store all the edges incident to a single node. To overcome this barrier we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph with additional desired properties: degrees in the subgraph are sufficiently small that nodes' neighborhoods can be stored on single machines, and solving the problem on the subgraph provides significant global progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known randomized algorithm of Luby [SICOMP'86], we obtain O(log Δ+log log n)$-round deterministic MPC algorithms for solving the fundamental problems of Maximal Matching and Maximal Independent Set with O(ne) space on each machine for any constant e > 0. Based on the recent work of Ghaffari et al. [FOCS'18], this additive O(log log n) factor is conditionally essential. These algorithms can also be shown to run in O(log Δ) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O(log2 Δ) rounds by Censor-Hillel et al. [DISC'17].

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We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/e)-round algorithm that finds a (1+e)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/e)-round algorithm that finds, with high probability, a (1+e)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/e)-round algorithm that finds, with high probability, a 8(1+e)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+e)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/e)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+e)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

We present O(loglog n) -round algorithms in the Massively Parallel Computation (MPC) model, with O (n) memory per machine, that compute a maximal independent set, a 1+e approximation of maximum matching, and a 2+eapproximation of minimum vertex cover, for any n-vertex graph and any constant \eps>0. These improve the state of the art as follows: Our MIS algorithm leads to a simple O(loglog Δ)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the O (√log Δ )-round algorithm of Ghaffari [PODC'17]. Our O(loglog n)-round (1+e)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log^2log n)-round (1+\eps)-approximation algorithm of Czumaj et al. [STOC'18] and $O(loglog n)-round (1+e)-approximation algorithm of Assadi et al. [arXiv'17]. Our O(loglog n)-round (2+e)-approximate minimum vertex cover algorithm improves on an O(loglog n)-round O(1)-approximation of Assadi et al. [arXiv'17].

We introduce a method for sparsifying distributed algorithms and exhibit how it leads to improvements that go past known barriers in two algorithmic settings of large-scale graph processing: Massively Parallel Computation (MPC), and Local Computation Algorithms (LCA). - MPC with Strongly Sublinear Memory: Recently, there has been growing interest in obtaining MPC algorithms that are faster than their classic $O(\log n)$-round parallel counterparts for problems such as MIS, Maximal Matching, 2-Approximation of Minimum Vertex Cover, and $(1+\epsilon)$-Approximation of Maximum Matching. Currently, all such MPC algorithms require $\tilde{\Omega}(n)$ memory per machine. Czumaj et al. [STOC'18] were the first to handle $\tilde{\Omega}(n)$ memory, running in $O((\log\log n)^2)$ rounds. We obtain $\tilde{O}(\sqrt{\log \Delta})$-round MPC algorithms for all these four problems that work even when each machine has memory $n^{\alpha}$ for any constant $\alpha\in (0, 1)$. Here, $\Delta$ denotes the maximum degree. These are the first sublogarithmic-time algorithms for these problems that break the linear memory barrier. - LCAs with Query Complexity Below the Parnas-Ron Paradigm: Currently, the best known LCA for MIS has query complexity $\Delta^{O(\log \Delta)} poly(\log n)$, by Ghaffari [SODA'16]. As pointed out by Rubinfeld, obtaining a query complexity of $poly(\Delta\log n)$ remains a central open question. Ghaffari's bound almost reaches a $\Delta^{\Omega\left(\frac{\log \Delta}{\log\log \Delta}\right)}$ barrier common to all known MIS LCAs, which simulate distributed algorithms by learning the local topology, a la Parnas-Ron [TCS'07]. This barrier follows from the $\Omega(\frac{\log \Delta}{\log\log \Delta})$ distributed lower bound of Kuhn, et al. [JACM'16]. We break this barrier and obtain an MIS LCA with query complexity $\Delta^{O(\log\log \Delta)} poly(\log n)$.

In this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let and be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3‐uniform hypergraph of order at least 8 and with maximum degree at most three, then and there is a connected 3‐uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on vertices, where denotes the maximum degree in H, then and this bound is asymptotically best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: Fast Δ-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al. [6], we prove that the randomized complexity of Δ-coloring a tree with maximum degree Δ is θ(logΔ log n), for any Δ ≥ 55, whereas its deterministic complexity is θ(logΔn) for any Δ ≥ 3. This also establishes a large separation between the deterministic complexity of Δ-coloring and (Δ+1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(logΔn) rounds can be transformed to run in O(log* n -- log*Δ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(logΔn) time deterministically. (This gives an alternate proof that deterministically Δ-coloring a tree with small Δ takes Ω(logΔn) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √log n}. This shows that a deterministic Ω(logΔn) lower bound for any problem (Δ-coloring a tree, for example) implies a randomized Ω(logΔ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(√log log n) terms in the complexities of the best MIS and (Δ+1)-coloring algorithms without also improving the 2O(√log n})-round Panconesi-Srinivasan algorithms.

Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set ((MIS)) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types -- (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of $\Omega(\Delta)$ update time (where $\Delta$ is the maximum degree in the graph) for \emph{vertex updates} presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes with approximation factors between 2 and 4 and corresponding running-time trade-offs. We have implemented our algorithms and reported the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances.

Let [Formula: see text] be a connected undirected graph. A vertex coloring [Formula: see text] of [Formula: see text] is an [Formula: see text]-vertex coloring if for each vertex [Formula: see text] in [Formula: see text], the number of different colors assigned to [Formula: see text] is at most [Formula: see text]. The [Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is the maximum number of colors which are used in an [Formula: see text]-vertex coloring of [Formula: see text]. In this paper, we provide sharp bounds for [Formula: see text] of a graph [Formula: see text] in terms of its vertex cover number, maximum degree and diameter, respectively. We also determine precise values for [Formula: see text] in some cases.

Locally checkable labeling problems (LCLs) are distributed graph problems in which a solution is globally feasible if it is locally feasible in all constant-radius neighborhoods. Vertex colorings, maximal independent sets, and maximal matchings are examples of LCLs. On the one hand, it is known that some LCLs benefit exponentially from randomness---for example, any deterministic distributed algorithm that finds a sinkless orientation requires Θ(log n) rounds in the LOCAL model, while the randomized complexity of the problem is Θ(log log n) rounds. On the other hand, there are also many LCLs in which randomness is useless. Previously, it was not known whether there are any LCLs that benefit from randomness, but only subexponentially. We show that such problems exist: for example, there is an LCL with deterministic complexity Θ(log2 n) rounds and randomized complexity Θ(log n log log n) rounds.

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

Twelve Countings with Rooted Plane Trees

As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams.In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems. Here, we are given a parameter k and the objective is to decide whether there is a solution bounded by k. By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of Maximal Matching and Vertex Cover. We consider various models for a graph stream on n nodes: the insertion-only model where the edges can only be added, and the dynamic model where edges can be both inserted and deleted. More formally, we show the following results:•In the insertion only model, there is a one-pass deterministic algorithm for the parameterized Vertex Cover problem which computes a sketch using Õ(k2) space1 such that at each timestamp in time Õ(2k) it can either extract a solution of size at most k for the current instance, or report that no such solution exists. We also show a tight lower bound of Ω(k2) for the space complexity of any (randomized) streaming algorithms for the parameterized Vertex Cover, even in the insertion-only model.•In the dynamic model, and under the promise that at each timestamp there is a maximal matching of size at most k, there is a one-pass Õ(k2)-space (sketch-based) dynamic algorithm that maintains a maximal matching with worst-case update time Õ(k2). This algorithm partially solves Open Problem 64 from [1]. An application of this dynamic matching algorithm is a one-pass Õ(k2)-space streaming algorithm for the parameterized Vertex Cover problem that in time Õ(2k) extracts a solution for the final instance with probability 1 – δ/no(1), where δ < 1. To the best of our knowledge, this is the first graph streaming algorithm that combines linear sketching with sequential operations that depend on the graph at the current time.•In the dynamic model without any promise, there is a one-pass randomized algorithm for the parameterized Vertex Cover problem which computes a sketch using Õ(nk) space such that in time Õ(nk + 2k) it can either extract a solution of size at most k for the final instance, or report that no such solution exists.

Let G be a graph. A set S⊆V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\subseteq V(G)$$\\end{document} is independent if its elements are pairwise nonadjacent. A vertex v∈V(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v \\in V(G)$$\\end{document} is shedding if for every independent set S⊆V(G)\\N[v]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\subseteq V(G) {\\setminus } N[v]$$\\end{document} there exists u∈N(v)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u \\in N(v)$$\\end{document} such that S∪{u}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S \\cup \\{u\\}$$\\end{document} is independent. An independent set S is maximal if it is not contained in another independent set. An independent set S is maximum if the size of every independent set of G is not bigger than |S|. The size of a maximum independent set of G is denoted α(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha (G)$$\\end{document}. A graph G is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is α(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha (G)$$\\end{document}. The graph G belongs to class W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} if every two disjoint independent sets in G are included in two disjoint maximum independent sets. If a graph belongs to the class W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document}, then it is well-covered. Finding a maximum independent set in an input graph is a well-known NP-hard problem. Recognizing well-covered graphs is co-NP-complete. Recently, it was proved that deciding whether an input graph belongs to the class W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} is co-NP-complete. However, when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} is still not known. In this article, we investigate the connection between shedding vertices and W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} graphs. On the one hand, we prove that recognizing shedding vertices is co-NP-complete. On the other hand, we find polynomial solutions for restricted cases of the problem. We also supply polynomial characterizations of several families of W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {W_2}$$\\end{document} graphs.

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs. When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l). 1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l. 2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0. 3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.