Chapter 6: Maximal Independent
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Towards Maximum Independent Sets on Massive Graphs
Towards Maximum Independent Sets on Massive Graphs Yu Liuy Jiaheng Lu x Hua Yangy Xiaokui Xiaoz Zhewei Weiy yDEKE, MOE and School of Information, Renmin University of China x Department of Computer Science, University of Helsinki, Finland zSchool of Computer Engineering, Nanyang Technological University, Singapore ABSTRACT Maximum independent set (MIS) is a fundamental problem in graph theory and it has important applications in many areas such as so- cial network analysis, graphical information systems and coding theory. The problem is NP-hard, and there has been numerous s- tudies on its approximate solutions. While successful to a certain degree, the existing methods require memory space at least linear in the size of the input graph. This has become a serious concern in !"#$!%&'!(#&)*+,+)*+)-#.+- /"#$!%&'0'#&)*+,+)*+)-#.+- view of the massive volume of today’s fast-growing graphs. Figure 1: An example to illustrate that fv , v g is a maximal inde- In this paper, we study the MIS problem under the semi-external 1 2 pendent set, but fv , v , v , v g is a maximum independent set. setting, which assumes that the main memory can accommodate 2 3 4 5 all vertices of the graph but not all edges. We present a greedy algorithm and a general vertex-swap framework, which swaps ver- duced subgraphs, minimum vertex covers, graph coloring, and tices to incrementally increase the size of independent sets. Our maximum common edge subgraphs, etc. Its significance is not solutions require only few sequential scans of graphs on the disk just limited to graph theory but also in numerous real-world ap- file, thus enabling in-memory computation without costly random plications, such as indexing techniques for shortest path and dis- disk accesses. -
Structural Parameterizations of Clique Coloring
Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e. -
Matroids You Have Known
26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction. -
Clique-Width and Tree-Width of Sparse Graphs
Clique-width and tree-width of sparse graphs Bruno Courcelle Labri, CNRS and Bordeaux University∗ 33405 Talence, France email: [email protected] June 10, 2015 Abstract Tree-width and clique-width are two important graph complexity mea- sures that serve as parameters in many fixed-parameter tractable (FPT) algorithms. The same classes of sparse graphs, in particular of planar graphs and of graphs of bounded degree have bounded tree-width and bounded clique-width. We prove that, for sparse graphs, clique-width is polynomially bounded in terms of tree-width. For planar and incidence graphs, we establish linear bounds. Our motivation is the construction of FPT algorithms based on automata that take as input the algebraic terms denoting graphs of bounded clique-width. These algorithms can check properties of graphs of bounded tree-width expressed by monadic second- order formulas written with edge set quantifications. We give an algorithm that transforms tree-decompositions into clique-width terms that match the proved upper-bounds. keywords: tree-width; clique-width; graph de- composition; sparse graph; incidence graph Introduction Tree-width and clique-width are important graph complexity measures that oc- cur as parameters in many fixed-parameter tractable (FPT) algorithms [7, 9, 11, 12, 14]. They are also important for the theory of graph structuring and for the description of graph classes by forbidden subgraphs, minors and vertex-minors. Both notions are based on certain hierarchical graph decompositions, and the associated FPT algorithms use dynamic programming on these decompositions. To be usable, they need input graphs of "small" tree-width or clique-width, that furthermore are given with the relevant decompositions. -
Greedy Sequential Maximal Independent Set and Matching Are Parallel on Average
Greedy Sequential Maximal Independent Set and Matching are Parallel on Average Guy E. Blelloch Jeremy T. Fineman Julian Shun Carnegie Mellon University Georgetown University Carnegie Mellon University [email protected] jfi[email protected] [email protected] ABSTRACT edge represents the constraint that two tasks cannot run in parallel, the MIS finds a maximal set of tasks to run in parallel. Parallel al- The greedy sequential algorithm for maximal independent set (MIS) gorithms for the problem have been well studied [16, 17, 1, 12, 9, loops over the vertices in an arbitrary order adding a vertex to the 11, 10, 7, 4]. Luby’s randomized algorithm [17], for example, runs resulting set if and only if no previous neighboring vertex has been in O(log jV j) time on O(jEj) processors of a CRCW PRAM and added. In this loop, as in many sequential loops, each iterate will can be converted to run in linear work. The problem, however, is only depend on a subset of the previous iterates (i.e. knowing that that on a modest number of processors it is very hard for these par- any one of a vertex’s previous neighbors is in the MIS, or know- allel algorithms to outperform the very simple and fast sequential ing that it has no previous neighbors, is sufficient to decide its fate greedy algorithm. Furthermore the parallel algorithms give differ- one way or the other). This leads to a dependence structure among ent results than the sequential algorithm. This can be undesirable in the iterates. -
Parameterized Algorithms for Recognizing Monopolar and 2
Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs∗ Iyad Kanj1, Christian Komusiewicz2, Manuel Sorge3, and Erik Jan van Leeuwen4 1School of Computing, DePaul University, Chicago, USA, [email protected] 2Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, [email protected] 3Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, [email protected] 4Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany, [email protected] October 16, 2018 Abstract A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB )-graph. There are many (ΠA, ΠB)-Recognition problems, in- cluding the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameter- ized by the number of maximal cliques in G[A]. We complement our algo- rithmic results with several hardness results for (ΠA, ΠB )-Recognition. arXiv:1702.04322v2 [cs.CC] 4 Jan 2018 1 Introduction A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB , is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB. There is an abundance of (ΠA, ΠB )-graph classes, and important ones include bipartite graphs (which admit a partition into two independent ∗A preliminary version of this paper appeared in SWAT 2016, volume 53 of LIPIcs, pages 14:1–14:14. -
Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) Fully Dynamic Maximal Independent Set in Polylogarithmic Update Time Soheil Behnezhad∗, Mahsa Derakhshan∗, MohammadTaghi Hajiaghayi∗, Cliff Stein† and Madhu Sudan‡ ∗University of Maryland {soheil,mahsa,hajiagha}@cs.umd.edu †Columbia University [email protected] ‡Harvard University [email protected] Abstract— We present the first algorithm for maintaining a time. As such, one can trivially maintain MIS by recomput- maximal independent set (MIS) of a fully dynamic graph—which ing it from scratch after each update, in O(m) time. In a undergoes both edge insertions and deletions—in polylogarith- pioneering work, Censor-Hillel, Haramaty, and Karnin [15] mic time. Our algorithm is randomized and, per update, takes 2 2 O(log Δ · log n) expected time. Furthermore, the algorithm presented a round-efficient randomized algorithm for MIS in 2 4 can be adjusted to have O(log Δ · log n) worst-case update- dynamic distributed networks. Implementing the algorithm time with high probability. Here, n denotes the number of of [15] in the sequential setting—the focus of this paper— vertices and Δ is the maximum degree in the graph. requires Ω(Δ) update-time (see [15, Section 6]) where Δ The MIS problem in fully dynamic graphs has attracted sig- is the maximum-degree in the graph which can be as large nificant attention after a breakthrough result of Assadi, Onak, as Ω(n) or even Ω(m) for sparse graphs. Improving this Schieber, and Solomon [STOC’18] who presented an algorithm bound was one of the major problems the authors left 3/4 with O(m ) update-time (and thus broke the natural Ω(m) open. -
Sub-Coloring and Hypo-Coloring Interval Graphs⋆
Sub-coloring and Hypo-coloring Interval Graphs? Rajiv Gandhi1, Bradford Greening, Jr.1, Sriram Pemmaraju2, and Rajiv Raman3 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 08102. E-mail: [email protected]. 2 Department of Computer Science, University of Iowa, Iowa City, Iowa 52242. E-mail: [email protected]. 3 Max-Planck Institute for Informatik, Saarbr¨ucken, Germany. E-mail: [email protected]. Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a “forbidden subgraph” characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it. 1 Introduction Given a graph G = (V, E), a k-sub-coloring of G is a partition of V into sub-color classes V1,V2,...,Vk; a subset Vi ⊆ V is called a sub-color class if it induces a union of disjoint cliques in G. -
Shrub-Depth a Successful Depth Measure for Dense Graphs Graphs Petr Hlinˇen´Y Faculty of Informatics, Masaryk University Brno, Czech Republic
page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Ingredients: joint results with J. Gajarsk´y,R. Ganian, O. Kwon, J. Neˇsetˇril,J. Obdrˇz´alek, S. Ordyniak, P. Ossona de Mendez Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction • Being close to a STAR { \•-depth" sparse dense tree-depth { containment in a structure ??? (will show) Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion". Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion". -
Minimum Dominating Set Approximation in Graphs of Bounded Arboricity
Minimum Dominating Set Approximation in Graphs of Bounded Arboricity Christoph Lenzen and Roger Wattenhofer Computer Engineering and Networks Laboratory (TIK) ETH Zurich {lenzen,wattenhofer}@tik.ee.ethz.ch Abstract. Since in general it is NP-hard to solve the minimum dominat- ing set problem even approximatively, a lot of work has been dedicated to central and distributed approximation algorithms on restricted graph classes. In this paper, we compromise between generality and efficiency by considering the problem on graphs of small arboricity a. These fam- ily includes, but is not limited to, graphs excluding fixed minors, such as planar graphs, graphs of (locally) bounded treewidth, or bounded genus. We give two viable distributed algorithms. Our first algorithm employs a forest decomposition, achieving a factor O(a2) approximation in randomized time O(log n). This algorithm can be transformed into a deterministic central routine computing a linear-time constant approxi- mation on a graph of bounded arboricity, without a priori knowledge on a. The second algorithm exhibits an approximation ratio of O(a log ∆), where ∆ is the maximum degree, but in turn is uniform and determinis- tic, and terminates after O(log ∆) rounds. A simple modification offers a trade-off between running time and approximation ratio, that is, for any parameter α ≥ 2, we can obtain an O(aα logα ∆)-approximation within O(logα ∆) rounds. 1 Introduction We are interested in the distributed complexity of the minimum dominating set (MDS) problem, a classic both in graph theory and distributed computing. Given a graph, a dominating set is a subset D of nodes such that each node in the graph is either in D, or has a direct neighbor in D. -