Showing papers on "Split graph published in 2017"
TL;DR: The proofs are based on a new game-theoretic characterization of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes and prove a “rank-preserving” version of Gaifman’s locality theorem.
Abstract: Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez [2010, 2011], form a large variety of classes of “sparse graphs” including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion.We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes (parameterized by the length of the input formula). At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption).As a by-product, we give an algorithmic construction of sparse neighborhood covers for nowhere dense graphs. This extends and improves previous constructions of neighborhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those.Our proofs are based on a new game-theoretic characterization of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a “rank-preserving” version of Gaifman’s locality theorem.
93 citations
01 Aug 2017
TL;DR: A novel iterative algorithm to determine the maximum clique by searching a k-clique in S starting from k = ωc +1 until S becomes [EQUATION], when more iterations benefit marginally, which is much more efficient and robust than previous solutions.
Abstract: Cliques refer to subgraphs in an undirected graph such that vertices in each subgraph are pairwise adjacent. The maximum clique problem, to find the clique with most vertices in a given graph, has been extensively studied. Besides its theoretical value as an NP-hard problem, the maximum clique problem is known to have direct applications in various fields, such as community search in social networks and social media, team formation in expert networks, gene expression and motif discovery in bioinformatics and anomaly detection in complex networks, revealing the structure and function of networks. However, algorithms designed for the maximum clique problem are expensive to deal with real-world networks.In this paper, we devise a randomized algorithm for the maximum clique problem. Different from previous algorithms that search from each vertex one after another, our approach RMC, for the randomized maximum clique problem, employs a binary search while maintaining a lower bound ωc and an upper bound [EQUATION] of ω(G). In each iteration, RMC attempts to find a ωt-clique where [EQUATION]. As finding ωt in each iteration is NP-complete, we extract a seed set S such that the problem of finding a ωt-clique in G is equivalent to finding a ωt-clique in S with probability guarantees (≥1−n−c). We propose a novel iterative algorithm to determine the maximum clique by searching a k-clique in S starting from k = ωc+1 until S becomes [EQUATION], when more iterations benefit marginally. As confirmed by the experiments, our approach is much more efficient and robust than previous solutions and can always find the exact maximum clique.
59 citations
04 Feb 2017
TL;DR: This work is supported by NSFC Grants No. 61370184, the MeCS platform of the University of Picardie Jules Verne and the HPC platform of Jianghan Univeristy.
Abstract: We describe an exact branch-and-bound algorithm for the maximum weight clique problem (MWC), called WLMC, that is especially suited for large vertex-weighted graphs. WLMC incorporates two original contributions: a preprocessing to derive an initial vertex ordering and to reduce the size of the graph, and incremental vertex-weight splitting to reduce the number of branches in the search space. Experiments on representative large graphs from real-world applications show that WLMC greatly outperforms relevant exact and heuristic MWC algorithms, and refute the prevailing hypothesis that exact MWC algorithms are less adequate for large graphs than heuristic algorithms.
54 citations
TL;DR: For these classes of graphs, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs and tight lower bounds for the size of such solution sets depending on the order of the input graph are given.
48 citations
TL;DR: It is shown that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation.
43 citations
25 Jul 2017
TL;DR: An improved randomized distributed algorithm for congested clique model that improves quadratically on the O(log Δ) algorithm of [Ghaffari, SODA'16], and a certain local sparsification technique for MIS, which is of independent interest.
Abstract: Computing a Maximal Independent Set (MIS) is a central problem in distributed graph algorithms. This paper presents an improved randomized distributed algorithm for congested clique model, defined as follows: Given a graph G=(V, E), initially each node knows only its neighbors. Communication happens in synchronous rounds over a complete graph, and per round each node can send O(log n) bits to each other node. We present a randomized algorithm that computes an MIS in O((log Δ)/(√(log n)) + 1 ) ≤ O(√(log Δ)) rounds of congested clique, with high probability. Here Δ denotes the maximum degree in the graph. This improves quadratically on the O(log Δ) algorithm of [Ghaffari, SODA'16]. The core technical novelty in this result is a certain local sparsification technique for MIS, which we believe to be of independent interest.
41 citations
TL;DR: The main result in this paper is to show the existence of an FPT algorithm when the authors parameterize the problem by the solution size k, and it is proved that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatics number of its triangle-free induced subgraphs.
Abstract: The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size k, when k is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size k. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turing-kernel. More precisely, the hard cases are instances of size $${O}(k^5)$$O(k5). As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem for bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.
41 citations
TL;DR: In this paper, a polynomial-time algorithm for the maximum weight independent set problem on outer-string graphs was proposed, where the polygonal arcs are restricted to single segments.
Abstract: Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each curve intersects the boundary of the disk. Outerstring graphs are among the most general classes of intersection graphs studied. To date, no polynomial time algorithm is known for any of the classical graph optimization problems on outerstring graphs; in fact, most are NP-hard. It is known that there is an intersection model for any outerstring graph that consists of polygonal arcs attached to a circle. However, this representation may require an exponential number of segments relative to the size of the graph.Given an outerstring graph and an intersection model consisting of polygonal arcs with a total of N segments, we develop an algorithm that solves the Maximum Weight Independent Set problem in O ( N 3 ) time. If the polygonal arcs are restricted to single segments, then outersegment graphs result. For outersegment graphs, we solve the Maximum Weight Independent Set problem in O ( n 3 ) time where n is the number of vertices in the graph. An outerstring graph has an intersection model consisting of polygonal arcs with a total of N segments.In O ( N 3 ) time the maximum weight independent set in an outerstring graph is found.The algorithm finds the maximum weight independent set in an outersegment graph in cubic time.
41 citations
TL;DR: This paper gives upper and lower bounds for the clique-width and NLC-width of the modified graphs in terms of the cliques- width and N LC- width of the involved graphs.
Abstract: Clique-width is a well-known graph parameter. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded clique-width. The same holds for NLC-width. In this paper we study the behavior of clique-width and NLC-width under various graph operations and graph transformations. We give upper and lower bounds for the clique-width and NLC-width of the modified graphs in terms of the clique-width and NLC-width of the involved graphs.
41 citations
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TL;DR: In this paper, the degree sequence of a random graph is related to a simple model of nearly independent binomial random variables, and it is shown that many interesting functions of the joint distribution of graph degrees, such as the median degree, become amenable to estimation.
Abstract: In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n\to\infty$.
40 citations
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TL;DR: In this paper, a polynomial-time algorithm was given for the maximum weight independent set problem on any P_6-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph.
Abstract: In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$.
TL;DR: It is shown that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs and every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph.
Abstract: We study the classical 0---1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570---581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. The PTAS is obtained by expanding a general result of Demaine et al. (Proceedings of 46th annual IEEE symposium on foundations of computer science, FOCS 2005, pp 637---646, 2005). The knapsack problem with forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.
TL;DR: This work presents two independent proofs that every connected series-parallel graph has a vertex that is common to all of its longest paths and describes how one such vertex can be found in linear time.
TL;DR: An algorithm is introduced to compute the minimum edit distance of a sequence of length m to any path in a node-labeled directed graph (V, E) in O( |V |+m|E|) time and O(|V |) space.
Abstract: Graphs are commonly used to represent sets of sequences. Either edges or nodes can be labeled by sequences, so that each path in the graph spells a concatenated sequence. Examples include graphs to represent genome assemblies, such as string graphs and de Bruijn graphs, and graphs to represent a pan-genome and hence the genetic variation present in a population. Being able to align sequencing reads to such graphs is a key step for many analyses and its applications include genome assembly, read error correction, and variant calling with respect to a variation graph. Given the wide range of applications of this basic problem, it is surprising that algorithms with optimal runtime are, to the best of our knowledge, yet unknown. In particular, aligning sequences to cyclic graphs currently represents a challenge both in theory and practice. Here, we introduce an algorithm to compute the minimum edit distance of a sequence of length m to any path in a node-labeled directed graph (V,E) in O(V+m|E|) time and O(|V|) space. The corresponding alignment can be obtained in the same runtime using O(√m|V|) space. The time complexity depends only on the length of the sequence and the size of the graph. In particular, it does not depend on the cyclicity of the graph, or any other topological features.
TL;DR: A new approach for maximal clique and k-plex enumeration is proposed, which identifies dense subgraphs by binary graph partitioning and can achieve the speedups of up to 10x over existing approaches on large graphs.
21 Jun 2017
TL;DR: The Token Sliding problem consists in deciding whether there exists a sequence of independent sets which transforms I into J so that every pair of consecutive independent sets of the sequence can be obtained via a single token move.
Abstract: Let I be an independent set of a graph G. Imagine that a token is located on every vertex of I. We can now move the tokens of I along the edges of the graph as long as the set of tokens still defines an independent set of G. Given two independent sets I and J, the Token Sliding problem consists in deciding whether there exists a sequence of independent sets which transforms I into J so that every pair of consecutive independent sets of the sequence can be obtained via a single token move. This problem is known to be PSPACE-complete even on planar graphs.
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TL;DR: In this article, the complexity of the Independent Feedback Vertex Set problem was shown to be polynomial-time solvable for H$-free graphs, where H$ contains a claw or cycle.
Abstract: The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $k\geq 0$, to delete at most $k$ vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the $3$-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for $H$-free graphs. We prove that it is NP-complete if $H$ contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $P_4$-free graphs. We show that it remains polynomial-time solvable for $P_5$-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most $k$ for a given integer $k\geq 0$. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for $H$-free graphs with the complexity of $3$-Colouring, Independent Odd Cycle Transversal and other related problems.
01 Jun 2017
TL;DR: It is shown that in the Erdös–Rényi random graph model, join is instance optimal in the left-deep-join framework under reasonable assumptions, and an algorithm to compute the optimal join plan is devised.
Abstract: Subgraph enumeration, which aims to find all the subgraphs of a large data graph that are isomorphic to a given pattern graph, is a fundamental graph problem with a wide range of applications. However, existing sequential algorithms for subgraph enumeration fall short in handling large graphs due to the involvement of computationally intensive subgraph isomorphism operations. Thus, some recent researches focus on solving the problem using MapReduce. Nevertheless, exiting MapReduce approaches are not scalable to handle very large graphs since they either produce a huge number of partial results or consume a large amount of memory. Motivated by this, in this paper, we propose a new algorithm $$\mathsf {Twin}$$Twin$$\mathsf {Twig}$$Twig$$\mathsf {Join}$$Join based on a left-deep-join framework in MapReduce, in which the basic join unit is a $$\mathsf {Twin}$$Twin$$\mathsf {Twig}$$Twig (an edge or two incident edges of a node). We show that in the Erdos---Renyi random graph model, $$\mathsf {Twin}$$Twin$$\mathsf {Twig}$$Twig$$\mathsf {Join}$$Join is instance optimal in the left-deep-join framework under reasonable assumptions, and we devise an algorithm to compute the optimal join plan. We further discuss how our approach can be adapted to handle the power-law random graph model. Three optimization strategies are explored to improve our algorithm. Ultimately, by aggregating equivalent nodes into a compressed node, we construct the compressed graph, upon which the subgraph enumeration is further improved. We conduct extensive performance studies in several real graphs, one of which contains billions of edges. Our approach significantly outperforms existing solutions in all tests.
TL;DR: The main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, all graphs H for which the class of H-free chordal graphs has bounded clique-width, and illustrates the usefulness of this classification for classifying other types of graph classes.
Abstract: A graph is H-free if it has no induced subgraph isomorphic to H. Brandstadt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstadt, Le, and Mosca erroneously claimed that the gem and co-gem are the only two 1-vertex P4-extensions H for which the class of H-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of H-free chordal graphs of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of inline image-free graphs has bounded clique-width via a reduction to K4-free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of H-free weakly chordal graphs.
01 Oct 2017
TL;DR: The white-box Ramsey problem is hard and this is true even if one is looking for a much smaller clique or independent set than the theorem guarantees, which follows from the hardness of problems such as factoring, discrete-log and learning with errors.
Abstract: Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so while examining a linear number of edges. If the graph is given by a black-box, where to figure out whether a certain edge exists the box should be queried, then a large number of queries must be issued. But what if one is given a program or circuit for computing the existence of an edge? This problem was raised by Buss and Goldberg and Papadimitriou in the context of TFNP, search problems with a guaranteed solution.We examine the relationship between black-box complexity and white-box complexity for search problems with guaranteed solution such as the above Ramsey problem. We show that under the assumption that collision resistant hash function exist (which follows from the hardness of problems such as factoring, discrete-log and learning with errors) the white-box Ramsey problem is hard and this is true even if one is looking for a much smaller clique or independent set than the theorem guarantees.In general, one cannot hope to translate all black-box hardness for TFNP into white-box hardness: we show this by adapting results concerning the random oracle methodology and the impossibility of instantiating it.Another model we consider is the succinct black-box, where there is a known upper bound on the size of the black-box (but no limit on the computation time). In this case we show that for all TFNP problems there is an upper bound on the number of queries proportional to the description size of the box times the solution size. On the other hand, for promise problems this is not the case.Finally, we consider the complexity of graph property testing in the white-box model. We show a property which is hard to test even when one is given the program for computing the graph. The hard property is whether the graph is a two-source extractor.
TL;DR: In this article, the authors give a simple eigenvalue characterization for a graph having perfect state transfer at time π / 2 and show that the hypercube is the sparsest Hadamard diagonalizable connected unweighted graph with perfect transfer.
TL;DR: It is shown that circle graphs have unbounded sim- width, and thus also unbounded mim-width, and it is proved that chordal graphs and co-comparability graphs have sim-width at most 1.
TL;DR: This paper provides several approximate min-max theorems relating cycle-transversals and cycle-packings of line graphs of subcubic triangle- free graphs and proves Jones Conjecture for claw-free graphs with maximum degree 4.
TL;DR: In this article, the authors show linear lower bounds for the mim-width of strongly chordal split graphs, co-comparability graphs, and circle graphs, up to constants.
TL;DR: In this article, the monotonicity of the hyper-Zagreb index under some graph transformations was studied, and the extremal graphs among -vertex trees (acyclic), unicyclic, and bicyclic graphs were determined.
Abstract: The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as , where is the degree of the vertex in a graph . In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs among -vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.
TL;DR: This study introduces vertex-neighbors based type-2 soft sets over ( set of all vertices of a graph) and (set of all edges of a graphs) and introduces some type- 2 soft operations in graphs by presenting several examples to demonstrate these new concepts.
Abstract: The correspondence between a vertex and its neighbors has an essential role in the structure of a graph. Type-2 soft sets are also based on the correspondence of primary parameters and underlying parameters. In this study, we present an application of type-2 soft sets in graph theory. We introduce vertex-neighbors based type-2 soft sets over (set of all vertices of a graph) and (set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. Finally, we describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.
TL;DR: It is proved that the class of claw-free cubic graphs has a vertex set that can be partitioned into two total dominating sets, and that this property also holds for a connected, cubic graph that is chordal or 4 -chordal.
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TL;DR: In this paper, a new random sampling strategy for k-bandlimited signals defined on graphs, based on determinantal point processes (DPP), is presented, which enables perfect recovery of bandlimited signals.
Abstract: We present a new random sampling strategy for k-bandlimited signals defined on graphs, based on determinantal point processes (DPP). For small graphs, ie, in cases where the spectrum of the graph is accessible, we exhibit a DPP sampling scheme that enables perfect recovery of bandlimited signals. For large graphs, ie, in cases where the graph's spectrum is not accessible, we investigate, both theoretically and empirically, a sub-optimal but much faster DPP based on loop-erased random walks on the graph. Preliminary experiments show promising results especially in cases where the number of measurements should stay as small as possible and for graphs that have a strong community structure. Our sampling scheme is efficient and can be applied to graphs with up to $10^6$ nodes.
TL;DR: It is shown that if H contains a certain multigraph as a minor, then H-graphs are GI -complete and the clique problem is APX -hard, and treewidth is used to show that both the k-clique and list k-coloring problems are FPT on H- graphs.
TL;DR: This work introduces the multi-layer coherent subgraph model, which defines clusters of vertices that are densely connected by edges with similar labels in a subset of the graph layers, and introduces the best-first search algorithm MiMAG.
Abstract: Detecting dense subgraphs such as cliques or quasi-cliques is an important graph mining problem. While this task is established for simple graphs, today's applications demand the analysis of more complex graphs: In this work, we consider a frequently observed type of graph where edges represent different types of relations. These multiple edge types can also be viewed as different "layers" of a graph, which is denoted as a "multi-layer graph". Additionally, each edge might be annotated by a label characterizing the given relation in more detail. By simultaneously exploiting all this information, the detection of more interesting subgraphs can be supported. We introduce the multi-layer coherent subgraph model, which defines clusters of vertices that are densely connected by edges with similar labels in a subset of the graph layers. We avoid redundancy in the result by selecting only the most interesting, non-redundant subgraphs for the output. Based on this model, we introduce the best-first search algorithm MiMAG. In thorough experiments, we demonstrate the strengths of MiMAG in comparison with related approaches on synthetic as well as real-world data sets.