Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Coudert, David; Ducoffe, Guillaume; Popa, Alexandru

doi:10.1137/1.9781611975031.176

Cited by 32 publications

(74 citation statements)

References 90 publications

(213 reference statements)

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“…As an appetizer we first consider an n-vertex split graph with clique-number log O(1) n, that is a notouriously hard case for diameter computation [8]. Given such a split graph G with stable set S and maximal clique K, we can pre-process G in linear-time so as to partition the vertices of S into twin classes: with two vertices in S being called twins if and only if they have the same neighbourhood in K (e.g., see [21]). If the VC-dimension of G is at most d then, by the Sauer-Shelah-Perles Lemma [53,54] the number of twin classes is an O(|K| d ) = log O(d) n. Therefore, after some linear-time preprocessing, we are left with computing the diameter on a graph of polylogarithmic order!…”

Section: Our Contributionsmentioning

confidence: 99%

“…We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time.…”

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confidence: 97%

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Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic-time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes -where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radius and center in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| − 1. number of edges on a shortest path. Beyond its many practical applications, this fundamental problem in Graph Theory has attracted a lot of attention in the fine-grained complexity study of polynomial-time solvable problems [1,4,8,15,18,22,25,32,52]. More precisely, for every n-vertex m-edge unweighted graph the textbook algorithm for computing its diameter runs in time O(nm). In a seminal paper [52] this roughly quadratic running-time was matched by a quadratic lower-bound, assuming the Strong Exponential-Time Hypothesis (SETH). We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time. Related workBefore we detail our contributions, we wish to mention a few recent (and not so recent) results that are most related to our approach.

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Section: Our Contributionsmentioning

confidence: 99%

mentioning

confidence: 97%

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Based on empirical studies, an O(mn) running time is claimed, where m is the number of edges in the graph. Furthermore, there are heuristics for computing the hyperbolicity of a given graph [14], and there are investigations of whether one can compute hyperbolicity in linear time when some graph parameters take small values [16,21].…”

Section: Introductionmentioning

confidence: 99%

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Chalopin

Chepoi

Dragan

et al. 2019

Discrete Comput Geom

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In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs, via a new parameter which we call rooted insize. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomialtime, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm (with an additive constant 1) for computing the hyperbolicity of an n-vertex graph G = (V, E) in optimal time O(n 2 ) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space.We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space (X, d) has such a geodesic spanning tree. 1 The O(·) notation hides polyloglog factors. 2 Informally, (y|z)w can be viewed as half the detour you make, when going over w to get from y to z.

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“…This is optimal assuming SETH. For the parameter modular-width, for short mw, there is an O(|V | + |E| + mw 3 )time algorithm in order to compute all the eccentricities [15]. Shrub-depth is sometimes regarded as an interesting competitor for modular-width [36].…”

Section: Open Problemmentioning

confidence: 99%

Fast Diameter Computation Within Split Graphs

Ducoffe

Habib

Viennot

2019

Combinatorial Optimization and Applications

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When can we compute the diameter of a graph in quasi linear time? We address this question for the class of split graphs, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either 2 or 3, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of a split graph in less than quadratic time. Therefore it is worth to study the complexity of diameter computation on subclasses of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded clique-interval number and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the VC-dimension and the stabbing number of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs:• For the k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if k = O(1), and even in quasi linear time if k = o(log n) and in addition a corresponding ordering is given. However, under SETH this cannot be done in truly subquadratic time for any k = ω(log n).• For the complements of k-clique-interval split graphs, we can compute their diameter in truly subquadratic time if k = O(1), and even in time O(km) if a corresponding ordering is given. Again this latter result is optimal under SETH up to polylogarithmic factors.Our findings raise the question whether a k-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for k = 1 and for some subclasses such as boundedtreewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -including the ones mentioned above -have a bounded clique-interval number.

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Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Cited by 32 publications

References 90 publications

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Fast Diameter Computation Within Split Graphs

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