Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

Yamazaki, Koichi

doi:10.3390/a11110173

Cited by 6 publications

(6 citation statements)

References 35 publications

(45 reference statements)

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“…pathwidth) of 3-manifolds (Corollary 14). This reduction, together with previous results [46,55,56], suggests that this problem may be computationally hard.…”

Section: Applicationsmentioning

confidence: 58%

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On the width of complicated JSJ decompositions

Huszár¹,

Spreer²

2023

Preprint

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Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we show that a "sufficiently complicated" JSJ decomposition of a 3-manifold enforces a "complicated structure" for all of its triangulations. More concretely, we show that, under certain conditions, the treewidth (resp. pathwidth) of the graph that captures the incidences between the pieces of the JSJ decomposition of an irreducible, closed, orientable 3-manifold M yields a linear lower bound on its treewidth tw(M) (resp. pathwidth pw(M)), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of M.We present several applications of this result. We give the first example of an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth. We construct Haken 3-manifolds with arbitrarily large treewidth-previously the existence of such 3-manifolds was only known in the non-Haken case. We also show that the problem of providing a constant-factor approximation for the treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to computing a constant-factor approximation for the treewidth (resp. pathwidth) of 3-manifolds.

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“…pathwidth) of 3-manifolds (Corollary 14). This reduction, together with previous results [46,55,56], suggests that this problem may be computationally hard.…”

Section: Applicationsmentioning

confidence: 58%

“…Computing a constant-factor approximation of treewidth (resp. pathwidth) for arbitrary graphs is known to be conditionally NP-hard under the Small Set Expansion Hypothesis [46,55,56]. For proving Corollary 14, however, we rely on the assumption that the graph has bounded degree.…”

Section: Lemma 9 ([6 Lemma 46]mentioning

confidence: 99%

On the width of complicated JSJ decompositions

Huszár¹,

Spreer²

2023

Preprint

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“…For many well-known graph classes a decomposition of bounded mim-width can be found in polynomial time. However, for general graphs it is known that computing mim-width is W[1]-hard and not in APX unless NP = ZPP [40], while Yamazaki [42] shows that under the small set expansion hypothesis it is not in APX unless P = NP. For dynamic programming algorithms as in this paper, to circumvent the assumption that we are given a decomposition, we want functions f , g and an algorithm that given a graph of mim-width OPT computes an f (O PT )-approximation to mim-width in time n g(O PT ) , i.e.…”

Section: Subset Odd Cycle Transveral (Soct)mentioning

confidence: 99%

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

2022

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The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\cdot n^{4}$$ 2 O ( r w 3 ) · n 4 , $$2^{O(q^2\log (q))}\cdot n^{4}$$ 2 O ( q 2 log ( q ) ) · n 4 , and $$n^{O(k^2)}$$ n O ( k 2 ) where rw is the rank-width, q the $${\mathbb {Q}}$$ Q -rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

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“…The investigation of graph width parameters finds extensive applications across diverse fields, such as matroid theory, lattice theory, theoretical computer science, game theory, network theory, artificial intelligence, graph theory, and discrete mathematics, as evidenced by numerous studies (for example, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]22,[28][29][30][31][32][33]). These graph width parameters are frequently explored in conjunction with obstruction, contributing to a robust body of research.…”

Section: Introductionmentioning

confidence: 99%

Filter for Submodular Partition Function: Connection to Loose Tangle

Fujita¹

2023

Preprint

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Loose Tangle is a concept in graph theory that has a dual relationship with branch-width which is well-known graph width parameter. Ultrafilter, a fundamental notion in mathematics, is similarly known to have a dual relationship with branch-width when extended to a connectivity system (X, f). In this compact paper, we revisit and contemplate the interplay between Loose Tangle and Filter through the lens of a submodular partition function.

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Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

Cited by 6 publications

References 35 publications

On the width of complicated JSJ decompositions

On the width of complicated JSJ decompositions

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

Filter for Submodular Partition Function: Connection to Loose Tangle

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