Distance-d Independent Set Problems for Bipartite and Chordal Graphs

Eto, Hiroshi; Guo, Fengrui; Miyano, Eiji

doi:10.1007/978-3-642-31770-5_21

Cited by 10 publications

(23 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• Leaf node i with X i = {v 0 }: This is the base case of our induction. There is only one d-scattered set K in V i of size κ = 1, for which (9)(10)(11)(12) is true, that includes v 0 and only one for κ = 0 that does not. In the following cases, we assume (our inductive hypothesis) that all entries of D i−1 (and D i−2 for join nodes) contain the correct number of sets K.…”

Section: Treewidth: Dynamic Programming Algorithmmentioning

confidence: 99%

“…, then there must be some vertex v j ∈ X i (on the path between the two) for which (11) was not true. For (10)(11)(12), observe that for vertices v j of low state s j , (10-11) must have been true for either i − 1 or i − 2 and (12) for the other, while for vertices v j of high state s j it suffices that (12) must have been true for both.…”

Section: Treewidth: Dynamic Programming Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Structurally parameterized d-scattered set

Katsikarelis

Lampis

Paschos

2022

Discrete Applied Mathematics

View full text Add to dashboard Cite

In d-Scattered Set we are given an (edge-weighted) graph and are asked to select at least k vertices, so that the distance between any pair is at least d, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following:• For any d ≥ 2, an O * (d tw )-time algorithm, where tw is the treewidth of the input graph and a tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set.• d-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if k is an additional parameter.• A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness.• A 2 O(td 2 ) -time algorithm parameterized by tree-depth (td), as well as a matching ETHbased lower bound, both for unweighted graphs.We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter > 0, runs in time O * ((tw/ ) O(tw) ) and returns a d/(1+ )-scattered set of size k, if a d-scattered set of the same size exists. arXiv:1709.02180v4 [cs.CC] 10 Nov 2018Our contribution: First, in Section 3 we present a lower bound of (d − ) tw · n O(1) on the complexity of any algorithm solving d-Scattered Set parameterized by tw, based on the Strong Exponential Time Hypothesis (SETH [19,20]). This result can be seen as a non-trivial extension of the bound of (2 − ) tw · n O(1) for Independent Set ([24]) for larger values of d, for which the construction is required to be much more compact in terms of encoded information per unit of treewidth. Next, in Section 4 we provide a dynamic programming algorithm of running time O * (d tw ), matching this lower bound, over a given tree decomposition of width tw. The algorithm actually solves the counting version of d-Scattered Set, making use of standard techniques (dynamic programming on tree decompositions), with an application of the fast subset convolution technique of [2] (or state changes [7,30]) to bring the running time down to match the size of the dynamic programming tables.Having thus identified the complexity of the problem with respect to tw, we next focus on the more restrictive parameters vc and fvs and we show in Section 5 that the edge-weighted d-Scattered Set problem parameterized by vc + k is W[1]-hard. If, on the other hand, all edge-weights are set to 1, then d-Scattered Set (the unweighted variant) parameterized by fvs + k is W[1]-hard. Our reductions also imply lower bounds based on the Exponential Time Hypothesis (ETH [19,20]), yet we do not believe these to be tight, due to the quadratic increase in parameter size (as the construction's focus lies on the edges). One observation we can make is that there are few cases where we can expect to obtain an FPT algorithm without bounding the va...

show abstract

Section: Treewidth: Dynamic Programming Algorithmmentioning

confidence: 99%

Section: Treewidth: Dynamic Programming Algorithmmentioning

confidence: 99%

Structurally parameterized d-scattered set

Katsikarelis

Lampis

Paschos

2022

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Let G poly be the class of graphs with at most poly(n) minimal separators, for some polynomial poly. We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, Distance-d Independent Set, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on G poly , for any even d. The problem is NP-hard on chordal graphs for any odd d ≥ 3 [12]. We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of G poly including chordal and circulararc graphs, and we discuss variants of independent domination problems.…”

mentioning

confidence: 98%

On Distance-d Independent Set and Other Problems in Graphs with “few” Minimal Separators

Montealegre

Todinca

2016

Graph-Theoretic Concepts in Computer Science

View full text Add to dashboard Cite

Abstract. Fomin and Villanger ([14], STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let G poly be the class of graphs with at most poly(n) minimal separators, for some polynomial poly. We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, Distance-d Independent Set, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on G poly , for any even d. The problem is NP-hard on chordal graphs for any odd d ≥ 3 [12]. We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of G poly including chordal and circulararc graphs, and we discuss variants of independent domination problems.

show abstract

“…For odd d ≥ 5, we will construct graph H from G as follows (again, a similar reduction is alluded to in the proof of Theorem 3.10 from [16] and partly also used for Corollary 1 from [11]): we make a vertex in H for each vertex of G and we also attach a distinct path of (d − 3)/2 edges to each of them. We then make a vertex for every edge of G, turn all these vertices into a clique and also connect each one to the two vertices of H representing its endpoints.…”

Section: Inapproximabilitymentioning

confidence: 99%

“…We improve upon these upper bounds by showing that any degree-based greedy approximation algorithm in fact achieves a ratio of O(∆ d/2 ), also matching our lower bound. We then turn our attention to bipartite graphs and show that d-Scattered Set can be approximated within a factor of 2 √ n in polynomial time also for even values of d, matching its known n 1/2− -inapproximability from [11] and complementing the known √ n-approximation for odd values of d from [16].…”

Section: Introductionmentioning

confidence: 97%

Improved (In-)Approximability Bounds for d-Scattered Set

Katsikarelis¹,

Lampis²,

Paschos³

2020

Approximation and Online Algorithms

View full text Add to dashboard Cite

In the d-Scattered Set problem we are asked to select at least k vertices of a given graph, so that the distance between any pair is at least d. We study the problem's (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization. Specifically, we show:• A lower bound of ∆ d/2 − on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree ∆ and an improved upper bound of O(∆ d/2 ) on the approximation ratio of any greedy scheme for this problem.• A polynomial-time 2 √ n-approximation for bipartite graphs and even values of d, that matches the known lower bound by considering the only remaining case.• A lower bound on the complexity of any ρ-approximation algorithm of (roughly) 2 n 1− ρd for even d and 2 n 1− ρ(d+ρ) for odd d (under the randomized ETH), complemented by ρ-approximation algorithms of running times that (almost) match these bounds.

show abstract

Distance-d Independent Set Problems for Bipartite and Chordal Graphs

Cited by 10 publications

References 15 publications

Structurally parameterized d-scattered set

Structurally parameterized d-scattered set

On Distance-d Independent Set and Other Problems in Graphs with “few” Minimal Separators

Improved (In-)Approximability Bounds for d-Scattered Set

Contact Info

Product

Resources

About