Optimality Program in Segment and String Graphs

Bonnet, Édouard; Rzążewski, Paweł

doi:10.1007/978-3-030-00256-5_7

Cited by 6 publications

(10 citation statements)

References 33 publications

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“…The algorithm and its analysis are very similar to the proof of Theorem 1, so we will only point out the di erences. The adaptation is inspired by the known subexponential-time algorithms for 3 C [5,14]. We remark that a quasi-polynomial-time algorithm for 3 C in P t -free graphs with running time n O(log 3 n) can be also derived from the work of Gartland and Lokshtanov [13], by an analogous adaptation of their approach.…”

Section: Partitioning Vertices: 3 Cmentioning

confidence: 99%

Quasi-polynomial-time algorithm for Independent Set in $P_t$-free graphs via shrinking the space of induced paths

Pilipczuk¹,

Pilipczuk²,

Rzążewski³

2020

Preprint

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Section: Partitioning Vertices: 3 Cmentioning

confidence: 99%

Quasi-polynomial-time algorithm for Independent Set in $P_t$-free graphs via shrinking the space of induced paths

Pilipczuk¹,

Pilipczuk²,

Rzążewski³

2020

Preprint

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“…As a stark contrast, they showed that 6 C does not admit a subexponential-time algorithm in segment graphs. This was later improved by Bonnet and Rzążewski [5] who showed that already 4 C cannot be solved in subexponential time in segment graphs, but 3 C admits a 2 O(n 2/3 log n) -algorithm in all string graphs. They also showed several positive and negative results concerning subexponential-time algorithms for segment and string graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, many real-life graphs have some underlying geometry [26,29,30]. Thus the complexity of graph problems restricted to various classes of geometric intersection graphs has been an active research topic [4,5,10,[19][20][21][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

“…For a class C of geometric objects, what is the maximum k (if any), such that for every graph H with mrc(H) k, the LH (H) problem admits a subexponential-time algorithm in intersection graphs of objects from C? Note that k from the question might not exist, as for example 4 C (and thus LH (K 4 )) does not admit a subexponential-time algorithm in segment graphs [5], while mrc(K 4 ) = 0.…”

Section: Introductionmentioning

confidence: 99%

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Computing list homomorphisms in geometric intersection graphs

Kisfaludi-Bak¹,

Okrasa²,

Rzążewski³

2022

Preprint

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A homomorphism from a graph G to a graph H is an edge-preserving mapping from V (G) to V (H).Let H be a xed graph with possible loops. In the list homomorphism problem, denoted by LH (H), the instance is a graph G, whose every vertex is equipped with a subset of V (H), called list. We ask whether there exists a homomorphism from G to H, such that every vertex from G is mapped to a vertex from its list.We study the complexity of the LH (H) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the question for what graphs H and for what types of geometric objects, the LH (H) problem can be solved in time subexponential in the number of vertices of the instance.We fully resolve this question for string graphs, i.e., intersection graphs of continuous curves in the plane. Quite surprisingly, it turns out that the dichotomy exactly coincides with the analogous dichotomy for graphs excluding a xed path as an induced subgraph [Okrasa, Rzążewski, STACS 2021].Then we turn our attention to subclasses of string graphs, de ned as intersections of fat objects. We observe that the (non)existence of subexponential-time algorithms in such classes is closely related to the size mrc(H) of a maximum re exive clique in H, i.e., maximum number of pairwise adjacent vertices, each of which has a loop. We study the maximum value of mrc(H) that guarantees the existence of a subexponential-time algorithm for LH (H) in intersection graphs of (i) convex fat objects, (ii) fat similarly-sized objects, and (iii) disks. In the rst two cases we obtain optimal results, by giving matching algorithms and lower bounds.Finally, we discuss possible extensions of our results to weighted generalizations of LH (H).

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“…Fox and Pach [10] gave for every > 0, a polynomial time algorithm for computing the Maximum Independent set of k-string graphs (intersection graphs of curves on the plane where two curves intersecting at most k times) with approximation ratio at most n . While Pawlik et al [18] proved that triangle-free segment graphs (intersection graphs of line segments on the plane) can have arbitrarily high Chromatic Number, Bonnet et al [4] gave a subexponential algorithm to color string graphs with three colors. In this paper, we study the MDS problem on string graphs and its subclasses.…”

Section: Introductionmentioning

confidence: 99%

Approximating Minimum Dominating Set on String Graphs

Chakraborty

Das

Mukherjee

2019

Lecture Notes in Computer Science

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In this paper, we give approximation algorithms for the Minimum Dominating Set (MDS) problem on string graphs and its subclasses. A path is a simple curve made up of alternating horizontal and vertical line segments. A k-bend path is a path made up of at most k + 1 line segments. An L-path is a 1-bend path having the shape 'L'. A vertically-stabbed-L graph is an intersection graph of L-paths intersecting a common vertical line. We give a polynomial time 8-approximation algorithm for MDS problem on vertically-stabbed-L graphs whose APX-hardness was shown by Bandyapadhyay et al. (MFCS, 2018). To prove the above result, we needed to study the Stabbing segments with rays (SSR) problem introduced by Katz et al. (Comput. Geom. 2005). In the SSR problem, the input is a set of (disjoint) leftwarddirected rays, and a set of (disjoint) vertical segments. The objective is to select a minimum number of rays that intersect all vertical segments. We give a O((n+m) log(n+m))-time 2-approximation algorithm for the SSR problem where n and m are the number of rays and segments in the input. A unit k-bend path is a k-bend path whose segments are of unit length. A graph is a unit B k -VPG graph if it is an intersection graph of unit k-bend paths. Any string graph is a unit-B k -VPG graph for some finite k. Using our result on SSR-problem, we give a polynomial time O(k 4 )-approximation algorithm for MDS problem on unit B k -VPG graphs for k ≥ 0.

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Optimality Program in Segment and String Graphs

Cited by 6 publications

References 33 publications

Quasi-polynomial-time algorithm for Independent Set in $P_t$-free graphs via shrinking the space of induced paths

Quasi-polynomial-time algorithm for Independent Set in $P_t$-free graphs via shrinking the space of induced paths

Computing list homomorphisms in geometric intersection graphs

Approximating Minimum Dominating Set on String Graphs

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