Structurally Parameterized d-Scattered Set

Abstract

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. IntroductionIn this paper we study the d-Scattered Set problem: given graph G = (V, E) and a metric weight function w : E → N + that gives the length of each edge, we are asked if there exists a set K of at least k selections from V , such that the distance between any pair v, u ∈ K is at least d(v, u) ≥ d, where d(v, u) denotes the shortest-path distance from v to u under weight function w. If w assigns weight 1 to all edges, the variant is called unweighted.The problem can already be seen to be hard, as it generalizes Independent Set (for d = 2), even to approximate (under standard complexity assumptions), i.e. the optimal k cannot be approximated to n 1− in polynomial time [19], while an alternative name is 28,13]. This hardness prompts the analysis of the problem when the input graph is of restricted structure, our aim being to provide a comprehensive account of the complexity of d-Scattered Set through various upper and lower bound results. Our viewpoint is parameterized: we consider the well-known structural parameters treewidth tw, tree-depth td, vertex cover number vc and feedback vertex set number fvs, that comprehensively express the intended restrictions on the input graph's structure (as they range in size and applicability), while we examine both the edge-weighted and unweighted variants of the problem. 1

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

“…Before we proceed further in the description of our reduction, let us give the basic ideas, which look like other SETH-based lower bounds from the literature [17,[20][21][22]24]. The constructed graph consists of a main part of n paths of length 4m, each divided into m sections.…”

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…It is also worth noting here that INDEPENDENT SET problem can be generalized to the d-SCATTERED SET problem where 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 [92]. Recently in [93] some lower and upper bounds on the approximation of the d-SCATTERED SET problem have been provided.…”

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

“…Furthermore, [14] shows the problem admits an EPTAS on (apex)-minor-free graphs, based on the theory of bidimensionality, while on a related result [22] offers an n O( √ n) -time algorithm for planar graphs, making use of Voronoi diagrams and based on ideas previously used to obtain geometric QPTASs. Finally, [21] presents tight upper/lower bounds on the structurally parameterized complexity of the problem, while [24] shows that it admits an almost linear kernel on every nowhere dense graph class.…”

“…The problem has been well-studied, also from the parameterized point of view [21,24], while approximability in polynomial time has already been considered for bipartite, regular and degree-bounded graphs [12,11], perhaps the natural candidate for the next intractability frontier. This paper aims to advance our understanding in this direction by providing the first lower bound on the approximation ratio of any polynomial-time algorithm as a function of the maximum degree of any vertex in the input graph, while also improving upon the known ratios to match this lower bound.…”