Structural parameters, tight bounds, and approximation for (k,r)-center

Abstract

In (k, r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically:• For any r ≥ 1, we show an algorithm that solves the problem in O * ((3r + 1) cw ) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for r = 1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw.• We strengthen previously known FPT lower bounds, by showing that (k, r)-Center is W[1]-hard parameterized by the input graph's vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs.• We show that the complexity of the problem parameterized by tree-depth is 2 Θ(td 2 ) , by showing an algorithm of this complexity and a tight ETH-based lower bound.We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth, which work efficiently independently of the values of k, r. In particular, we give algorithms which, for any > 0, run in time O * ((tw/ ) O(tw) ), O * ((cw/ ) O(cw) ) and return a (k, (1 + )r)-center if a (k, r)-center exists, thus circumventing the problem's W-hardness.

Structurally parameterized d-scattered set

“…d ≥ O(log n)), even for the unweighted case and more restricted parameters. Figure 1 illustrates the relationships between considered parameters and summarizes our results.Related work: Our work can be considered as a continuation of the investigations in [21], where the (k, r)-Center problem is similarly studied with respect to several well-known structural parameters and a number of fine-grained upper/lower bounds is presented, while some of the techniques employed for our SETH lower bound are also present in [8].The SETH-based lower bound of (2 − ) tw · n O(1) on the running time of any algorithm for Independent Set parameterized by tw comes from [24]. For d-Scattered Set, Halldórsson et al [17] showed a tight inapproximability ratio of n 1− for even d and n 1/2− for odd d, while Eto et al [13] showed that on r-regular graphs the problem is APX-hard for r, d ≥ 3, while also providing polynomial-time O(r d−1 )-approximations and a polynomial-time approximation scheme (PTAS) for planar graphs.…”

“…Related work: Our work can be considered as a continuation of the investigations in [21], where the (k, r)-Center problem is similarly studied with respect to several well-known structural parameters and a number of fine-grained upper/lower bounds is presented, while some of the techniques employed for our SETH lower bound are also present in [8].…”

The Parameterized Hardness of the k-Center Problem in Transportation Networks

“…They show that the problem is FPT on unweighted map graphs for the combined parameter (k, ρ). Also for the tree-depth, k-Center is FPT [21]. Another parameter related to transportation networks is the skeleton dimension, for which it was recently shown [8] that, under ETH, no 2 2 o( √ s) · n O(1) time algorithm can compute a (2 − ε)-approximation for any ε > 0, if the skeleton dimension is s. It is not known whether this parameter yields any approximation schemes when combined with for instance k, as is the case for the highway dimension.…”

“…In fact it is even W [2]-hard [15] to compute a (2 − ε)-approximation for any ε > 0, and thus parametrizing by k does not help to overcome the polynomial-time inapproximability. For structural parameters such as the vertex-cover number or the feedback-vertex-set number the problem remains W[1]-hard [21], even when combining with the parameter k. For each of the two more general structural parameters treewidth and cliquewidth, an efficient parameterized approximation scheme (EPAS) was shown to exist [21], i.e., a (1 + ε)-approximation can be computed in f (ε, w) · n O(1) time for any ε > 0, if w is either the treewidth or the cliquewidth, and n is the number of vertices.…”

“…Later, Becker et al [7] used the framework introduced by Feldmann et al [16] to show that whenever c > 4 there is an EPAS for k-Center parameterized by ε, k, and h. Note that the highway dimension is always upper bounded by the vertex-cover number, as every edge of any non-trivial path is incident to a vertex cover. Hence the aforementioned W[1]-hardness result by Katsikarelis et al [21] for the combined parameter k and the vertex-cover number proves that it is necessary to approximate the optimum when using k and h as the combined parameter. When parametrizing only by the highway dimension but not k, it is not even known if a parameterized approximation scheme (PAS) exists, i.e., an f (ε, h) · n g(ε) time (1 + ε)-approximation algorithm for some computable functions f, g. However, under the Exponential Time Hypothesis (ETH) [9], by [15] there is no algorithm with doubly exponential 2 2 o( √ h) · n O(1) runtime computing a (2 − ε)-approximation for any ε > 0.…”