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 2Θ(td2), 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 ϵ>0, run in time O∗((tw∕ϵ)O(tw)), O∗((cw∕ϵ)O(cw)) and return a (k,(1+ϵ)r)-center if a (k,r)-center exists, thus circumventing the problem’s W-hardness.
Highlights
In this paper we study the (k, r)-Center problem: given a graph G = (V, E) and a weight function w : E → N+ which satisfies the triangle inequality and defines the length of eachLeibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany50:2 Parameterized (k, r)-Center edge, we are asked if there exists a set K of at most k vertices of V, so that ∀u ∈ V \ K we have minv∈K d(v, u) ≤ r, where d(v, u) denotes the shortest-path distance from v to u under weight function w
We prove the following: (k, r)-Center can be solved in time O∗((3r + 1)cw), but it cannot be solved in time O∗((3r+1− )cw) for any r ≥ 1, unless the Strong Exponential Time Hypothesis (SETH) [26, 27] fails
We remark that (k, r)-Center is a rare example of a problem that turns out to be hard parameterized by vc. We complement these lower bounds by an FPT algorithm for the unweighted case, running in time O∗(5vc). (k, r)-Center can be solved in time O∗(2O(td2)) for unweighted graphs, but if it can be solved in time O∗(2o(td2)), the ETH is false
Summary
We remark that (k, r)-Center is a rare example of a problem that turns out to be hard parameterized by vc We complement these lower bounds by an FPT algorithm for the unweighted case, running in time O∗(5vc). We remark that this is a somewhat uncommon example of a parameterized problem whose parameter dependence turns out to be exponential in the square of the parameter These results, together with the recent work of [10] showing tight bounds of O∗((2r + 1)tw) on the problem’s complexity parameterized by tw, give a complete and often fine-grained, picture on (k, r)-Center for the most important graph parameters. +O(μ)-approximations for (k, r)-Center, where μ is the tree-breadth or cluster diameter in a layering partition of the input graph, while [18] gives a polynomial-time bicriteria approximation scheme for graphs of bounded genus
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
