Parameterized Approximation Algorithms for Some Location Problems in Graphs

Abstract

Abstract. We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) r-Domination problem and the (Connected) p-Center problem for unweighted and undirected graphs. Given a graph G, we show how to construct a (connected) ris a minimum (connected) r-dominating set of G and µ is our graph parameter, which is the tree-breadth or the cluster diameter in a layering partition of G. Additionally, we show that a +O(µ)-approximation for the (Connected) p-Center problem on G can be computed in polynomial time. Our interest in these parameters stems from the fact that in many real-world networks, including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others, and in many structured classes of graphs these parameters are small constants.

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

“…For the edge-weighted variant, [24] shows that a (2 − )-approximation is W [2]-hard for parameter k and NP-hard for graphs of highway dimension h = O(log 2 n), while also offering a 3/2-approximation algorithm of running time 2 O(kh log(h)) • n O (1) , exploiting the similarity of this problem with that of solving Dominating Set on graphs of bounded vc. Finally, for unweighted graphs, [37] provides efficient (linear/polynomial) algorithms computing (r + O(µ))-dominating sets and +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 [22] gives a polynomial-time bicriteria approximation scheme for graphs of bounded genus.…”