Implicit branching and parameterized partial cover problems

Abstract

Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems have been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems.

Subexponential algorithms for partial cover problems

“…We have given the first subexponential algorithms for Partial Vertex Cover and Partial r-Dominating Set problems on planar and apex minor free graphs, answering an open problem in [1]. Our results were based on a simple but powerful observation relating lexicographically least solutions and r-dominating sets of size at most k. This allowed us to significantly improve the running time of several algorithm presented in [1] in an elegant way.…”

“…Our results were based on a simple but powerful observation relating lexicographically least solutions and r-dominating sets of size at most k. This allowed us to significantly improve the running time of several algorithm presented in [1] in an elegant way. Through this process, we have also expanded the list of problems tractable using the irrelevant vertex argument and it would be nice to apply this technique for other problems in planar and other classes of sparse graphs.…”

“…Partial cover problems have been investigated extensively and are well understood in the context of polynomial time approximation [2,4,3,5,16,18] and parameterized complexity [1,4,24,25,23,27]. In this paper we study partial cover problems defined on graphs namely Partial Vertex Cover and Partial r-Dominating Set from the view point of parameterized algorithms.…”

“…The P-r-DS problem being a generalization of Dominating Set is also known to be W [2]-hard on general graphs when parameterized by the cover size. Amini et al [1] considered these problems with the size of the cover k being the parameter and initiated a study of these problem on sparse graphs namely planar graphs, apex minor free graphs and H-minor free graphs. They obtained algorithms with running time 2 O(k) n O(1) for PVC and Pr-DS and left an open question of whether these problems have an algorithm with running time 2 o(k) n O(1) , like their non partial counterpart on planar graphs or more generally on H-minor free graphs.…”

Parameterized Complexity of Edge Interdiction Problems

“…For instance, the Partial Vertex Cover (k-PVC) problem asks for k vertices maximizing the number of covered edges. Partial covering problems have been studied intensively not only because they generalize classical covering problems, but also because of many real life applications, see for example [2,4,1,7].…”