Subexponential algorithms for partial cover problems

Abstract

Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et. al. [FSTTCS 08 ] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely H-minor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O(k) n O(1) .During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on H-minor free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. [FSTTCS 08 ] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O( √ k) n O(1) not only on planar graphs but also on much larger classes of graphs, namely, apex-minor free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler.

Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams

“…In all these problems, there is a relation between the size of the largest grid minor and the size of the optimum solution, which allows us to bound the treewidth of the graph in terms of the parameter of the problem. More recently, subexponential parameterized algorithms have been explored also for problems where there is no such straightforward parameter-treewidth bound: for example, for Partial Vertex Cover and Dominating Set [22], k-Internal Out-Branching and k-Leaf Out-Branching [17], Multiway Cut [33], Subset TSP [34], Strongly Connected Steiner Subgraph [9], Steiner Tree [42,43]. For some of these problems, it is easy to see that they are fixed-parameter tractable on planar graphs, and the challenge is to make the dependence on k subexponential, e.g., to obtain 2 O( A similar "square root phenomenon" has been observed in the case of geometric problems: it is usual to see a square root in the exponent of the running time of algorithms for NP-hard problems defined in the 2-dimensional Euclidean plane.…”

Planar k-Path in Subexponential Time and Polynomial Space

“…Here we are given a graph G, positive integers k and t and we look for a subset S ⊆ V (G) such that |S| ≤ k and the number of edges incident to S is at least t. This problem is not known to admit a polynomial kernel, even on planar graphs. However using the approach described in this paper for k-Path and combining it with an algorithm of Fomin et al [13] that in polynomial time either finds a solution for an instance (G, k, t) or obtains an equivalent instance (G , k, t) such that tw(G ) ≤ O( √ k), one can give a subexponential time, polynomial space parameterized algorithm for k-Partial Vertex Cover on apex-minorfree graphs.…”

Parameterized and approximation complexity of Partial VC Dimension

“…Partial VC Dimension belongs to the category of partial versions of common decision problems, in which, instead of satisfying the problem's constraint task for all elements (here, all 2 k equivalence classes), we ask whether we can satisfy a certain number, ℓ, of these constraints. See for example the papers [21,34,41] that study some partial versions of standard decision problems, such as Set Cover, Vertex Cover or Dominating Set.…”