It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining a polynomial kernel for Proper Interval Vertex Deletion. This resolves an open question of van Bevern, Komu...

In the previous lecture we covered polynomial time reductions and approximation algorithms for vertex cover and set cover problems. By reductions we showed that SAT, 3SAT, Independent Set, Vertex Cover, Integer Programming, and Clique problems are NP-Hard. In this lecture we will continue to cover approximation algorithms for maximum coverage and metric TSP problems. We will also cover Strong N...

i∈F ci ≥ k for all F ∈ F(∆). If c is a (0, 1)-vector, then c may be identified with the subset C = {i ∈ [n] : ci 6= 0} of [n]. It is clear that c is a 1-cover if and only if C is a vertex cover of ∆ in the classical sense, that is, C ∩ F 6= ∅ for all F ∈ F(∆). Let S = K[x1, . . . , xn] be a polynomial ring in n variables over a field K. Let Ak(∆) denote the K-vector space generated by all monom...

Many NP-complete problems on planar graphs are “fixed-parameter tractable:” Recent theoretical work provided tree decomposition based fixed-parameter algorithms exactly solving various parameterized problems on planar graphs, among others Vertex Cover, in time O(c √ kn). Here, c is some constant depending on the graph problem to be solved, n is the number of graph vertices, and k is the problem...

It is well-known that the Vertex Cover problem is in P on bipartite graphs, however; the computational complexity of the Partial Vertex Cover problem on bipartite graphs is open. In this paper, we first show that the Partial Vertex Cover problem is NP-hard on bipartite graphs. We then identify an interesting special case of bipartite graphs, for which the Partial Vertex Cover problem can be sol...

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Tran...

Abstract The computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks run time solve way smaller than even best known FPT-approaches can explain. We link these observations two p...

Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. Again, the minimum vertex cover problem is to find a vertex cover with the smallest number of vertices. Consider a k-partite graph G = (V, E) with v...

We study the generalization of covering problems to partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example , in k-set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-set cover, if each element occurs in at most f sets, then we derive a primal-dual f-approxima...

The Nemhauser–Trotter theorem provides an algorithm which is frequently used as a subroutine in approximation algorithms for the classical Vertex Cover problem. In this paper we present an extension of this theorem so it fits a more general variant of Vertex Cover, namely, the Generalized Vertex Cover problem, where edges are allowed not to be covered at a certain predetermined penalty. We show...