Let G = (V,E) be an undirected graph with arc weights w : V → R+. Define xv for each vertex v as follows: xv = 1, if v is in the vertex cover; xv = 0, if v is not chosen. Our goal is to find min ( ∑ v∈V wvxv), such that xu + xv ≥ 1, ∀e = (u, v) ∈ E, xv ∈ {0, 1}. However, we can’t solve Integer Linear Programming (ILP) problems in polynomial time. So we have to use Linear Programming (LP) to app...

We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP ⊆ coNP/poly), for planar graphs Guo and Niedermeier [ICALP’08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS’11] to 4k....

15.

We construct a Π2 enumeration degree which is a strong minimal cover.

An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress.

We study several set cover problems in low dimensional geometric settings. Specifically, we describe a PTAS for the problem of computing a minimum cover of given points by a set of weighted fat objects. Here, we allow the objects to expand by some prespecified δ-fraction of their diameter. Next, we show that the problem of computing a minimum weight cover of points by weighted halfplanes (witho...

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem Q if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we sh...

We introduce the cross-composition framework for proving kernelization lower bounds. A classical problem L and/or-cross-composes into a parameterized problem Q if it is possible to efficiently construct an instance of Q with polynomially bounded parameter value that expresses the logical and or or of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a re...

Abstract The NP-complete Vertex Cover problem asks to cover all edges of a graph by small (given) number vertices. It is among the most prominent graph-algorithmic problems. Following recent trend in studying temporal graphs (a sequence graphs, so-called layers, over same vertex set but, time, changing edge sets), we initiate study Multistage . Herein, given graph, goal find for each layer and ...

The class of D-dotted interval (D-DI) graphs is the class of intersection graphs of arithmetic progressions with jump (common difference) at most D. We consider various classical graphtheoretic optimization problems in D-DI graphs of arbitrarily, but fixed, D. We show that Maximum Independent Set, Minimum Vertex Cover, and Minimum Dominating Set can be solved in polynomial time in this graph cl...