We study low degree graph problems such as Maximum Independent Set and Minimum Vertex Cover. The goal is to improve approximation lower bounds for them and for a number of related problems like Max-B-Set Packing, Min-B-Set Cover, and Max-B-Dimensional Matching, B 3. We prove, for example, that it is NP-hard to achieve an approximation factor of 95 94 for Max-3-DM, and a factor of 48 47 for Max-4-DM. In both cases the hardness result applies even to instances with exactly two occurrences of each element.

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

The Steiner tree problem on weighted graphs seeks a minimum weight subtree containing a given subset of the vertices (terminals). We show that it is NP-hard to approximate the Steiner tree problem within a factor 96/95. Our inapproximability results are stated in a parametric way, and explicit hardness factors would be improved automatically providing gadgets and/or expanders with better parameters. IntroductionConsider a graph G = (V, E) with weight function w : E → R + on the edges and a set of required vertices T ⊆ V , called the terminals. A Steiner tree T is a subtree of G that spans all vertices in T (using vertices in V \ T as well) and its weight is defined by w(T ) = e∈E(T ) w(e).The Steiner Tree problem (STP) is to find a Steiner tree of minimum weight. Steiner trees are important in various applications, for example, in VLSI design, wirelength estimation, and network routing.An instance of the Steiner tree problem is called quasi-bipartite if there is no edge within the set V \ T , and uniformly quasi-bipartite if it is quasi-bipartite and all edges incident to the same non-terminal vertex have the same weight.

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. [Blow-Ups, Win/Win's and crown rules: some new directions in FPT, in: We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so-called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some additional constraints. It is described how crown decompositions and strong crown decompositions suitable for such problems can be computed in polynomial time. For weighted König-Egerváry graphs (G, w) we observe that the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be efficiently computed.Further, for some specific classes of graphs, simple algorithms for the MIN-VC problem with a constant approximation factor r < 2 are provided. On the other hand, we conclude that for the regular graphs, or for the Hamiltonian connected graphs, the problem is as hard to approximate as for general graphs.It is demonstrated how the results about strong crown reductions can be used to achieve a linear size problem kernel for some related vertex cover problems. Feasible solution:A vertex cover C for G, i.e., a subset C ⊆ V such that for each e ∈ E, e ∩ C = ∅. Goal: To minimize the weight w(C) := u∈C w(u) of the vertex cover C. The unweighted version of the Minimum Vertex Cover problem (shortly, MIN-VC) is the special case of MIN-w-VC with uniform weights w(u) = 1 for each u ∈ V .As the Minimum Vertex Cover problem cannot be solved exactly in polynomial time, unless P = NP, approaches have concentrated on the design of polynomial time approximation algorithms. In spite of a great deal of efforts, the tight bound on its approximability by a polynomial time algorithm is left open. Recall that the problem has a simple 2-approximation algorithm and, for any constant r < 2, no r-approximation algorithm is known, even in the unweighted case. Currently the best lower bound on polynomial time approximability is 10 √ 5 − 21 ≈ 1.36067, due to Dinur and Safra [17]. They proved that achieving a smaller approximation factor is NP-hard.Recently, there has been increasing interest and progress in lowering the exponential running time of algorithms that solve NP-hard optimization problems, like MIN-VC, exactly [11,32]. The theory of parametrized computation and fixed parameter tractability is a newly developed approach dealing with exact algorithms for such intractable problems. Many hard problems can be associated with a parameter in such a way that the problems are tractable when the parameter is fixed or varies within a small range. Such parametrized problems are now known as fixed parameter tractable (FPT) [18]. The parametrized version of the Vertex Cover problem is a well known FPT problem and has received considerable interest:Para...

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