Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3

In this paper we propose an O(n1.0892) algorithm solving the Maximum Independent Set problem for graphs with maximum degree 3 improving the previously best upper bound of O(n1.0977). A useful secondary effect of the proposed algorithm is that being applied to 2k kernel, it improves the upper bound on the parameterized complexity of the Vertex Cover problem for graphs with maximum degree 3 (VC-3). In particular, the new upper bound for the VC-3 problem is O(k1.1864+n), improving the previously best upper bound of O(k2∗k1.194+n). The presented results have a methodological interest because, to the best of our knowledge, this is the first time when a new parameterized upper bound is obtained through design and analysis of an exact exponential algorithm

Vertex Cover Gets Faster and Harder on Low Degree Graphs

The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon. We consider the problem of hitting all axis-parallel slabs induced by a point set P, and show that it is equivalent to the problem of finding a vertex cover on a graph whose edge set is the union of two Hamiltonian Paths. We show the latter problem to be NP-complete, and we also give an algorithm to find a vertex cover of size at most k, on graphs of maximum degree four, whose running time is 1.2637^k n^O(1)

Branch-and-Reduce Exponential/FPT Algorithms in Practice: A Case Study of Vertex Cover

We investigate the gap between theory and practice for exact branching algorithms. In theory, branch-and-reduce algorithms currently have the best time complexity for numerous important problems. On the other hand, in practice, state-of-the-art methods are based on different approaches, and the empirical efficiency of such theoretical algorithms have seldom been investigated probably because they are seemingly inefficient because of the plethora of complex reduction rules. In this paper, we design a branch-and-reduce algorithm for the vertex cover problem using the techniques developed for theoretical algorithms and compare its practical performance with other state-of-the-art empirical methods. The results indicate that branch-and-reduce algorithms are actually quite practical and competitive with other state-of-the-art approaches for several kinds of instances, thus showing the practical impact of theoretical research on branching algorithms.Comment: To appear in ALENEX 201

An O∗(1.0821n)O^*(1.0821^n)-Time Algorithm for Computing Maximum Independent Set in Graphs with Bounded Degree 3

We give an $O^*(1.0821^n)$-time, polynomial space algorithm for computing Maximum Independent Set in graphs with bounded degree 3. This improves all the previous running time bounds known for the problem