We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O * ((2.618) k ) algorithm for the problem. Here k is the excess of the vertex cover size over the LP optimum, and we write O * (f (k)) for a time complexity of the form O(f (k)n O(1) ), where f (k) grows exponentially with k. We proceed to show that a more sophisticated branching algorithm achieves a runtime of O * (2.3146 k ).Following this, using known and new reductions, we give O * (2.3146 k ) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an O * (1.5214 k ) algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The most notable improvement is the new bound for Odd Cycle Transversal -this is the first algorithm which beats the dependence on k of the seminal O * (3 k ) algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k − O(log k) vertices. Our kernel is simpler than previously known kernels achieving the same size bound. Vertex CoverInstance: An undirected graph G and a positive integer k. Parameter: k.Problem: Does G have a vertex cover of of size at most k?We start with a few basic definitions regarding parameterized complexity. For decision problems with input size n, and a parameter k, the goal in parameterized complexity is to design an algorithm with runtime f (k)n O(1) where f is a function of k alone, as contrasted with a trivial n k+O(1) algorithm. Problems which admit such algorithms are said to be fixed parameter tractable (FPT). The theory of parameterized complexity was developed by Downey and Fellows [6]. For recent developments, see the book by Flum and Grohe [7].Vertex Cover was one of the first problems that was shown to be FPT [6]. After a long race, the current best algorithm for Vertex Cover runs in time O(1.2738 k + kn) [3]. However, when k < m, the size of the maximum matching, the Vertex Cover problem is not interesting, as the answer is trivially NO. Hence, when m is large (for example when the graph has a perfect matching), the running time bound of the standard FPT algorithm is not practical, as k, in this case, is quite large. This led to the following natural "above guarantee" variant of the Vertex Cover problem. Above Guarantee Vertex Cover (agvc)Instance: An undirected graph G, a maximum matching M and a positive integer k. Parameter: k − |M |.Problem: Does G have a vertex cover of of size at most k?In addition to being a natural parameterization of the classical Vertex Cover pr...

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We apply the Column Construction Method (Varadarajan et al., Proceedings of the Fifteenth Annual ACM-SIAM Symposium On Discrete Algorithms, pp. 562-571, 2004) to a minimal clique cover of an interval graph to obtain a new proof that First-Fit is 8-competitive for online coloring interval graphs. This proof also yields a new discovery that in each minimal clique cover of an interval graph G, there is a clique of size ω(G) 8 .

We present an improved online algorithm for coloring interval graphs with bandwidth. This problem has recently been studied by Adamy and Erlebach and a 195-competitive online strategy has been presented. We improve this by presenting a 10-competitive strategy. To achieve this result, we use variants of an optimal online coloring algorithm due to Kierstead and Trotter.

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