N. S. Narayanaswamy

Faster Parameterized Algorithms Using Linear Programming

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*(2.618k) 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)nO(1)). We proceed to show that a more sophisticated branching algorithm achieves a running time of O*(2.3146k). Following this, using previously known as well as new reductions, we give O*(2.3146k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, and Almost 2-SAT, and O*(1.5214k) algorithms for König Vertex Deletion and Vertex Cover parameterized by the size of the smallest odd cycle transversal and König vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for Odd Cycle Transversal—this is the first algorithm that improves on the dependence on k of the seminal O*(3k) 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 − clog k vertices. Our kernel is simpler than previously known kernels achieving the same size bound.

LP can be a cure for Parameterized Problems

We investigate the parameterized complexity of Vertex Cover parameterized above the optimum value of the linear programming (LP) relaxation of the integer linear programming formulation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that even the most straightforward branching algorithm (after some preprocessing) results in an O (2.6181 r ) algorithm for the problem where r is the excess of the vertex cover size over the LP optimum. 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.

A Note on First-Fit Coloring of Interval Graphs

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

An improved algorithm for online coloring of intervals with bandwidth

\frac{\omega(G)}{8}

Dynamic Storage Allocation and On-Line Colouring Interval Graphs

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A new characterization of matrices with the consecutive ones property

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.

An Optimal Lower Bound for Resolution with 2-Conjunctions

We present an improved on-line algorithm for colouring interval graphs with bandwidth. This problem has recently been studied in [1] and a 195-competitive online strategy has been presented. We improve this by presenting a 10-competitive strategy. To achieve this result, we use the online colouring algorithm presented in [8,9]. We also present a new analysis of a polynomial time 3-approximation algorithm for Dynamic Storage Allocation(DSA) using features of the optimal on-line algorithm for colouring interval graphs [8,9].

Parameterized Algorithms for (r;l)-Partization

We consider the following constraint satisfaction problem: Given a set F of subsets of a finite set S of cardinality n, and an assignment of intervals of the discrete set {1,...,n} to each of the subsets, does there exist a bijection f:S->{1,...,n} such that for each element of F, its image under f is same as the interval assigned to it. An interval assignment to a given set of subsets is called feasible if there exists such a bijection. In this paper, we characterize feasible interval assignments to a given set of subsets. We then use this result to characterize matrices with the Consecutive Ones Property (COP), and to characterize matrices for which there is a permutation of the rows such that the columns are all sorted in ascending order. We also present a characterization of set systems which have a feasible interval assignment.

Analysis of algorithms for an online version of the convoy movement problem

A lower bound is proved for refutations of certain clause sets in a generalization of Resolution that allows cuts on conjunctions of width 2. The hard clauses are the Tseitin graph formulas for a class of logarithmic degree expander graphs. The bound is optimal in the sense that it is truly exponential in the number of variables.

Approximability of Connected Factors

We consider the (r;l)-Partization problem of nding a set of at most k vertices whose deletion results in a graph that can be partitioned into r independent sets and l cliques. Restricted to perfect graphs and split graphs, we describe sequacious xed-parameter tractability results for (r; 0)-Partization, parameterized by k and r. For (r;l)-Partization where r + l = 2, we describe an O (2 k ) algorithm for perfect graphs. We then study the parameterized complexity hardness of a generalization of the Above Guarantee Vertex Cover by a reduction from (r;l)-Partization.