Iyad A. Kanj

Vertex Cover

Recently, there has been increasing interest and progress in lowering the worst-case time complexity for well-known NP-hard problems, particularly for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated, and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn+1.2852k) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree.

Improved upper bounds for vertex cover

This paper presents an O(1.2738^k+kn)-time polynomial-space algorithm for Vertex Cover improving the previous O(1.286^k+kn)-time polynomial-space upper bound by Chen, Kanj, and Jia. Most of the previous algorithms rely on exhaustive case-by-case branching rules, and an underlying conservative worst-case-scenario assumption. The contribution of the paper lies in the simplicity, uniformity, and obliviousness of the algorithm presented. Several new techniques, as well as generalizations of previous techniques, are introduced including: general folding, struction, tuples, and local amortized analysis. The algorithm also improves the O(1.2745^kk^4+kn)-time exponential-space upper bound for the problem by Chandran and Grandoni.

Tight lower bounds for certain parameterized NP-hard problems

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n/sup o(k)/poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t - l)-st level W[t

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted SAT, dominating set, hitting set, set cover, and feature set, cannot be solved in time n/sup o(k)/poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W[l] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-SAT (for any fixed q /spl ges/ 2), clique, and independent set, cannot be solved in time n/sup o(k)/ unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n/sup k/ poly(m) or O(n/sup k/).

Strong computational lower bounds via parameterized complexity

Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving

Linear FPT reductions and computational lower bounds

\mathcal{NP}

Vertex Cover: Further Observations and Further Improvements

-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by

Seeing the trees and their branches in the network is hard

2k

Improved parameterized upper bounds for vertex cover

, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by

Parameterized Algorithms for Feedback Vertex Set

335k