Ge Xia

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}

Seeing the trees and their branches in the network is hard

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

Improved parameterized upper bounds for vertex cover

2k

The Stretch Factor of the Delaunay Triangulation Is Less than 1.998

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

Parametric duality and kernelization: lower bounds and upper bounds on kernel size

335k

Labeled Search Trees and Amortized Analysis: Improved Upper Bounds for NP-Hard Problems

. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless