Li-Hsuan Chen

Exact algorithms for problems related to the densest k-set problem

Abstract Many graph concepts such as cliques, k-clubs, and k-plexes are used to define cohesive subgroups in a social network. The concept of densest k-set is one of them. A densest k-set in an undirected graph G = ( V , E ) is a vertex set S ⊆ V of size k such that the number of edges in the subgraph of G induced by S is maximum among all subgraphs of G induced by vertex sets of size k. One can obtain a densest k-set of G in O ( k 2 n k ) time by exhaustive-search technique for an undirected graph of n vertices and a number k n . However, if the value of k approaches n / 2 , the running time of the exhaustive-search algorithm is O ⁎ ( 2 n ) . Whether there exists an O ⁎ ( c n ) -time algorithm with the fixed constant c 2 to find a densest k-set in an undirected graph of n vertices remains open in the literature. In this paper, we point out that the densest k-set problem and a class of problems related to the concept of densest k-sets can be solved in time O ⁎ ( 1.7315 n ) .

Fixed-parameter algorithms for vertex cover P 3

Abstract Let P l denote a path in a graph G = ( V , E ) with l vertices. A vertex cover P l set C in G is a vertex subset such that every P l in G has at least a vertex in C . The Vertex Cover P l problem is to find a vertex cover P l set of minimum cardinality in a given graph. This problem is NP-hard for any integer l ⩾ 2 . The parameterized version of Vertex Cover P l problem called k - Vertex Cover P l asks whether there exists a vertex cover P l set of size at most k in the input graph. In this paper, we give two fixed parameter algorithms to solve the k - Vertex Cover P 3 problem. The first algorithm runs in time O ∗ ( 1.796 4 k ) in polynomial space and the second algorithm runs in time O ∗ ( 1.748 5 k ) in exponential space. Both algorithms are faster than previous known fixed-parameter algorithms.

A 5k kernel for P 2 -packing in net-free graphs

Given a graph G = (V, E), a P2-packing P is a collection of vertex disjoint copies of P2s in G where a P2 is a simple path with three vertices and two edges. The kP2-Packing problem asks whether there exists a P2-packing of size k in G by taking graph G and a fixed parameter k as the input. This problem is NP-hard for net-free graphs. In this paper, we give a kernelization algorithm for the kP2-Packing problem in net-free graphs. We show that in polynomial time our kernelization algorithm obtains a size-5k kernel which is smaller than those kernels found by previous known kernelization algorithms.

Moderately exponential time algorithms for the maximum induced matching problem

An induced matching

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

M\subseteq E

An improved parameterized algorithm for the p-cluster vertex deletion problem

M⊆E in a graph

An

G=(V, E)