On efficient fixed-parameter algorithms for weighted vertex cover By Rolf Niedermeier & Peter Rossmanith Presentation by Peerapol Bhuaratnarunkon April 29 th, 2004

Overview Unweighted Vertex Cover (UVC) Weighted Vertex Cover (WVC) Integer-WVC Real-WVC Dynamic Programming

Unweighted Vertex Cover O( k +kn) by Chen et al. - “Vertex folding” O( k +kn) by Niedermeier and Rossmanith - “Dynamic Programming”

Weighted Vertex Cover find a vertex cover with total weight less than or equal to k 3 Variants of WVC Integer-WVC, Real-WVC, General-WVC,

Integer-WVC Can be solved as fast as UVC Only additive term polynomial in k 1.) Branch at all vertices whose weight is at least 6 Branching vector (1,6) Branching number is good enough to compete the best UVC algorithm

Integer-WVC (con’t) 2.) Transform Integer-WVC to UVC Integer-WVC(G, k) iff UVC(G’, k) Let t(k,n) be time to solve UVC For UVC of a cluster instance, t(k,wn)=O(t(k,kn)) where w=O(1) i, weight 3 j, weight 1 WVC instance UVC instance cluster i’ cluster j’

Real-WVC 1) If no vertex with degree > 2, use Linear time dynamic programming* 2.1) If there is a vertex of degree > 4, branch on the vertex Branching vector (1,4) or better 2.2) If there is a degree-1 vertex* Branching vector (1,4) – or better

Real-WVC (con’t) 2.3) If there is triangle* Branching vector (3,4,3) – or better 2.4) If there is no triangle* Branching vector (3,4,3) – or better Can be solved in time O( k +kn)

Dynamic Programming Reduce exponential running time Use exponential space Store all induced subgraphs of size βk vertices in database Solve them & Store optimal solutions if size at most βk/2 Branching can stop earlier when size of search tree is as small as βk/2

Dynamic Programming (con’t) Apply to UVC Can’t directly apply to the fastest algorithm of Chen et al. Apply to the 2 nd fastest algorithm of Niedermeier et al. Achieve running time O( k +kn) Use O(1.275 k +kn) space So, Integer-UVC can be solved equally fast

Dynamic Programming (con’t) Apply to Real-WVC Achieve running time O( k +kn) Use O(1.363 k +kn) space

Reference R. Niedermeier and P. Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. In Journal of Algorithms 47, pages 63-77, Thank you !