On efficient fixed-parameter algorithms for weighted vertex cover

1. On efficient fixed-parameter algorithms forweighted vertex cover By Rolf Niedermeier & Peter Rossmanith Presentation by Peerapol BhuaratnarunkonApril 29th, 2004

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

3. Unweighted Vertex Cover • O(1.2852 k+kn) by Chen et al. - “Vertex folding” • O(1.2832 k+kn) by Niedermeier and Rossmanith- “Dynamic Programming”

4. 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,

5. 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 1.2852 is good enough to compete the best UVC algorithm

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

7. 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) - 1.3803 or better • 2.2) If there is a degree-1 vertex* • Branching vector (1,4) – 1.3803 or better

8. Real-WVC (con’t) • 2.3) If there is triangle* • Branching vector (3,4,3) – 1.3954 or better • 2.4) If there is no triangle* • Branching vector (3,4,3) – 1.3954 or better • Can be solved in time O(1.3954 k+kn)

9. 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

10. Dynamic Programming (con’t) • Apply to UVC • Can’t directly apply to the fastest algorithm of Chen et al. • Apply to the 2nd fastest algorithm of Niedermeier et al. • Achieve running time O(1.2832 k+kn) • Use O(1.275 k+kn) space • So, Integer-UVC can be solved equally fast

11. Dynamic Programming (con’t) • Apply to Real-WVC • Achieve running time O(1.3788 k+kn) • Use O(1.363 k+kn) space

12. Reference • R. Niedermeier and P. Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. In Journal of Algorithms 47, pages 63-77, 2003. Thank you !