Approximation algorithms via contraction decomposition

We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO+ 04, DHK05], and it generalizes a similar result for "compression" (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contraction-closed problems (whose optimal solution only improves under contraction), a much more general class than minor-closed problems. We prove that any contraction-closed problem satisfying just a few simple conditions has a PTAS in boundedgenus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimumweight c-edge-connected submultigraph on bounded-genus graphs, improving a...

• Type: Journal
• Year: 2010
• Where: COMBINATORICA
• Authors: Erik D. Demaine, MohammadTaghi Hajiaghayi, Bojan Mohar
• Keywords: COMBINATORICA 2010, Contraction-closed Problem, General H-minor-free Graphs, Machine Learning, Planar Graphs