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Approximation algorithms via contraction decomposition

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Abstract

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 [4,20, 10, 17], and it generalizes a similar result for “compression” (a variant of contraction) in planar graphs [29]. 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 bounded-genus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimum-weight c-edge-connected submultigraph on bounded-genus graphs, improving and generalizing previous algorithms of [24, 1, 29, 25, 8, 5]. We also highlight the only main difficulty in extending our results to general H-minor-free graphs.

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Authors and Affiliations

  1. MIT, Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA, 02139, USA

    Erik D. Demaine

  2. Computer Science Department, University of Maryland, A. V. Williams Bldg., Room 3249, College Park, MD, 20742, USA

    MohammadTaghi Hajiaghayi

  3. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Bojan Mohar

Authors
  1. Erik D. Demaine
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  2. MohammadTaghi Hajiaghayi
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  3. Bojan Mohar
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Corresponding author

Correspondence to Erik D. Demaine.

Additional information

A preliminary version of this paper appeared in Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, January 2007.

on leave from: Department of Mathematics University of Ljubljana 1000 Ljubljana Slovenia

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Demaine, E.D., Hajiaghayi, M. & Mohar, B. Approximation algorithms via contraction decomposition. Combinatorica 30, 533–552 (2010). https://doi.org/10.1007/s00493-010-2341-5

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