**Reference**:- Erik D. Demaine, MohammadTaghi Hajiaghayi, and Bojan Mohar, “Approximation Algorithms via Contraction Decomposition”,
*Combinatorica*, volume 30, number 5, 2010, pages 533–552. **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 [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 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 [GKP95, AGK^{+}98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general*H*-minor-free graphs. **Copyright**:- Copyright held by the authors.
**Length**:- The paper is 15 pages.
**Availability**:- The paper is available in PostScript (464k), gzipped PostScript (193k), and PDF (239k).
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**Related papers**:- ContractionTSP_SODA2007 (Approximation Algorithms via Contraction Decomposition)

See also other papers by Erik Demaine.

Last updated July 23, 2024 by Erik Demaine.