Approximation algorithms via contraction decomposition

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.

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

“…The advantage of our methodology is that handling weighted graphs and subsettype problems are naturally incorporated, and thus it might be possible to combine all the steps for a potential PTAS into a single framework for H-minorfree graphs based on what we presented in this work. Hence, whereas our work is an important step towards this generalization, still a number of hard challenges remain; see also [8] for a further discussion on this matter. …”

“…The first is a generalization of the framework of Klein [15] for planar graphs that is based on finding a spanner for a problem, a subgraph containing a nearly optimal solution having length O(OPT). In Section 4.1 we show how to find such a spanner and in Section 4.2 we generalize Klein's framework to higher genus graphs using the techniques of Demaine et al [8]. In the second method, dynamic programming is done over the bricks of the mortar graph.…”

“…For embedded graphs with genus > 0, the concept of bounded dual radius does not apply in the same way. We deal with treewidth directly and obtain the following algorithm: we apply the Contraction Decomposition Theorem 13 [8] to B ÷ (MG) and contract a set of edges S ⋆ in B ÷ (MG). However, we apply a special weight to portal edges so as to prevent them from being included in S ⋆ .…”

A Randomized Rounding Approach to the Traveling Salesman Problem

“…Polynomial-time approximation schemes (PTAS) have been found for Euclidean [2], planar [24,3,28], or low-genus metrics [16,15] instances. However, the problem is known to be MAX SNP-hard [33] even when the distances one or two (a.k.a (1, 2)-TSP).…”

Computing the Shortest Essential Cycle

“…For example, algorithms that repeatedly cut a given surface along short, topologically nontrivial cycles have been used for removing topological noise from graphical models [21], finding short cut graphs for surface parametrization [18], computing shortest paths in a given homotopy class [12], approximating optimal traveling salesman tours in surface-embedded graphs [14], probabilistically embedding high-genus graphs into planar graphs [26,2], drawing abstract graphs in the plane with the fewest possible crossings [28], and testing isomorphism between graphs of fixed genus [27]. These and other applications have motivated a series of algorithms for computing optimal cycles with various topological properties [35,31,18,7,4,29,5,8,6,9].…”