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

Abstract

We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu (2007) from planar graphs to bounded-genus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to bounded-genus graphs.

Minor-Free Graphs Have Light Spanners

“…Klein [15] improved upon this running time to 2 O( 1 2 ) n by modifying the PTAS framework, using the same light spanner. Borradaile, Demaine and Tazari generalized Klein's EPTAS to bounded genus graphs [4].…”

“…Except for TSP, many connectivity problems [4] have PTASes for bounded genus graphs but are not known to have PTASes for H-minor-free graphs -for example, subset TSP and Steiner tree. The PTASes for these problems rely on having a light subgraph that approximates the optimal solution within 1 + (and hence is spanner-like).…”

Shortest non-trivial cycles in directed surface graphs

“…A surface is non-orientable if it contains a subset homeomorphic to the Möbius band, and orientable otherwise. 1 Janiga and Koubek actually claim an algorithm to compute the minimum (s, t)-cut, but their algorithm has a subtle error [35], which may lead to an incorrect result when the minimum (t, s)-cut is smaller than the minimum (s, t)-cut.…”

“…Examples include algorithms for probabilistically embedding high-genus graphs into planar graphs [3,30], drawing abstract graphs in the plane with few crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees [1], and removing topological noise from surface models [27,47]. In all these applications, cutting along the shortest possible cycle is preferred or even required.…”

“…The shortest non-trivial directed cycle in an annular graph is dual to either the minimum (s, t)-cut or the minimum (t, s)-cut in the directed planar dual graph, whichever has smaller capacity, where s and t are the dual vertices corresponding to the boundaries of the annulus. Janiga and Koubek [34] describe an adaptation of Reif's algorithm that can find this cycle 1 in O(n log 2 n/ log log n) time; the running time of their algorithm can be improved to O(n log n) using more recent algorithms for shortest paths [29,38]. The same running time can also be achieved by recent planar maximum flow algorithms [2,20,45].…”