An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

TL;DR

This paper introduces an efficient partitioning oracle for bounded-treewidth graphs that significantly reduces query complexity, enabling faster property testing and approximation algorithms for various graph problems.

Contribution

It presents the first polynomial-query partitioning oracle for bounded-treewidth graphs, improving efficiency over previous exponential-query methods.

Findings

Oracle makes only O(poly(1/epsilon)) queries per partition query.

Enables poly(1/epsilon)-time property testing for graph classes like k-outerplanar and series-parallel.

Allows approximation of maximum matching, vertex cover, and dominating set sizes within epsilon*n additive error.

Abstract

Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any epsilon > 0, a partitioning oracle provides query access to a fixed partition of the input bounded-degree minor-free graph, in which every component has size poly(1/epsilon), and the number of edges removed is at most epsilon*n, where n is the number of vertices in the graph. However, the oracle of Hassidimet al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/epsilon)) queries to the input graph to answer each query about the partition. Examples of bounded-treewidth graph classes include k-outerplanar graphs for fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/epsilon)-time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/epsilon)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive epsilon*n in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/epsilon) time whether the input bounded-treewidth graph is k-colorable or perfect.