Baker game and polynomial-time approximation schemes

TL;DR

This paper formalizes Baker's technique as a game, proving that certain optimization problems have polynomial-time approximation schemes on graph classes where the game can be won quickly, extending prior results without complex theorems.

Contribution

It introduces a Baker game framework and shows all proper minor-closed graph classes allow PTAS for first-order expressible problems via this game approach.

Findings

Baker game can be won in a constant number of rounds on proper minor-closed classes.

First-order expressible monotone problems admit PTAS on these classes.

The approach avoids using the minor structure theorem.

Abstract

Baker devised a technique to obtain approximation schemes for many optimization problems restricted to planar graphs; her technique was later extended to more general graph classes. In particular, using the Baker's technique and the minor structure theorem, Dawar et al. gave Polynomial-Time Approximation Schemes (PTAS) for all monotone optimization problems expressible in the first-order logic when restricted to a proper minor-closed class of graphs. We define a Baker game formalizing the notion of repeated application of Baker's technique interspersed with vertex removal, prove that monotone optimization problems expressible in the first-order logic admit PTAS when restricted to graph classes in which the Baker game can be won in a constant number of rounds, and prove without use of the minor structure theorem that all proper minor-closed classes of graphs have this property.