Baker game and polynomial-time approximation schemes

Abstract

Baker [1] 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. [5] 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.

PTAS for Sparse General-Valued CSPs

“…Dvořák [Dvo20] defined a Baker class as any class of graphs that can be dealt with in the above ways. Informally, a class of graphs is Baker if any graph in the class can be reduced to an empty graph by a bounded number of operations: either removing a single vertex, or selecting some layering λ and continuing separately with every interval in that layering (each subgraph induced by λ −1 (I) for intervals I of at most some size).…”

“…Dvořák's definition considers graphs with a total ordering of their vertex set. Roughly speaking, this is to restrict the definition to "monotone" strategies, where the single vertices to be deleted are decided upfront -this restriction won't be important for us, but we will state it as in [Dvo20] (let us also remark that layerings are not restricted by the ordering). Finally, the definition is made a bit complicated by the fact that the number of consecutive layers we may need to include in an interval may depend on how deep we go (how many iterations of the game are done).…”

“…To remedy this, Dvořák [Dvo20] later defined Baker classes and proved that (an effective version of) this condition suffices to provide a PTAS to all monotone optimisation problems expressible in first-order logic (including of course Vertex Cover). Very roughly, a class of graphs is Baker if one can reduce each graph in it to the empty graph by a bounded number of the following steps: either remove a single vertex, or select a breadth-first-search layering and recurse into all subgraphs that can be induced by a few consecutive layers.…”

“…Dvořák proved that the family of Baker classes still includes all the examples discussed above; on the other hand, it is included in the family of fractionally-treewidth-fragile classes (and hence less general, though the inclusion is not known to be strict). It is worth mentioning that proper minor-closed graph classes can be shown to be Baker (and fractionally-treewidth-fragile) relatively easily, without using the Graph Minor Structure theorem, in contrast to the earlier, less general definitions (see [Dvo20] for details).…”

Approximation schemes for bounded distance problems on fractionally treewidth-fragile graphs

“…• All of the problems mentioned above (an more) are known to admit PTAS in planar graphs by a layering argument of Baker [3]; this approach can be extended to some related graph classes, including all proper minor-closed classes [6,12].…”