FPT is characterized by useful obstruction sets

Abstract

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-PATHWIDTH, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions. ACM Reference Format:Fellows, M. R. and Jansen, B. M. P. 2014. FPT is characterized by useful obstruction sets: Connecting algorithms, kernels, and quasi-orders ACM Trans.

Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

“…In several settings (cf. [12]), the optimal kernel size matches the size of minimal obstructions in a problem-specific partial order. This is the case for d-nae-sat, whose kernel with O(n d−1 ) clauses matches the fact that critically 3-chromatic d-uniform hypergraphs have at most O(n d−1 ) hyperedges.…”