2009/10/23 by Stefan Kratsch, Kratsch, Stefan, Magnus Wahlström +2
Computer Science · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Constraint Satisfaction and Optimization #F.2.2 #FOS: Computer and information sciences #cs.CC
paper · pdf · doi:10.48550/arxiv.0910.4518
arxiv created 2009/10/23 · openalex publication_date 2009/10/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A parameterized problem consists of a classical problem and an additional component, the so-called parameter. This point of view allows a formal definition of preprocessing: Given a parameterized instance (I,k), a polynomial kernelization computes an equivalent instance (I',k') of size and parameter bounded by a polynomial in k. We give a complete classification of Min Ones Constraint Satisfaction problems, i.e., Min Ones SAT(Γ), with respect to admitting or not admitting a polynomial kernelization (unless NP ⊆ coNP/poly). For this we introduce the notion of mergeability. If all relations of the constraint language Γare mergeable, then a new variant of sunflower kernelization applies, based on non-zero-closed cores. We obtain a kernel with O(kd+1) variables and polynomial total size, where d is the maximum arity of a constraint in Γ, comparing nicely with the bound of O(kd-1) vertices for the less general and arguably simpler d-Hitting Set problem. Otherwise, any relation in Γthat is not mergeable permits us to construct a log-cost selection formula, i.e., an n-ary selection formula with O(log n) true local variables. From this we can construct our lower bound using recent results by Bodlaender et al. as well as Fortnow and Santhanam, proving that there is no polynomial kernelization, unless NP ⊆ coNP/poly and the polynomial hierarchy collapses to the third level.