Tight Kernel Bounds for Problems on Graphs with Small Degeneracy
Marek Cygan, Fabrizio Grandoni, Danny Hermelin · ACM Transactions on Algorithms · 2017
Kernelization is a strong and widely applied technique in parameterized complexity. In a nutshell, a kernelization algorithm for a parameterized problem transforms in polynomial time a given instance of the problem into an equivalent instance whose size depends solely on the parameter. Recent years have seen major advances in the study of both upper and lower bound techniques for kernelization, and by now this area has become one of the major research threads in parameterized complexity. In this article, we consider kernelization for problems on d -degenerate graphs, that is, graphs such that any subgraph contains a vertex of degree at most d . This graph class generalizes many classes of graphs for which effective kernelization is known to exist, for example, planar graphs, H -minor free graphs, and H -topological-minor free graphs. We show that for several natural problems on d -degenerate graphs the best-known kernelization upper bounds are essentially tight. In particular, using intricate constructions of weak compositions, we prove that unless coNP ⊆ NP/poly: • D ominating S et has no kernels of size O ( k ( d −1)( d −3)−ε ) for any ε > 0. The current best upper bound is O ( k (d+1) 2 ). • I ndependent D ominating S et has no kernels of size O ( k d −4−ε ) for any ε > 0. The current best upper bound is O ( k d +1 ). • I nduced M atching has no kernels of size O ( k d −3−ε ) for any ε > 0. The current best upper bound is O ( k d ). To the best of our knowledge, D ominating S et is the the first problem where a lower bound with superlinear dependence on d (in the exponent) can be proved. In the last section of the article, we also give simple kernels for C onnected V ertex C over and C apacitated V ertex C over of size O ( k d ) and O ( k d +1 ), respectively. We show that the latter problem has no kernels of size O ( k d −ε ) unless coNP ⊆ NP/poly by a simple reduction from d -E xact S et C over (the same lower bound for C onnected V ertex C over on d -degenerate graphs is already known).