Optimal state amalgamation is NP-hard

A state amalgamation of a directed graph is a node contraction which is only permitted under certain configurations of incident edges. In symbolic dynamics, state amalgamation and its inverse operation, state splitting, play a fundamental role in the theory of subshifts of finite type (SFT): any conjugacy between SFTs, given as vertex shifts, can be expressed as a sequence of symbol splittings followed by a sequence of symbol amalgamations. It is not known whether determining conjugacy between SFTs is decidable. We focus on conjugacy via amalgamations alone and consider the simpler problem of deciding whether one can perform $k$ consecutive amalgamations from a given graph. This problem also arises when using symbolic dynamics to study continuous maps, where one seeks to coarsen a Markov partition in order to simplify it. We show that this state amalgamation problem is NP-complete by reduction from the hitting set problem, thus giving further evidence that classifying SFTs up to conjugacy may be undecidable.

A Hybrid CPU-GPU-MIC Algorithm for the Hitting Set Problem

We present a hybrid exact algorithm for the Hitting Set Problem (HSP) for highly heterogeneous CPU-GPU-MIC platforms. With several techniques that permit an efficient exploitation of each architecture, low communication cost and effective load balancing, we were able to solve large HSP instances in reasonable time, achieving good performance and scalability. We obtained speedups of up to 25.32 in comparison with using two six-core CPUs and exact HSP solutions for instances with tens of thousands of variables in less than 5 hours. These results reinforce the statement that heterogeneous clusters of CPUs, GPUs and MICs can be used efficiently for high-performance computing.

3-Hitting set on bounded degree hypergraphs: Upper and lower bounds on the kernel size

We study upper and lower bounds on the vertex-kernel size for the 3-HITTING SET problem on hypergraphs of degree at most 3, denoted 3-3-HS. We first show that, unless P = NP, 3-3-HS on 3-uniform hypergraphs does not have a vertex-kernel of size at most 35k/19 > 1.8421k. We then give a 4k - k0.2692 vertex-kernel for 3-3-hs that is computable in time O(k2). We do not assume that the hypergraph is 3-uniform for the vertex-kernel upper bound results. This result improves the upper bound of 4k on the vertex-kernel size for 3-3-HS, implied by the results of Wahlström.