We introduce the Nondeterministic Strong Exponential Time Hypothesis (NSETH) as a natural extension of the Strong Exponential Time Hypothesis (SETH). We show that both refuting and proving NSETH would have interesting consequences. In particular we show that disproving NSETH would give new nontrivial circuit lower bounds. On the other hand, NSETH implies non-reducibility results, i.e. the absence of (deterministic) fine-grained reductions from SAT to a number of problems. As a consequence we conclude that unless this hypothesis fails, problems such as 3-sum, APSP and model checking of a large class of first-order graph properties cannot be shown to be SETH-hard using deterministic or zero-error probabilistic reductions.

Abstract. The best known approximation ratio for the shortest superstring problem is 2 11 23 (Mucha, 2012). In this note, we improve this bound for the case when the length of all input strings is equal to r, for r ≤ 7. For example, for strings of length 3 we get a 1 1 3 -approximation. An advantage of the algorithm is that it is extremely simple both to implement and to analyze. Another advantage is that it is based on de Bruijn graphs. Such graphs are widely used in genome assembly (one of the most important practical applications of the shortest common superstring problem). At the same time these graphs have only few applications in theoretical investigations of the shortest superstring problem.

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph G to graph H cannot be done in timeWe also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of |V (H)| o(|V (H)|) -time algorithm deciding if graph G is a subgraph of H. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph G to graph H cannot be done in timeWe also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of |V (H)| o(|V (H)|) -time algorithm deciding if graph G is a subgraph of H. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.

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