Nearly Optimal Time Bounds for kPath in Hypergraphs
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In the kPath problem the input is a graph G and an integer k, and the goal is to decide whether G contains a simple path of length k. The fastest algorithm known for the directed version of this problem runs in time 2^kpoly(n) [Williams, IPL, 2009] and for the undirected version in time 1.66^kpoly(n) [Björklund et al., JCSS, 2014], while the only known lower bound for these problems is 2^Ω(k)poly(n) assuming the Exponential Time Hypothesis (ETH), and finding any conditional lower bound with explicit constant in the exponent has been an important open problem. Motivated by this challenge, we study the kHyperPath problem which is a generalization of kPath to finding tight paths in r-uniform hypergraphs on n nodes and m hyperedges, namely, a sequence of k nodes such that for every r consecutive nodes in the sequence there is an edge in the graph containing them. This problem was investigated in its cycle variant for large values of the uniformity parameter r∈ [ω(k),k) [Lincoln et al., SODA 2018], where a conditional lower bound of Õ(n^k) was presented. Considering smaller values of r, for every integer r≥ 3 we show an algorithm with running time (2+ε)^k m^O(r r/ε) for kHyperPath even for the directed version of this problem, which additionally has a permutation for each edge, and requires the path to comply with those permutations. This running time is nearly optimal as a (2-γ)^kpoly(m)-time algorithm with γ independent of r is probably impossible by our main result, as follows. We present a hierarchy of conditional lower bounds assuming the Set Cover Conjecture, where for even undirected kHyperPath with explicit constants in the exponents, where for growing values of r the problem requires time that approaches 2^kpoly(m).
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