Consequences of APSP, triangle detection, and 3SUM hardness for separation between determinism and non-determinism

Abstract

Let NDTIME(f(n),g(n)) denote the class of problems solvable in O(g(n)) time by a multi-tape Turing machine using an f(n)-bit non-deterministic oracle, and let DTIME(g(n)) = NDTIME(0, g(n)). We show that if the all-pairs shortest paths problem (APSP) for directed graphs with N vertices and integer edge weights within a super-exponential range { −2Nk+o(1),....,2Nk+o(1) }, k≥1 does not admit a truly subcubic algorithm then for any ∈>0, NDTIME([ 1/2 log2 n ], n)⊆DTIME(n1+12+k−∈). If the APSP problem does not admit a truly subcubic algorithm already when the edge weights are of moderate size then we obtain an even stronger implication, namely that for any ∈>0, NDTIME([ 1/2 log2 n ], n)⊆DTIME(n1.5−∈). Similarly, we show that if the triangle detection problem (DT) in a graph on N vertices does not admit a truly sub-Nω -time algorithm then for any ∈>0, NDTIME([ 1/2 log2 n ], n)⊆DTIME(nw/2−∈), where ω stands for the exponent of fast matrix multiplication. For the more general problem of detecting a minimum weight ℓ-clique (MWCℓ) in a graph with edge weights of moderate size, we show that the non-existence of truly sub−Nℓ−time algorithm yields for any ∈>0, NDTIME((ℓ−2)[ 12 log2n ],n)⊆DTIME(n1+ℓ−22−∈). Next, we show that if 3SUM for N integers in { −2Nk+o(1),....2Nk+o(1) } for some k≥0, does not admit a truly subquadratic algorithm then for any ∈>0, NDTIME([ log2n ],n)⊆DTIME(n1+11+k−∈). Finally, we observe that the Exponential Time Hypothesis (ETH) implies NDTIME([ k log2n ],n)⊆DTIME(n) for some k>0, while the strong ETH (SETH) yields for any ∈>0, NDTIME([ log2n ],n)⊆DTIME(n2−ε). For comparison, the strongest known result on separation between non-deterministic and deterministic time only asserts NDTIME(O(n),n)⊆DTIME(n).