More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture

TL;DR

This paper explores the implications of falsifying the OV-conjecture and SETH, showing that such a failure would lead to unexpectedly fast algorithms for certain problems, challenging current complexity assumptions.

Contribution

It provides new evidence strengthening the hardness assumptions by demonstrating that their failure implies significant algorithmic breakthroughs.

Findings

If OV conjecture fails, faster algorithms exist for Zero-Weight-k-Clique and Min-Weight-k-Clique.

Failure of OV conjecture implies SAT for sparse TC1 circuits can be solved in sub-exponential time.

Weighted Clique conjecture would imply the OV conjecture, linking these hardness assumptions.

Abstract

The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension states there is no $\epsilon>0$ for which an $O(N^{2-\epsilon})\mathrm{poly}(D)$ time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size $N$ that contains $D$-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed $\epsilon>0$ such that: (1) For all $d$ and all large enough $k$, there is a randomized algorithm that takes $O(n^{(1-\epsilon)k})$ time to solve the Zero-Weight-$k$-Clique and Min-Weight-$k$-Clique problems on $d$-hypergraphs with $n$ vertices. As a consequence, the OV-conjecture is implied by the Weighted Clique conjecture. (2) For all $c$, the satisfiability of sparse TC1 circuits on $n$ inputs (that is, circuits with $cn$ wires, depth $c\log n$, and negation, AND, OR, and threshold gates) can be computed in time ${O((2-\epsilon)^n)}$.