Intersection Non-emptiness and Hardness Within Polynomial Time

Abstract

We establish strong connections between the non-emptiness of intersection problem for two and three DFA’s over a binary alphabet and the triangle finding and 3sum problems. In particular, we introduce efficient reductions from triangle finding to non-emptiness of intersection for two DFA’s over a binary alphabet and from 3sum to non-emptiness of intersection for three DFA’s over a binary alphabet. Additionally, in our main result, we show that for every $$\alpha \ge 2$$ , non-emptiness of intersection for three DFA’s over a unary alphabet can be solved in $$O(n^{\frac{\alpha }{2}})$$ time if and only if triangle finding can be solved in $$O(n^{\alpha })$$ time.