I am interested in the following. Let the inputs of the circuit correspond to the arcs of a directed graph. The circuit has to output 1 iff there is a directed path from a given node S to a given node T such that all arcs on that path have input value 1.

I can construct Boolean circuits that have size linear in nm where n is the number of vertices in the graph and m is the number of arcs. I would like to know if smaller circuits are possible. Especially, circuits that are linear in n+m.

  • 1
    $\begingroup$ Bad news: no lower bound of the form $\Omega(nm)$ is known, even for <i>monotone</i> circuits. The upper bound $O(nm)$ comes directly from Bellman-Ford DP algorithm. Actually, Floyd-Warshall gives a monotone circuit of size $O(n^3)$ even for the "all pairs connectivity" function, where existence of paths between all pairs $s$ and $t$ are computed (so, we already need $\Omega(n^2)$ output gates!). Here $\Omega(n^3)$ lower bound is already known (even for graphs with two layers); see e.g. Sect. 6.8 of Wegener's book. $\endgroup$ – Stasys Dec 22 '14 at 11:26

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.