Apparently this was due to Pătraşcu, but in this report on the ECCC server, Viola states that 3XOR can be reduced to listing triangles.

Assume that given a graph in adjacency list format, with $m$ edges, $z$ triangles, $O(m)$ nodes, one can list $\min\{z,m\}$ triangles in time $m^{(4/3)-\varepsilon}$ for $\epsilon>0$ constant. Then one can solve 3XOR on a set of size $n$ in time $n^{2-\epsilon'}$ with error $1\%$ for a constant $\epsilon'>0.$

My question is the reverse. If one were to have an efficient algorithm for 3XOR, what could one say about efficiently listing triangles?


1 Answer 1


[Copying here the answer from here]

In this paper, Corollary 1, we show that if you solve 3XOR in close to linear time then you can list $t$ triangles in a graph with $m$ edges in time $m\cdot t^{1/3},$ ignoring lower-order terms.


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