Finding Triangles or Independent Sets; and Other Dual Pair Approximations

We revisit the algorithmic problem of finding a triangle in a graph (\textsc{Triangle Detection}), and examine its relation to other problems such as \textsc{3Sum}, \textsc{Independent Set}, and \textsc{Graph Coloring}. We obtain several new algorithms: \smallskip (I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study the range of a conjecture of P\v{a}tra\c{s}cu (2010) regarding the triangle detection problem. \smallskip (II) An algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n)$ (ie, linear) time: (i)~compute a $\Omega(1/\sqrt{n})$-approximation of a maximum independent set in $G$ or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks. \smallskip (III) An algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n^{3/2})$ time: (i)~compute an $\sqrt{n}$-approximation for \textsc{Graph Coloring} of $G$ or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks on dense graphs, with $m =\omega(n^{9/8})$. \smallskip (IV) The second and third results suggest the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the \emph{dual pair} concept we introduce. We discuss and provide several instances of dual-pair approximation.