Finding Triangles or Independent Sets; and Other Dual Pair Approximations

We revisit the algorithmic problem of finding a triangle in a graph (Triangle Detection), and examine its relation to other problems such as 3Sum, Independent Set, and Graph Coloring. We obtain several new algorithms: (I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study a question of Pˇatraşcu (2010) regarding the triangle detection problem. (II) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚 + 𝑛) (i.e., linear) time: (i) compute a Ω(1/√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. (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 √𝑛-approximation for 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 = (𝑛 9/8 ). (IV) Results (II) and (III) above 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 dual pair concept we introduce. We provide several instances of dual-pair approximation, relating Longest Path, (1,2)-TSP, and other NP-hard problems.