Approximately Counting Triangles in Sublinear Time

We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a {\em sublinear-time\/} algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries and neighbor queries. We show that for any given approximation parameter $0<\epsilon<1$, the algorithm provides an estimate $\widehat{t}$ such that with high constant probability, $(1-\epsilon)\cdot t< \widehat{t}<(1+\epsilon)\cdot t$, where $t$ is the number of triangles in the graph $G$. The expected query complexity of the algorithm is $\!\left(\frac{n}{t^{1/3}} + \min\left\{m, \frac{m^{3/2}}{t}\right\}\right)\cdot {\rm poly}(\log n, 1/\epsilon)$, where $n$ is the number of vertices in the graph and $m$ is the number of edges, and the expected running time is $\!\left(\frac{n}{t^{1/3}} + \frac{m^{3/2}}{t}\right)\cdot {\rm poly}(\log n, 1/\epsilon)$. We also prove that $\Omega\!\left(\frac{n}{t^{1/3}} + \min\left\{m, \frac{m^{3/2}}{t}\right\}\right)$ queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to polylogarithmic factors in $n$ (and the dependence on $1/\epsilon$).Comment: To appear in the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015

Towards Resistance Sparsifiers

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+\epsilon)$-resistance sparsifier of size $\tilde O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\Omega(n/\epsilon^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers

Between Treewidth and Clique-width

Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]. We therefore seek a structural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits, branch decompositions and the work of Vatshelle [18] on Maximum Matching-width, we consider the graph parameter sm-width which lies between treewidth and clique-width. Some graph classes of unbounded treewidth, like distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized by sm-width

Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem