Section: New Results

Graph and Probabilistic Algorithms

Finding the shortest path in a directed graph is one of the most important combinatorial optimization problems, having applications in a wide range of fields. In its basic version, however, the problem fails to represent situations in which the value of the objective function is determined not only by the choice of each single arc, but also by the combined presence of pairs of arcs in the solution. In this work [40] we model these situations as a Quadratic Shortest Path Problem, which calls for the minimization of a quadratic objective function subject to shortest-path constraints. We prove strong NP-hardness of the problem and analyze polynomially solvable special cases, obtained by restricting the distance of arc pairs in the graph that appear jointly in a quadratic monomial of the objective function. Based on this special case and problem structure, we devise fast lower bounding procedures for the general problem and show computationally that they clearly outperform other approaches proposed in the literature in terms of its strength.

A fundamental result by Karger (SODA 1994) states that for any $\lambda$-edge-connected graph with $n$ nodes, independently sampling each edge with probability $p=\Omega (logn/\lambda )$ results in a graph that has edge connectivity $\Omega \left(\lambda p\right)$, with high probability. In [27] we prove the analogous result for vertex connectivity, when sampling vertices. We show that for any $k$-vertex-connected graph $G$ with $n$ nodes, if each node is independently sampled with probability $p=\Omega \left(\sqrt{logn/k}\right)$, then the subgraph induced by the sampled nodes has vertex connectivity $\Omega \left(k{p}^2\right)$, with high probability. This bound improves upon the recent results of Censor-Hillel et al. (SODA 2014) and is existentially optimal.

This is a joint work with Keren Censor-Hillel (Technion), Mohsen Ghaffari (MIT), Bernhard Haeupler (Carnegie Mellon U.), and Fabian Kuhn (U. of Freiburg).