Interdiction Problems on Planar Graphs

Interdiction problems are leader-follower games in which the leader is allowed to delete a certain number of edges from the graph in order to maximally impede the follower, who is trying to solve an optimization problem on the impeded graph. We introduce approximation algorithms and strong NP-completeness results for interdiction problems on planar graphs. We give a multiplicative $(1 + \epsilon)$-approximation for the maximum matching interdiction problem on weighted planar graphs. The algorithm runs in pseudo-polynomial time for each fixed $\epsilon > 0$. We also show that weighted maximum matching interdiction, budget-constrained flow improvement, directed shortest path interdiction, and minimum perfect matching interdiction are strongly NP-complete on planar graphs. To our knowledge, our budget-constrained flow improvement result is the first planar NP-completeness proof that uses a one-vertex crossing gadget.Comment: 25 pages, 9 figures. Extended abstract in APPROX-RANDOM 201

Attraction time for strongly reinforced walks

We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarr\`{e}s [Ann. Probab. (2007), to appear], that the attracting edge emerges with probability 1 whenever the underlying graph is locally bounded. We study the asymptotic behavior of the tail distribution of the (random) time of attraction. In particular, we obtain exact (up to a multiplicative constant) asymptotics if the underlying graph has two edges. Next, we show some extensions in the setting of finite graphs, and infinite graphs with bounded degree. As a corollary, we obtain the fact that if the reinforcement weight has the form $w(k)=k^{\rho}$, $\rho>1$, then (universally over finite graphs) the expected time to attraction is infinite if and only if $\rho\leq1+\frac{1+\sqrt{5}}{2}$.Comment: Published in at http://dx.doi.org/10.1214/08-AAP564 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Weyl Tensors, Strongly Regular Graphs, Multiplicative Characters, and a Quadratic Matrix Equation

Let $S$ be a real symmetric $n\times n$-matrix with zeros on the diagonal and let $\theta$ be a real number. We show that the set of quadratic equations $$\sum_{k}S_{i,k}S_{k,j}+S_{i,j}^2=\theta S_{i,j},\text { for }i<j,$$ has solutions $(S,\theta)$, with $S\neq0$, in all dimensions $n\geq 4$. % The equations originate from a question about the Riemannian curvature tensor. Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields