Assigning channels via the meet-in-the-middle approach

We study the complexity of the Channel Assignment problem. By applying the meet-in-the-middle approach we get an algorithm for the $\ell$-bounded Channel Assignment (when the edge weights are bounded by $\ell$) running in time $O^*((2\sqrt{\ell+1})^n)$. This is the first algorithm which breaks the $(O(\ell))^n$ barrier. We extend this algorithm to the counting variant, at the cost of slightly higher polynomial factor. A major open problem asks whether Channel Assignment admits a $O(c^n)$-time algorithm, for a constant $c$ independent of $\ell$. We consider a similar question for Generalized T-Coloring, a CSP problem that generalizes \CA. We show that Generalized T-Coloring does not admit a $2^{2^{o\left(\sqrt{n}\right)}} {\rm poly}(r)$-time algorithm, where $r$ is the size of the instance.Comment: SWAT 2014: 282-29

A Probabilistic Bound on the Basic Role Mining Problem and Its Applications

Abstract In this paper we describe a new probabilistic approach to the role engineering process for RBAC. In particular, we address the issue of minimizing the number of roles, problem known in literature as the Basic Role Mining Problem (basicRMP). We leverage the equivalence of the above issue with the vertex coloring problem. Our main result is the proof that the minimum number of roles is sharply concentrated around its expected value. A further contribution is to show how this result can be applied as a stop condition when striving to find out an approximation for the basicRMP. We also show that the proposal can be used to decide whether it is advisable to undertake the efforts to renew an RBAC state. Note that both these applications can result in a substantial saving of resources. A thorough analysis using advanced probabilistic tools supports our results. Finally, further relevant research directions are also highlighted.

Integer realizations of disk and segment graphs

A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. Every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the integer grid and equal (integer) radius; and every segment graph can be realized by segments whose endpoints lie on the integer grid. Here we show that there exist disk graphs on n vertices such that in every realization by integer disks at least one coordinate or radius is 22Ω(n) and on the other hand every disk graph can be realized by disks with integer coordinates and radii that are at most 22O(n) ; and we show the analogous results for unit disk graphs and segment graphs. For (unit) disk graphs this answers a question of Spinrad, and for segment graphs this improves over a previous result by Kratochvíl and Matoušek

The number of disk graphs

A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labeled unit disk and disk graphs on nn vertices. We show that the number of unit disk graphs on nn vertices is n2n⋅α(n)nn2n⋅α(n)n and the number of disk graphs on nn vertices is n3n⋅β(n)nn3n⋅β(n)n, where α(n)α(n) and β(n)β(n) are Θ(1)Θ(1). We conjecture that there exist constants α,βα,β such that the number of unit disk graphs is n2n⋅(α+o(1))nn2n⋅(α+o(1))n and the number of disk graphs is n3n⋅(β+o(1))nn3n⋅(β+o(1))n

Martingales and Locality in Distributed Computing

\ud We use Martingale inequalities to give a simple and uniform analysis of two families of distributed randomised algorithms for edge colouring graphs