Martin Koutecký

Fixed Parameter Complexity of Distance Constrained Labeling and Uniform Channel Assignment Problems

We study computational complexity of the class of distance-constrained graph labeling problems from the fixed parameter tractability point of view. The parameters studied are neighborhood diversity and clique width.

Parameterized Resiliency Problems via Integer Linear Programming

We introduce an extension of decision problems called resiliency problems. In resiliency problems, the goal is to decide whether an instance remains positive after any (appropriately defined) perturbation has been applied to it. To tackle these kinds of problems, some of which might be of practical interest, we introduce a notion of resiliency for Integer Linear Programs (ILP) and show how to use a result of Eisenbrand and Shmonin (Math. Oper. Res., 2008) on Parametric Linear Programming to prove that ILP Resiliency is fixed-parameter tractable (FPT) under a certain parameterization. To demonstrate the utility of our result, we consider natural resiliency versions of several concrete problems, and prove that they are FPT under natural parameterizations. Our first results concern a four-variate problem which generalizes the Disjoint Set Cover problem and which is of interest in access control. We obtain a complete parameterized complexity classification for every possible combination of the parameters. Then, we introduce and study a resiliency version of the Closest String problem, for which we extend an FPT result of Gramm et al. (Algorithmica, 2003). We also consider problems in the fields of scheduling and social choice. We believe that many other problems can be tackled by our framework.

Parameterized shifted combinatorial optimization

Abstract Shifted combinatorial optimization is a new nonlinear optimization framework broadly extending standard combinatorial optimization, involving the choice of several feasible solutions simultaneously. This framework captures well studied and diverse problems, from sharing and partitioning to so-called vulnerability problems. In particular, every standard combinatorial optimization problem has its shifted counterpart, typically harder. Already with explicitly given input set SCO may be NP -hard. Here we initiate a study of the parameterized complexity of this framework. First we show that SCO over an explicitly given set parameterized by its cardinality may be in XP , FPT or P , depending on the objective function. Second, we study SCO over sets definable in MSO logic (which includes, e.g., the well known MSO-partitioning problems). Our main results are that SCO over MSO definable sets is in XP parameterized by the MSO formula and treewidth (or clique-width) of the input graph, and W [1] -hard even under further severe restrictions.

Opinion Diffusion and Campaigning on Society Graphs

We study the effects of campaigning, where the society is partitioned into voter clusters and a diffusion process propagates opinions in a network connecting the clusters. Our model is very powerful and can incorporate many campaigning actions, various partitions of the society into clusters, and very general diffusion processes. Perhaps surprisingly, we show that computing the cheapest campaign for rigging a given election can usually be done efficiently, even with arbitrarily-many voters. Moreover, we report on certain computational simulations.

Parameterized Shifted Combinatorial Optimization

Shifted combinatorial optimization is a new nonlinear optimization framework which is a broad extension of standard combinatorial optimization, involving the choice of several feasible solutions at a time. This framework captures well studied and diverse problems ranging from so-called vulnerability problems to sharing and partitioning problems. In particular, every standard combinatorial optimization problem has its shifted counterpart, which is typically much harder. Already with explicitly given input set the shifted problem may be NP-hard. In this article we initiate a study of the parameterized complexity of this framework. First we show that shifting over an explicitly given set with its cardinality as the parameter may be in XP, FPT or P, depending on the objective function. Second, we study the shifted problem over sets definable in MSO logic (which includes, e.g., the well known MSO partitioning problems). Our main results here are that shifted combinatorial optimization over MSO definable sets is in XP with respect to the MSO formula and the treewidth (or more generally clique-width) of the input graph, and is W[1]-hard even under further severe restrictions.

Parameterized complexity of distance labeling and uniform channel assignment problems

Abstract We rephrase the Distance labeling problem as a specific uniform variant of the Channel Assignment problem and show that the latter one is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, the Distance labeling problem is FPT when parameterized by the neighborhood diversity, the maximum p i and k . This is indeed a more general answer to an open question of Fiala et al.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. Finally, we show that the uniform variant of the Channel Assignment problem becomes NP -complete when generalized to graphs of bounded clique width.