Daniel Stefankovic

Recognizing string graphs in NP

A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph [18, 20]. These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NP-complete, since Kratochvíl [12] showed that the recognition problem is NP-hard. The result has consequences for the computational complexity of problems in graph drawing, and topological inference.

Removing even crossings

An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Toth proved that a graph can always be redrawn so that its even edges are not involved in any intersections. We give a new and significantly simpler proof of the stronger statement that the redrawing can be done in such a way that no new odd intersections are introduced. We include two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowskis theorem), and the new result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.

Decidability of string graphs

We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvil and Matousek (J. Combin. Theory. Ser. B 53 (1991).) and settles the long-standing open problem of the decidability of string graph recognition (Bell System Tech. J. 45 (1996) 1639; Open Problem at Fifth Hungarian Collogium on Combinatories, 1976). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Proceedings of the 14th International Joint Conference on Artificial Intelligence, 1995, p. 901). The general theory of Euler diagrams turns out to be as hard as second-order arithmetic.

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Recent inapproximability results of Sly ( 2010 ), together with an approximation algorithm presented by Weitz ( 2006 ), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λ c (

Adaptive simulated annealing: A near-optimal connection between sampling and counting

\mathbb{T}_{\Delta}

Negative examples for sequential importance sampling of binary contingency tables

) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented an FPTAS for the partition function when λ c (

Pitfalls of Heterogeneous Processes for Phylogenetic Reconstruction

\mathbb{T}_{\Delta}

On the computational complexity of Nash equilibria for (0, 1) bimatrix games

) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists e Δ > 0 such that (unless RP = NP) there is no FPRAS for approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λ c (

Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting

\mathbb{T}_{\Delta}

Phylogeny of Mixture Models: Robustness of Maximum Likelihood and Non-Identifiable Distributions

) c (