András Sebö

On Metric Generators of Graphs

We study generators of metric spaces--sets of points with the property that every point of the space is uniquely determined by the distances from their elements. Such generators put a light on seemingly different kinds of problems in combinatorics that are not directly related to metric spaces. The two applications we present concern combinatorial search: problems on false coins known from the borderline of extremal combinatorics and information theory; and a problem known from combinatorial optimization--connected joins in graphs.We use results on the detection of false coins to approximate the metric dimension (minimum size of a generator for the metric space defined by the distances) of some particular graphs for which the problem was known and open. In the opposite direction, using metric generators, we show that the existence of connected joins in graphs can be solved in polynomial time, a problem asked in a survey paper of Frank. On the negative side we prove that the minimization of the number of components of a join is NP-hard.We further explore the metric dimension with some problems. The main problem we are led to is how to extend an isometry given on a metric generator of a metric space.

On Integer Multiflow Maximization

Generalizing the two-commodity flow theorem of Rothschild and Whinston [Oper. Res., 14 (1966), pp. 377--387] and the multiflow theorem of Lovasz [Acta Mat. Akad. Sci. Hungaricae, 28 (1976), pp. 129--138] and Cherkasky [Ekonom.-Mat. Metody, 13 (1977), pp. 143--151], Karzanov and Lomonosov [Mathematical Programming, O. I. Larichev, ed., Institute for System Studies, 1978, pp. 59--66] in 1978 proved a min-max theorem on maximum multiflows. Their original proof is quite long and technical and relies on earlier investigations into metrics. The main purpose of the present paper is to provide a relatively simple proof of this theorem. Our proof relies on the locking theorem, which is another result of Karzanov and Lomonosov, and the polymatroid intersection theorem of Edmonds [Combinatorial Structures and Their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schonheim, eds., Gordon and Breach, 1970, pp. 69--87]. For completeness, we also provide a simplified proof of the locking theorem. Finally, we introduce the notion of a node demand problem and, as another application of the locking theorem, we derive a feasibility theorem concerning it. The presented approach gives rise to (combinatorial) polynomial-time algorithms.

Improving on the 1.5-Approximation of a Smallest 2-Edge Connected Spanning Subgraph

We give a

A QUICK PROOF OF SEYMOUR'S THEOREM ON t-JOINS

\frac{17}{12}

Optimal cuts in graphs and statistical mechanics

-approximation algorithm for the following NP-hard problem: Given a simple undirected graph, find a 2-edge connected spanning subgraph that has the minimum number of edges. The best previous approximation guarantee was

THE SCHRIJVER SYSTEM OF ODD JOIN POLYHEDRA

\frac{3}{2}

Minsquare Factors and Maxfix Covers of Graphs

. If the well-known

The Path-Packing Structure of Graphs

\frac{4}{3}

On Combinatorial Properties of Binary Spaces

conjecture for the metric traveling salesman problem holds, then the optimal value (minimum number of edges) is at most

Finding the t -join structure of graphs

\frac{4}{3}