Tamás Király

On decomposing a hypergraph into k connected sub-hypergraphs

By applying the matroid partition theorem of J. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69 (1965) 67) to a hypergraphic generalization of graphic matroids, due to Lorea (Cahiers Centre Etudes Rech. Oper. 17 (1975) 289), we obtain a generalization of Tuttes disjoint trees theorem for hypergraphs. As a corollary, we prove for positive integers k and q that every (kq)-edge-connected hypergraph of rank q can be decomposed into k connected sub-hypergraphs, a well-known result for q = 2. Another by-product is a connectivity-type sufficient condition for the existence of k edge-disjoint Steiner trees in a bipartite graph.

On the orientation of graphs and hypergraphs

Graph orientation is a well-studied area of combinatorial optimization, one that provides a link between directed and undirected graphs. An important class of questions that arise in this area concerns orientations with connectivity requirements. In this paper we focus on how similar questions can be asked about hypergraphs, and we show that often the answers are also similar many known graph orientation theorems can be extended to hypergraphs, using the familiar uncrossing techniques. Our results also include a short proof and an extension of a theorem of Khanna et al. (Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Alogrithm, 2001, pp. 663-671), and a new orientation theorem that provides a characterization for (2k + 1)-edge-connected graphs.

Degree bounded matroids and submodular flows

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: We are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e)≤g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a submodular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, and the task is to find a minimum cost 0–1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v is between f(v) and g(v). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e)−2Δ+1≤|B∩e|≤g(e)+2Δ−1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ−1. For the second problem we can find a 0–1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v)−1 and g(v)+1. These results can be applied to obtain approximation algorithms for several combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree problem, the Minimum Bounded Degree Spanning Tree Union problem, the Minimum Bounded Degree Directed Cut Cover problem, and the Minimum Bounded Degree Graph Orientation problem.

On Polyhedra Related to Even Factors

As a common generalization of matchings and matroid intersection, W.H. Cunningham and J.F. Geelen introduced the notion of path-matching, which they generalized even further by introducing even factors of weakly symmetric digraphs. Later, a purely combinatorial approach to even factors was given by Gy. Pap and L. Szegő, who showed that the maximum even factor problem remains tractable in the class of hardly symmetric digraphs. The present paper shows a direct polyhedral way to derive weighted integer min-max formulae generalizing those previous results.

A new approach to splitting-off

A new approach to undirected splitting-off is presented in this paper. We study the behaviour of splitting-off algorithms when applied to the problem of covering a symmetric skew-supermodular set function by a graph. This hard problem is a natural generalization of many solved connectivity augmentation problems, such as local edgeconnectivity augmentation of graphs, global arc-connectivity augmentation of mixed graphs with undirected edges, or the node-to-area connectivity augmentation problem in graphs. Using a simple lemma we characterize the situation when a splitting-off algorithm can be stuck. This characterization enables us to give very simple proofs for the classical results mentioned above. Finally we apply our observations in generalizations of the above problems: we consider two connectivity augmentation problems in hypergraphs. The first is the local edgeconnectivity augmentation of undirected hypergraphs by hyperedges of minimum total size without increasing the rank. The second is global arc-connectivity augmentation of mixed hypergraphs by adding hyperedges of minimum total size without increasing the rank. We show that a greedy approach works in (almost) all of these cases.

Combined connectivity augmentation and orientation problems

Two important branches of graph connectivity problems are connectivity augmentation, which consists of augmenting a graph by adding new edges so as to meet a specified target connectivity, and connectivity orientation, where the goal is to find an orientation of an undirected or mixed graph that satisfies some specified edge-connection property. In the present work, an attempt is made to link the above two branches, by considering degree-specified and minimum cardinality augmentation of graphs so that the resulting graph admits an orientation satisfying a prescribed edge-connection requirement, such as (k,l)-edge-connectivity. The results are obtained by combining the supermodular polyhedral methods used in connectivity orientation with the splitting off operation, which is a standard tool in solving augmentation problems.

Stable Multicommodity Flows

We extend the stable flow model of Fleiner to multicommodity flows. In addition to the preference lists of agents on trading partners for each commodity, every trading pair has a preference list on the commodities that the seller can sell to the buyer. A blocking walk (with respect to a certain commodity) may include saturated arcs, provided that a positive amount of less preferred commodity is traded along the arc. We prove that a stable multicommodity flow always exists, although it is PPAD-hard to find one.

A note on kernels and Sperner's Lemma

The kernel-solvability of perfect graphs was first proved by Boros and Gurvich, and later Aharoni and Holzman gave a shorter proof. Both proofs were based on Scarfs Lemma. In this note we show that a very simple proof can be given using a polyhedral version of Sperners Lemma. In addition, we extend the Boros-Gurvich theorem to h-perfect graphs and to a more general setting.

Total Dual Integrality of Rothblum's Description of the Stable-Marriage Polyhedron

Rothblum showed that the convex hull of the stable matchings of a bipartite preference system can be described by an elegant system of linear inequalities. In this paper we prove that the description given by Rothblum is totally dual integral. We give a constructive proof based on the results of Gusfield and Irving on rotations, which gives rise to a strongly polynomial algorithm for finding an integer optimal dual solution.

Degree bounded forest covering

We prove that for an undirected graph with arboricity at most k+e, its edges can be decomposed into k forests and a subgraph with maximum degree ⌈ke+1/1-e⌉. The problem is solved by a linear programming based approach: we first prove that there exists a fractional solution to the problem, and then use a result on the degree bounded matroid problem by Kiraly, Lau and Singh [5] to get an integral solution.