Zoltán Király

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.

Better and Simpler Approximation Algorithms for the Stable Marriage Problem

We first consider the problem of finding a maximum stable matching if incomplete lists and ties are both allowed, but ties only for one gender. For this problem we give a simple, linear time 3/2-approximation algorithm, improving on the best known approximation factor 5/3 of Irving and Manlove [5]. Next, we show how this extends to the Hospitals/Residents problem with the same ratio if the residents have strict orders. We also give a simple linear time algorithm for the general problem with approximation factor 5/3, improving the best known 15/8-approximation algorithm of Iwama, Miyazaki and Yamauchi [7]. For the cases considered in this paper it is NP-hard to approximate within a factor of 21/19 by the result of Halldorsson et al. [3]. Our algorithms not only give better approximation ratios than the cited ones, but are much simpler and run significantly faster. Also we may drop a restriction used in [5] and the analysis is substantially more moderate.

Linear Time Local Approximation Algorithm for Maximum Stable Marriage

We consider a two-sided market under incomplete preference lists with ties, where the goal is to find a maximum size stable matching. The problem is APX-hard, and a 3/2-approximation was given by McDermid [1]. This algorithm has a non-linear running time, and, more importantly needs global knowledge of all preference lists. We present a very natural, economically reasonable, local, linear time algorithm with the same ratio, using some ideas of Paluch [2]. In this algorithm every person make decisions using only their own list, and some information asked from members of these lists (as in the case of the famous algorithm of Gale and Shapley). Some consequences to the Hospitals/Residents problem are also discussed.

Graph orientations with edge-connection and parity constraints

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge-connected graph with |V| + |E| even has a (k-1)-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge-disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s.

On the Combinatorics of Projective Mappings

We consider composition sets of one-dimensional projective mappings and prove that small composition sets are closely related to Abelian subgroups.

Packing paths of length at least two

We give a simple proof for Kanekos theorem which gives a sufficient and necessary condition for the existence of vertex disjoint paths in a graph, each of length at least two, that altogether cover all vertices of the original graph. Moreover we generalize this theorem and give a formula for the maximum number of vertices that can be covered by such a path system.

On-line 3-chromatic graphs—II: critical graphs

Abstract On-line coloring of a graph is the following process. The graph is given vertex by vertex (with adjacencies to the previously given vertices) and for the actual vertex a color different from the colors of the neighbors must be irrevocably assigned. The on-line chromatic number of a graph G, χ ∗ (G) is the minimum number of colors needed to color on-line the vertices of G (when it is given in the worst order). A graph G is on-line k-critical if χ ∗ (G)=k , but χ ∗ (G′) for all proper induced subgraphs G′ ⊂ G. We show that there are finitely many (51) connected on-line 4-critical graphs but infinitely many disconnected ones. This implies that the problem whether χ ∗ (G) ⩽ 3 is polynomially solvable for connected graphs but leaves open whether this remains true without assuming connectivity. Using the structure descriptions of connected on-line 3-chromatic graphs we obtain one algorithm which colors all on-line 3-chromatic graphs with 4 colors. It is a tight result. This is a companion paper of [1] in which we analyze the structure of triangle-free on-line 3-chromatic graphs.

Parity Constrained k-Edge-Connected Orientations

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset T ⊆ V with |E|+|T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k + 2)-edge-connected graph with |V|+|E| even has a k-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k+1 edge-disjoint paths from s to every node and k edge-disjoint paths from every node to s.

A Network Coding Algorithm for Multi-Layered Video Streaming

Multi-layered video streaming considers different quality requirements of the receivers. Network coding has been shown to be a useful tool to increase throughput of multi-layered service compared to simple multicasting. Kim et al. gave a simple effective algorithm using network coding. We generalize their approach and give an algorithm that solves the problem for two layers optimally for certain natural objective functions and prove NP-hardness of the problem for some other objectives, as well as for more than two layers. We also give some new heuristics for three layers.

On-Line 3-Chromatic Graphs I. Triangle-Free Graphs

This is the first half of a two-part paper devoted to on-line 3-colorable graphs. Here on-line 3-colorable triangle-free graphs are characterized by a finite list of forbidden induced subgraphs. The key role in our approach is played by the family of graphs which are both triangle- and (2K2 + K1)-free. Characterization of this family is given by introducing a bipartite modular decomposition concept. This decomposition, combined with the greedy algorithm, culminates in an on-line 3-coloring algorithm for this family. On the other hand, based on the characterization of this family, all 22 forbidden subgraphs of on-line 3-colorable triangle-free graphs are determined. As a corollary, we obtain the 10 forbidden subgraphs of on-line 3-colorable bipartite graphs. The forbidden subgraphs in the finite basis characterization are on-line 4-critical, i.e., they are on-line 4-chromatic but their proper induced subgraphs are on-line 3-colorable. The results of this paper are applied in the companion paper [Discrete Math., 177 (1997), pp. 99--122] to obtain the finite basis characterization of connected on-line 3-colorable graphs (with 51 4-critical subgraphs). However, perhaps surprisingly, connectivity (or the triangle-free property) is essential in a finite basis characterization: there are infinitely many on-line 4-critical graphs.