arXiv subjects

Tamás Király

Publications and source records attributed to Tamás Király.

26 records · Page 2Linked to original sources

arXiv2022

Hypergraph characterization of split matroids

We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being closed under duality and taking minors. Furthermore, these matroids proved to be useful in giving exact asymptotic bounds for the dimension of the Dressian, and also implied new results on the rays of the tropical Grassmannians. In the present paper, we introduce the notion of elementary split matroids, a subclass of split matroids that contains all connected split matroids. We give a hypergraph characterization of elementary split matroids in terms of independent sets, and show that the proposed class is closed not only under duality and taking minors but also truncation. We further show that, in contrast to split matroids, the proposed class can be characterized by a single forbidden minor. As an application, we provide a complete list of binary split matroids.

math.CO↗

arXiv2022

Approximation by Lexicographically Maximal Solutions in Matching and Matroid Intersection Problems

We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly $2$. Furthermore, we prove that if the ratio is less than $2$, say $α$, then a lexicographically maximal solution achieves $(α/2)$-approximation, and this bound is tight.

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arXiv2019

Popular Branchings and Their Dual Certificates

Let $G$ be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching $B$ is popular if $B$ does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if $G$ admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that "approximately popular" branchings always exist.

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arXiv2019

Scheduling with Non-Renewable Resources: Minimizing the Sum of Completion Times

The paper considers single-machine scheduling problems with a non-renewable resource. In this setting, we are given a set jobs, each of which is characterized by a processing time, a weight, and the job also has some resource requirement. At fixed points in time, a certain amount of the resource is made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that at any time their resource requirement does not exceed the available amounts of resources. The objective that we consider here is the minimization of the sum of weighted completion times. We give polynomial approximation algorithms and complexity results for single scheduling machine problems. In particular, we show strong NP-hardness of the case of unit resource requirements and weights ($1|rm=1,a_j=1|\sum C_j$), thus answering an open question of Györgyi and Kis. We also prove that the schedule corresponding to the Shortest Processing Time First ordering provides a $3/2$-approximation for the same problem. We give simple constant factor approximations and a more complicated PTAS for the case of $0$ processing times ($1|rm=1,p_j=0|\sum w_jC_j$). We close the paper by proposing a new variant of the problem in which the resource arrival times are unknown. A $4$-approximation is presented for this variant, together with an $(4-\varepsilon)$-inapproximability result.

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arXiv2019

Equitable Partitions into Matchings and Coverings in Mixed Graphs

Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings. In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching--covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the problems of partitioning a mixed graph into matching forests or mixed edge covers, so that all parts are equal with respect to some criterion, such as edge/arc numbers or total sizes. Moreover, we provide the best possible multicriteria equalization.

math.CO↗

arXiv2018

Improving the Integrality Gap for Multiway Cut

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of $k$ terminal nodes, and the goal is to partition the node set of the graph into $k$ non-empty parts each containing exactly one terminal so that the total weight of the edges crossing the partition is minimized. The multiway cut problem for $k\ge 3$ is APX-hard. For arbitrary $k$, the best-known approximation factor is $1.2965$ due to [Sharma and Vondrák, 2014] while the best known inapproximability factor is $1.2$ due to [Angelidakis, Makarychev and Manurangsi, 2017]. In this work, we improve on the lower bound to $1.20016$ by constructing an integrality gap instance for the CKR relaxation. A technical challenge in improving the gap has been the lack of geometric tools to understand higher-dimensional simplices. Our instance is a non-trivial $3$-dimensional instance that overcomes this technical challenge. We analyze the gap of the instance by viewing it as a convex combination of $2$-dimensional instances and a uniform 3-dimensional instance. We believe that this technique could be exploited further to construct instances with larger integrality gap. One of the ingredients of our proof technique is a generalization of a result on \emph{Sperner admissible labelings} due to [Mirzakhani and Vondrák, 2015] that might be of independent combinatorial interest.

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arXiv2017

Global and fixed-terminal cuts in digraphs

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut. 1. The fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show a tight approximability factor of $2$ for the fixed-terminal node-weighted double cut. We show that the global node-weighted double cut cannot be approximated to a factor smaller than $3/2$ under the Unique Games Conjecture (UGC). 2. The fixed-terminal edge-weighted bicut is known to have a tight approximability factor of $2$. We show that the global edge-weighted bicut is approximable to a factor strictly better than $2$, and that the global node-weighted bicut cannot be approximated to a factor smaller than $3/2$ under UGC. 3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of $4/3$ for the node-weighted $3$-cut problem. Second, we show that for constant $k$, there exists an efficient algorithm to solve the minimum $\{s,t\}$-separating $k$-cut problem. Our techniques for the algorithms are combinatorial, based on LPs and based on enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.

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arXiv2015

Blocking optimal $k$-arborescences

Given a digraph $D=(V,A)$ and a positive integer $k$, an arc set $F\subseteq A$ is called a \textbf{$k$-arborescence} if it is the disjoint union of $k$ spanning arborescences. The problem of finding a minimum cost $k$-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost $k$-arborescence. For $k=1$, the problem was solved in [A. Bernáth, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general $k$ that has polynomial running time if $k$ is fixed.

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