arXiv subjects

Tamás Király

Publications and source records attributed to Tamás Király.

At least 19 recordsLinked to original sources

arXiv2026

$\{s,t\}$-Separating Principal Partition Sequence of Submodular Functions

Narayanan showed the existence of the principal partition sequence of a submodular function, a structure with numerous applications in areas such as clustering, fast algorithms, and approximation algorithms. In this work, motivated by two applications, we develop a theory of $\{s,t\}$-separating principal partition sequence of a submodular function. We define this sequence, show its existence, and design a polynomial-time algorithm to construct it. We show two applications: (1) approximation algorithm for the $\{s,t\}$-separating submodular $k$-partitioning problem for monotone and posimodular functions and (2) polynomial-time algorithm for the hypergraph orientation problem of finding an orientation that simultaneously has strong connectivity at least $k$ and $(s,t)$-connectivity at least $\ell$.

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arXiv2026

The NTU Partitioned Matching Game for International Kidney Exchange Programs

Motivated by the real-world problem of international kidney exchange (IKEP), recent literature introduced a generalized transferable utility matching game featuring a partition of the vertex set of a graph into players, and analyzed its complexity. We explore the non-transferable utility (NTU) variant of the game, where the utility of players is given by the number of their matched vertices. Our motivation for studying this problem is twofold. First, the NTU version is arguably a more natural model of the international kidney exchange program, as the utility of a participating country mostly depends on how many of its patients receive a kidney, which is non-transferable by nature. Second, the special case where each player has two vertices, which we call the NTU matching game with couples, is interesting in its own right and has intriguing structural properties. We study the core of the NTU game, which suitably captures the notion of stability of an IKEP, as it precludes incentives to deviate from the proposed solution for any possible coalition of the players. We prove computational complexity results about the weak and strong cores under various assumptions on the players. In particular, we show that if every player has two vertices, then the weak core is always nonempty, and the existence of a strong core solution can be decided in polynomial time. Moreover, one can efficiently optimize on the strong core. In contrast, it is NP-hard to decide whether the strong core is empty when each player has three vertices. We also show that if the number of players is constant, then the non-emptiness of the weak and strong cores is polynomial-time decidable, and we can find a minimum-cost core solution in polynomial time.

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arXiv2025

Matroid Intersection under Minimum Rank Oracle

In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the ranks of the given set in the two matroids. For the unweighted matroid intersection problem, we show how to construct a necessary part of the exchangeability graph, which enables us to emulate the standard augmenting path algorithm. For the weighted problem, the tractability is open in general. Nevertheless, we describe several special cases where tractability can be achieved, and we discuss potential approaches and the challenges encountered. On the positive side, we present a solution for the case where no circuit of one matroid is contained within a circuit of the other. Additionally, we propose a fixed-parameter tractable algorithm, parameterized by the maximum size of a circuit of one matroid. We also show that a lexicographically maximal common independent set can be found by the same approach, which leads to a nontrivial approximation ratio for finding a maximum-weight common independent set. On the negative side, we prove that the approach employed for the tractable cases above involves an NP-hard problem in the general case. We also show that if we consider the generalization to polymatroid intersection, even the unweighted problem is hard under the minimum rank oracle.

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arXiv2025

Rainbow Arborescence Conjecture

The famous Ryser--Brualdi--Stein conjecture asserts that every $k \times k$ Latin square contains a partial transversal of size $k-1$. Since its appearance, the conjecture has attracted significant interest, leading to several proposed generalizations. One of the most notable of these, by Aharoni, Kotlar, and Ziv, conjectures that $k$ disjoint common bases of two matroids of rank $k$ have a common independent partial transversal of size $k-1$. Although simple counterexamples show that the size $k-1$ above cannot be improved to $k$ (i.e., a transversal instead of a partial transversal), it is remarkable that no such counterexample is known for the special case of spanning arborescences. This motivated the formulation of the Rainbow Arborescence Conjecture: any graph on $n$ vertices formed by the union of $n-1$ spanning arborescences contains an arborescence using exactly one arc from each. We prove several partial results on this conjecture. We show that the computational problem of testing the existence of such an arborescence with a fixed root is NP-complete, verify the conjecture in several special cases, and study relaxations of the problem. In particular, we establish the validity of the conjecture when the underlying undirected graph is a cycle; this also yields a new result on systems of distinct representatives for intervals on a cycle.

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arXiv2025

The Rainbow Arborescence Problem on Cycles

The rainbow arborescence conjecture posits that if the arcs of a directed graph with $n$ vertices are colored by $n-1$ colors such that each color class forms a spanning arborescence, then there is a spanning arborescence that contains exactly one arc of every color. We prove that the conjecture is true if the underlying undirected graph is a cycle.

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arXiv2025

A new approach to bipartite stable matching optimization

As a common generalization of previously solved optimization problems concerning bipartite stable matchings, we describe a strongly polynomial network flow based algorithm for computing $\ell$ disjoint stable matchings with minimum total cost. The major observation behind the approach is that stable matchings, as edge sets, can be represented as certain cuts of an associated directed graph. This allows us to use results on disjoint cuts directly to answer questions about disjoint stable matchings. We also provide a construction that represents stable matchings as maximum-size antichains in a partially ordered set (poset), which enables us to apply the theorems of Dilworth, Mirsky, Greene and Kleitman directly to stable matchings. Another consequence of these approaches is a min-max formula for the minimum number of stable matchings covering all stable edges.

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arXiv2025

Approximating Submodular Matroid-Constrained Partitioning

The submodular partitioning problem asks to minimize, over all partitions $P$ of a ground set $V$, the sum of a given submodular function $f$ over the parts of $P$. The problem has seen considerable work in approximability, as it encompasses multiterminal cuts on graphs, $k$-cuts on hypergraphs, and elementary linear algebra problems such as matrix multiway partitioning. This research has been divided between the fixed terminal setting, where we are given a set of terminals that must be separated by $P$, and the global setting, where the only constraint is the size of the partition. We investigate a generalization that unifies these two settings: minimum submodular matroid-constrained partition. In this problem, we are additionally given a matroid over the ground set and seek to find a partition $P$ in which there exists some basis that is separated by $P$. We explore the approximability of this problem and its variants, reaching the state of the art for the special case of symmetric submodular functions, and provide results for monotone and general submodular functions as well.

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arXiv2024

Maximum Stable Matching with Matroids and Partial Orders

The Stable Marriage problem (SM), solved by the famous deferred acceptance algorithm of Gale and Shapley (GS), has many natural generalizations. If we allow ties in preferences, then the problem of finding a maximum stable matching becomes NP-hard, and the best known approximation ratio is 1.5 (McDermid 2009, Paluch 2011, Z. Király 2012), achievable by running GS on a cleverly constructed modified instance. Another elegant generalization of SM is the matroid kernel problem introduced by Fleiner (2001), which is solvable in polynomial time using an abstract matroidal version of GS. Our main result is a simple 1.5-approximation algorithm for the matroid kernel problem when preferences are given as interval orders -- a broad subclass of partial orders that covers many applications beyond preferences with ties. In addition, for the bipartite matching case, we show that the output of our algorithm also 1.5-approximates the LP-optimum of the relaxation of the corresponding Integer Program, which shows that the integrality gap is at most 1.5 for the interval order case. To contrast this with hardness results, we show that if arbitrary partial orders are allowed in the preferences, then even in the bipartite matching case, the problem becomes hard to approximate within a factor better than 2 assuming the Unique Games Conjecture, and the integrality gap becomes 2.

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arXiv2024

Popular Maximum-Utility Matchings with Matroid Constraints

We investigate weighted settings of popular matching problems with matroid constraints. The concept of popularity was originally defined for matchings in bipartite graphs, where vertices have preferences over the incident edges. There are two standard models depending on whether vertices on one or both sides have preferences. A matching $M$ is popular if it does not lose a head-to-head election against any other matching. In our generalized models, one or both sides have matroid constraints, and a weight function is defined on the ground set. Our objective is to find a popular optimal matching, i.e., a maximum-weight matching that is popular among all maximum-weight matchings satisfying the matroid constraints. For both one- and two-sided preferences models, we provide efficient algorithms to find such solutions, combining algorithms for unweighted models with fundamental techniques from combinatorial optimization. The algorithm for the one-sided preferences model is further extended to a model where the weight function is generalized to an M$^\natural$-concave utility function. Finally, we complement these tractability results by providing hardness results for the problems of finding a popular near-optimal matching. These hardness results hold even without matroid constraints and with very restricted weight functions.

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arXiv2024

Finding Spanning Trees with Perfect Matchings

We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight spanning tree among those containing a perfect matching. On the positive side, we design a simple greedy algorithm for the case when the graph is complete (or complete bipartite) and the edge weights take at most two values. On the negative side, the problem is NP-hard even when the graph is complete (or complete bipartite) and the edge weights take at most three values, or when the graph is cubic, planar, and bipartite and the edge weights take at most two values. We also consider an interesting variant. We call a tree strongly balanced if on one side of the bipartition of the vertex set with respect to the tree, all but one of the vertices have degree $2$ and the remaining one is a leaf. This property is a sufficient condition for a tree to have a perfect matching, which enjoys an additional property. When the underlying graph is bipartite, strongly balanced spanning trees can be written as matroid intersection, and this fact was recently utilized to design an approximation algorithm for some kind of connectivity augmentation problem. The natural question is its tractability in nonbipartite graphs. As a negative answer, it turns out NP-hard to test whether a given graph has a strongly balanced spanning tree or not even when the graph is subcubic and planar.

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arXiv2024

Multiway Cuts with a Choice of Representatives

In this paper, we study several generalizations of multiway cut where the terminals can be chosen as \emph{representatives} from sets of \emph{candidates} $T_1,\ldots,T_q$. In this setting, one is allowed to choose these representatives so that the minimum-weight cut separating these sets \emph{via their representatives} is as small as possible. We distinguish different cases depending on (A) whether the representative of a candidate set has to be separated from the other candidate sets completely or only from the representatives, and (B) whether there is a single representative for each candidate set or the choice of representative is independent for each pair of candidate sets. For fixed $q$, we give approximation algorithms for each of these problems that match the best known approximation guarantee for multiway cut. Our technical contribution is a new extension of the CKR relaxation that preserves approximation guarantees. For general $q$, we show $o(\log q)$-inapproximability for all cases where the choice of representatives may depend on the pair of candidate sets, as well as for the case where the goal is to separate a fixed node from a single representative from each candidate set. As a positive result, we give a $2$-approximation algorithm for the case where we need to choose a single representative from each candidate set. This is a generalization of the $(2-2/k)$-approximation for k-cut, and we can solve it by relating the tree case to optimization over a gammoid.

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arXiv2024

Splitting-off in Hypergraphs

The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász (1974) and Mader (1978) showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of $k$-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.

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arXiv2024

Hypergraph Connectivity Augmentation in Strongly Polynomial Time

We consider hypergraph network design problems where the goal is to construct a hypergraph that satisfies certain connectivity requirements. For graph network design problems where the goal is to construct a graph that satisfies certain connectivity requirements, the number of edges in every feasible solution is at most quadratic in the number of vertices. In contrast, for hypergraph network design problems, we might have feasible solutions in which the number of hyperedges is exponential in the number of vertices. This presents an additional technical challenge in hypergraph network design problems compared to graph network design problems: in order to solve the problem in polynomial time, we first need to show that there exists a feasible solution in which the number of hyperedges is polynomial in the input size. The central theme of this work is to show that certain hypergraph network design problems admit solutions in which the number of hyperedges is polynomial in the number of vertices and moreover, can be solved in strongly polynomial time. Our work improves on the previous fastest pseudo-polynomial run-time for these problems. In addition, we develop strongly polynomial time algorithms that return near-uniform hypergraphs as solutions (i.e., every pair of hyperedges differ in size by at most one). As applications of our results, we derive the first strongly polynomial time algorithms for (i) degree-specified hypergraph connectivity augmentation using hyperedges, (ii) degree-specified hypergraph node-to-area connectivity augmentation using hyperedges, and (iii) degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.

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arXiv2023

The popular assignment problem: when cardinality is more important than popularity

We consider a matching problem in a bipartite graph $G=(A\cup B,E)$ where nodes in $A$ are agents having preferences in partial order over their neighbors, while nodes in $B$ are objects without preferences. We propose a polynomial-time combinatorial algorithm based on LP duality that finds a maximum matching or assignment in $G$ that is popular among all maximum matchings, if there exists one. Our algorithm can also be used to achieve a trade-off between popularity and cardinality by imposing a penalty on unmatched nodes in $A$. We also provide an $O^*(|E|^k)$ algorithm that finds an assignment whose unpopularity margin is at most $k$; this algorithm is essentially optimal, since the problem is $\mathsf{NP}$-complete and $\mathsf{W}_l[1]$-hard with parameter $k$. We also prove that finding a popular assignment of minimum cost when each edge has an associated binary cost is $\mathsf{NP}$-hard, even if agents have strict preferences. By contrast, we propose a polynomial-time algorithm for the variant of the popular assignment problem with forced/forbidden edges. Finally, we present an application in the context of housing markets.

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arXiv2023

Solving the Maximum Popular Matching Problem with Matroid Constraints

We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama (2020) and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become coNP-hard, even in the $b$-matching case.

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arXiv2023

Matroid Intersection under Restricted Oracles

Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the rank sum, common independence, and maximum rank oracles. We give a strongly polynomial-time algorithm for weighted matroid intersection under the rank sum oracle. In the common independence oracle model, we prove that the unweighted matroid intersection problem is tractable when one of the matroids is a partition matroid, and that even the weighted case is solvable when one of the matroids is an elementary split matroid. Finally, we show that the common independence and maximum rank oracles together are strong enough to realize the steps of our algorithm under the rank sum oracle.

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arXiv2022

On the complexity of packing rainbow spanning trees

One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. Bérczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spanning trees, an extension of the problem of packing arborescences in directed graphs which was answered by Edmonds' seminal result on disjoint arborescences. We complement his result by showing that it is NP-complete to decide whether an edge-colored graph contains two disjoint rainbow spanning trees. Our complexity result holds even for the very special case when the graph is the union of two spanning trees and each color class contains exactly two edges. As a corollary, we give a negative answer to a question on the decomposition of oriented $k$-partition-connected digraphs.

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arXiv2022

Manipulating the outcome of stable matching and roommates problems

The stable marriage and stable roommates problems have been extensively studied due to their high applicability in various real-world scenarios. However, it might happen that no stable solution exists, or stable solutions do not meet certain requirements. In such cases, one might be interested in modifying the instance so that the existence of a stable outcome with the desired properties is ensured. We focus on three different modifications. In stable roommates problems with all capacities being one, we give a simpler proof to show that removing an agent from each odd cycle of a stable partition is optimal. We further show that the problem becomes NP-complete if the capacities are greater than one, or the deleted agents must belong to a fixed subset of vertices. Motivated by inverse optimization problems, we investigate how to modify the preferences of the agents as little as possible so that a given matching becomes stable. The deviation of the new preferences from the original ones can be measured in various ways; here we concentrate on the $\ell_1$-norm. We show that, assuming the Unique Games Conjecture, the problem does not admit a better than $2$ approximation. By relying on bipartite-submodular functions, we give a polynomial-time algorithm for the bipartite case. We also show that a similar approach leads to a 2-approximation for general graphs. Last, we consider problems where the preferences of agents are not fully prescribed, and the goal is to decide whether the preference lists can be extended so that a stable matching exists. We settle the complexity of several variants, including cases when some of the edges are required to be included or excluded from the solution.

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