arXiv subjects

András Sebő

Publications and source records attributed to András Sebő.

14 recordsLinked to original sources

arXiv2026

Matchings: Source, Goal and Faithful Companion

Matchings were among the earliest motivations for graph theory. They subsequently remained a central goal, inspiring the development of new mathematical tools that went well beyond problems directly concerning matchings. These tools proved widely applicable, accompanying the growth of graph theory over the past century. A milestone in this trajectory is W.T. Tutte, The factorisation of linear graphs, J. Lond. Math. Soc. (1), 22, no. 2, (1947), 107-111., which firmly embedded graph theory - through matchings - into the body of classical mathematics, in particular, linear algebra and polynomials. We revisit this article of Tutte, presenting its original content, sketching some aspects of its impact until some recent progress, and trace one of its subsequent lines of development finally leading to a new contribution answering an open challenge and extending known results.

math.CO↗

arXiv2025

On the Boxicity of Line Graphs and of Their Complements

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs.'' The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. As every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $NP$-hard: for the existence and optimization of interval-order subgraphs of line graphs, or of interval completions and the boxicity of their complement, if the boxicity is bounded. We finally extend our approach to upper and lower bounding the boxicity of line graphs.

math.CO↗

arXiv2024

Packing, Hitting, and Colouring Squares

Given a finite family of squares in the plane, the packing problem asks for the maximum number $ν$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $τ$ of points hitting all of them. Clearly, $τ\ge ν$. Both problems are known to be NP-hard, even for families of axis-parallel unit squares. The main results of this work provide the first non-trivial bounds for the $τ/ ν$ ratio for not necessarily axis-parallel squares. We establish an upper bound of $6$ for unit squares and $10$ for squares of varying sizes. The worst ratios we can provide with examples are $3$ and $4$, respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval $[\frac{3}{2},2]$ for unit squares and $[\frac{3}{2},4]$ for squares of varying sizes. The methods we introduced for the $τ/ν$ ratio can also be used to relate the chromatic number $χ$ and clique number $ω$ of squares by bounding the $χ/ω$ ratio by $6$ for unit squares and $9$ for squares of varying sizes. The $τ/ ν$ and $χ/ω$ ratios have already been bounded before by a constant for "fat" objects, the fattest and simplest of which are disks and squares. However, while disks have received significant attention, specific bounds for squares have remained essentially unexplored. This work intends to fill this gap.

cs.CG↗

arXiv2023

Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs''. The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.

math.CO↗

arXiv2023

Odd Paths, Cycles and $T$-joins: Connections and Algorithms

Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization" (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem and some multiflow problems. We clarify the interconnections between some of these problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in undirected graphs without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lovász (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''). On the other hand, we provide an efficient algorithm to the closely related and well-studied Minimum-weight Odd $T$-Join problem for non-negative weights: our algorithm runs in FPT time parameterized by $c$, where $c$ is the number of connected components in some efficiently computed minimum-weight $T$-join. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is still only conjectured to be polynomial-time solvable. The analogous problems for digraphs are also considered.

cs.CC↗

arXiv2023

Jump-systems of $T$-paths

Jump systems are sets of integer vectors satisfying a simple axiom, generalizing matroids, also delta-matroids, and well-kown combinatorial examples such as degree sequences of subgraphs of a graph. It is useful to know if a set of vectors defined from combinatorial structures is a jump system: this has consequences for optimizing on the set, or on some derived sets of vectors. In this note we are mainly concerned in telling our proof of the following more than two decades old fact and its original, elementary proof for an example different from degree sequences: {\em Given an udirected graph $G=(V,E)$ and $T\subseteq V$, the vectors $m$ indexed by $T$ for which there exist a set of openly disjoint $T$-paths so that each $t\in T$ is the endpoint of exactly $m(t)$ paths forms a jump system. The same holds for edge-disjoint $T$-paths.} We are also exhibiting the context and some consequences of this fact, with some pointers to recent developments, among them ro another proof by Iwata and Yokoi, to some related jump system intersection theorems and to some open problems.

math.CO↗

arXiv2021

How many matchings cover the nodes of a graph?

Given an undirected graph, are there $k$ matchings whose union covers all of its nodes, that is, a matching-$k$-cover? A first, easy polynomial solution from matroid union is possible, as already observed by Wang, Song and Yuan (Mathematical Programming, 2014). However, it was not satisfactory neither from the algorithmic viewpoint nor for proving graphic theorems, since the corresponding matroid ignores the edges of the graph. We prove here, simply and algorithmically: all nodes of a graph can be covered with $k\ge 2$ matchings if and only if for every stable set $S$ we have $|S|\le k\cdot|N(S)|$. When $k=1$, an exception occurs: this condition is not enough to guarantee the existence of a matching-$1$-cover, that is, the existence of a perfect matching, in this case Tutte's famous matching theorem (J. London Math. Soc., 1947) provides the right `good' characterization. The condition above then guarantees only that a perfect $2$-matching exists, as known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953). Some results are then deduced as consequences with surprisingly simple proofs, using only the level of difficulty of bipartite matchings. We give some generalizations, as well as a solution for minimization if the edge-weights are non-negative, while the edge-cardinality maximization of matching-$2$-covers turns out to be already NP-hard. We have arrived at this problem as the line graph special case of a model arising for manufacturing integrated circuits with the technology called `Directed Self Assembly'.

math.CO↗

arXiv2020

Integer Plane Multiflow Maximisation : Flow-Cut Gap and One-Quarter-Approximation

In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-cut-gap. Planarity means here that the union of the supply and demand graph is planar. We first prove that there exists a multiflow of value at least half of the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer one of value at least half of the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing again at most half of the value, in polynomial time, achieving a $1/4$-approximation algorithm for maximum integer multiflows in the plane, and an integer-flow-cut gap of $8$.

cs.DS↗

arXiv2019

Ear-Slicing for Matchings in Hypergraphs

We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a $3$-regular, $3$-uniform hypergraph into at most one triangle (hyperedge of size $3$) and edges (subsets of size $2$ of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of Cornuéjols' and Pulleyblank's. The existence of such a packing can be considered to be a hypergraph variant of Petersen's theorem on perfect matchings, and leads to a simple proof for a sharpening of Lu's theorem on antifactors of graphs.

cs.DM↗

arXiv2019

Color-critical Graphs and Hereditary Hypergraphs

A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary hypergraph, generalizing the chromatic number. We then show examples of applying the results to new problems and indicate the way to algorithms and refined complexity results for all these examples at the same time.

math.CO↗

arXiv2018

The Salesman's Improved Paths: 3/2+1/34 Integrality Gap and Approximation Ratio

We give a new, strongly polynomial-time algorithm and improved analysis for the metric $s-t$ path TSP. It finds a tour of cost less than 1.53 times the optimum of the subtour elimination LP, while known examples show that 1.5 is a lower bound for the integrality gap. A key new idea is the deletion of some edges of Christofides' trees, which is then accompanied by novel arguments of the analysis: edge-deletion disconnects the trees, which are then partly reconnected by `parity correction'. We show that the arising `connectivity correction' can be achieved for a minor extra cost. On the one hand this algorithm and analysis extend previous tools such as the best-of-many Christofides algorithm. On the other hand, powerful new tools are solicited, such as a flow problem for analyzing the reconnection cost, and the construction of a set of more and more restrictive spanning trees, each of which can still be found by the greedy algorithm. We show that these trees can replace the convex combination of spanning trees in the best-of-may Christofides algorithm. These new methods lead to improving the integrality ratio and approximation guarantee below 1.53, as it is already sketched in the preliminary shortened version of this article that appeared in FOCS 2016. The algorithm and analysis have been significantly simplified in the current article, and details of proofs and explanations have been added.

cs.DM↗

arXiv2017

Layers and Matroids for the Traveling Salesman's Paths

Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive "generalized Gao-trees". We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the $s-t$ path TSP.

cs.DM↗

arXiv2015

Ear-decompositions and the complexity of the matching polytope

The complexity of the matching polytope of graphs may be measured with the maximum length $β$ of a starting sequence of odd ears in an ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its facets are defined by 2-connected factor-critical graphs, which have an odd ear-decomposition (according to a theorem of Lovász). In particular, $β(G) \leq 1$ if and only if the matching polytope of the graph $G$ is completely described by non-negativity, star and odd-circuit inequalities. This is essentially equivalent to the h-perfection of the line-graph of $G$, as observed by Cao and Nemhauser. The complexity of computing $β$ is apparently not known. We show that deciding whether $β(G)\leq 1$ can be executed efficiently by looking at any ear-decomposition starting with an odd circuit and performing basic modulo-2 computations. Such a greedy-approach is surprising in view of the complexity of the problem in more special cases by Bruhn and Schaudt, and it is simpler than using the Parity Minor Algorithm. Our results imply a simple polynomial-time algorithm testing h-perfection in line-graphs (deciding h-perfection is open in general). We also generalize our approach to binary matroids and show that computing $β$ is a Fixed-Parameter-Tractable problem (FPT).

math.CO↗

arXiv2012

Shorter Tours by Nicer Ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a generalization, the connected-$T$-join problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the graphic $s$-$t$-path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

cs.DM↗