arXiv subjects

Gábor Kun

Publications and source records attributed to Gábor Kun.

7 recordsLinked to original sources

arXiv2026

Dichotomy for orderings?

Fagin defined the class $NP$ by the means of Existential Second-Order logic. Feder and Vardi expressed it (up to polynomial equivalence) by special fragments of Existential Second-Order logic (SNP), while the authors used forbidden expanded substructures (cf. lifts and shadows). Consequently, for such problems there is no dichotomy, unlike for CSPs. We prove that ordering problems for graphs defined by finitely many forbidden ordered subgraphs capture the full power of the class $NP$, that is, any language in the class $NP$ is polynomially equivalent to an ordering problem. In particular, we refute a conjecture of Hell, Mohar and Rafiey that dichotomy holds for this class. On the positive side, we confirm the conjecture of Duffus, Ginn and Rödl that ordering problems defined by a single obstruction which is a biconnected ordered graph are $NP$-complete if the graph is not complete. We initiate the study of meta-theorems for classes which have the full power of the class $NP$. For example, homomorphism problems (or CSPs) do not have full power (similarly to coloring problems). On the other hand, we show that problems defined by the existence of an ordering, which avoids certain ordered patterns, have full power. We find it surprising that such simple structures can express the full power of $NP$. A principal tool for obtaining these results is the Sparse Incomparability Lemma in many of its variants, which are classical results in the theory of homomorphisms of graphs and structures. We prove it here in the setting of ordered stuctures as a Temporal Sparse Incomparability Lemma. This is a non-trivial result, even in the random setting, and a deterministic algorithm requires more effort. Interestingly, our proof involves the Lovász Local Lemma.

cs.CC↗

arXiv2025

Graphings with few circulations

In 2021, motivated by graph limit theory Lovász extended most of the theory of flows to a measure theoretic setting. Using this framework, the first author constructed $d$-regular treeings that are measurably bipartite, and have no nonzero measurable circulations, that is, flows without sources or sinks. In particular, these treeings do not admit a measurable perfect matching. In this paper, we develop tools to build $d$-regular treeings where the space of circulations is exactly $k$-dimensional for any positive integer $k$. As applications, we construct 1) a treeing with a single balanced orientation, but no Schreier decoration; 2) a treeing with a single Schreier decoration; 3) and a treeing with a proper edge $d$-coloring, but no further perfect matchings. The first answers a question raised by Lovász, as this particular balanced orientation does not decompose as a linear combination of finite cycles and infinite paths.

math.CO↗

arXiv2024

The measurable Hall theorem fails for treeings

We construct, for every $d \geq 3$, a $d$-regular acyclic measurably bipartite graphing that admits no measurable perfect matching, resolving a problem of Kechris and Marks. A dense variant of our construction yields a coupling of two standard Borel probability measure spaces whose support contains no deterministic coupling, though the conditional probabilities of the coupling measure are atomless. This refutes a conjecture of Gurel-Gurevich and Peled.

math.CO↗

arXiv2024

Posets are easily testable

Alon and Shapira proved that every monotone class (closed under taking subgraphs) of undirected graphs is strongly testable, that is, under the promise that a given graph is either in the class or $\varepsilon$-far from it, there is a test using a constant number of samples (depending on $\varepsilon$ only) that rejects every graph not in the class with probability at least one half, and always accepts a graph in the class. However, their bound on the number of samples is quite large since they heavily rely on Szemerédi's regularity lemma. We study the case of posets and show that every monotone class of posets is easily testable, that is, a polynomial (of $\varepsilon^{-1}$) number of samples is sufficient. We achieve this via proving a polynomial removal lemma for posets. We give a simple classification: for every monotone class of posets, there is an $h$ such that the class is indistinguishable (every large enough poset in one class is $\varepsilon$-close to a poset in the other class) from the class of $C_h$-free posets, where $C_h$ denotes the chain with $h$ elements. This allows us to test every monotone class of posets using $O(\varepsilon^{-1})$ samples. The test has a two-sided error, but it is almost complete: the probability of refuting a poset in the class is polynomially small in the size of the poset. The analogous results hold for comparability graphs, too.

math.CO↗

arXiv2024

On pattern-avoiding permutons

The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order $k$ have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most $(k-1)$ many, and this bound is sharp. We use this to give a simple proof of the `permutation removal lemma'.

math.CO↗

arXiv2022

The uniform Gardner conjecture and rounding Borel flows

We study groups which satisfy Gardner's equidecomposition conjecture for uniformly distributed sets. We prove that an amenable group has this property if and only if it does not admit $(\mathbb{Z}/2\mathbb{Z}) *(\mathbb{Z}/2\mathbb{Z})$ as a quotient by a finite subgroup. Our technical contribution is an algorithm for rounding Borel flows for actions of amenable groups.

math.LO↗

arXiv2021

Matchings in Benjamini-Schramm convergent graph sequences

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini-Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d-regular bipartite graphs for any fixed d at least 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of d-regular bipartite graphs converging to the d-regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d-regular bipartite graphs. We show that the limit equals to the exponent in Schrijver's lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen-Onak (also Elek--Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.

math.CO↗