Searcharxiv⌕ Search

arXiv subjects

Eli Berger

Publications and source records attributed to Eli Berger.

At least 19 recordsLinked to original sources
arXiv2026

Global and local degree conditions for matchability

A corollary of Hall's marriage theorem is that a sufficient condition for a list $(V_1, \ldots ,V_m)$ of sets to have a system of distinct representatives is that $|V_i|\ge deg_{\{V_1, \ldots ,V_m\}}(v)$ for every $i\in [m]$ and $v \in \bigcup_{i\in [m]}V_i$. This we dub a {\em global} condition. A folklore result is that a {\em local} condition - that the inequality holds for pairs $i,v$ for which $v \in V_i$ - suffices. These are special cases of a general type of results - large sets, whose elements are sparse in some sense, have a system of representatives that is independent in a related graph. We study two such scenarios, in both of which each $V_i$ is replaced by a $k$-uniform hypergraph $H_i$, the representatives are hyperedges, and distinctness is replaced by disjointness. In one setting the sparsity is measured by the degrees of vertices in the hypergraphs, in the other by the degrees of vertices in the line graph. The proofs use the topological version of Hall's theorem. In particular, we shall use a lower bound on the topological connectivity of the independence complex of a graph, defined by vector representations. We also provide short proofs of the local infinite version, known as the ``Milner-Shelah theorem''.

math.CO↗
arXiv2026

A sharp bound on the integrality gap in the 3-set cover problem

Given a hypergraph with edges of size at most 3, the 3-set cover problem asks to determine the minimum size of a family of edges which covers the vertex set. As the problem is NP-hard, it is natural to consider its fractional (linear programming) relaxation, which provides a lower bound on the value of the optimal solution. The ratio between the actual value and that of the fractional relaxation is called the integrality gap. A classic bound of Lov\'asz implies that the integrality gap in this problem is at most 11/6. This has been improved to 5/3 by Fujito and Okumura. Here we prove that the integrality gap is at most 3/2, which is best possible. A corollary of this result is that the vertex set of any 3-uniform, regular hypergraph on n vertices can be covered by n/2 (or fewer) edges. This solves the k=3 case of a problem of de A. Moreira and Kohayakawa. As another application, we derive a certain variant of the Gale-Shapley stable marriage theorem for triples.

math.CO↗
arXiv2026

When all directed cycles have the same weight

A digraph $G$ is weightable if its edges can be weighted with real numbers such that the total weight in each directed cycle equals 1. There are several equivalent conditions: that $G$ admits a 0/1-weighting with the same property, or that $G$ contains no subdivided "double-cycle" as a subdigraph, or that for every triple of vertices, all directed cycles containing all three pass through them in the same cyclic order. And there is quite a rich supply of such digraphs: for instance, any digraph drawn in the plane such that each of its directed cycles rotates clockwise around the origin is weightable (let us call such digraphs "circular"), and there are weightable planar digraphs with much more complicated structure than this. Until now the general structure of weightable digraphs was not known, and that is our objective in this paper. We will show that: - there is a construction that builds every planar weightable digraph from circular digraphs; and - there is a (different) construction that builds every weightable digraph from planar ones. We derive a poly-time algorithm to test if a digraph is weightable.

math.CO↗
arXiv2025

A common generalization to strengthenings of Drisko's Theorem for intersections of two matroids

Let $\mathcal{M}$ and $\mathcal{N}$ be two matroids on the same ground set $V$. Let $A_1,\dots,A_{2n-1}$ be sets which are independent in both $\mathcal{M}$ and $\mathcal{N}$, satisfying $|A_i|\geq \textrm{min}(i,n)$ for all $i$. We show that there exists a partial rainbow set of size $n$, which is independent in both $\mathcal{M}$ and $\mathcal{N}$. This is a common generalization of rainbow matching results for bipartite graphs by Aharoni, Berger, Kotlar, and Ziv, and for the intersection of two matroid by Kotlar and Ziv.

math.CO↗
arXiv2024

Looms

A pair $(A,B)$ of hypergraphs is called orthogonal if $|a \cap b|=1$ for every pair of edges $a \in A$ and $b \in B$. An orthogonal pair of hypergraphs is called a loom if each of its two members is the set of minimum covers of the other. Looms appear naturally in the context of a conjecture of Gyárfás and Lehel on the covering number of cross-intersecting hypergraphs. We study their properties and ways of construction, and prove special cases of a conjecture that if true would imply the Gyárfás--Lehel conjecture.

math.CO↗
arXiv2024

Coloring the intersection of two matroids

A result [The intersection of a matroid and a simplicial complex, Trans. Amer. Math. Soc. 358] from 2006 of Aharoni and the first author of this paper states that for any two positive integers $p,q$, where $p$ divides $q$, if a matroid $\mathcal{M}$ is $p$-colorable and a matroid $\mathcal{N}$ is $q$-colorable then $\mathcal{M} \cap \mathcal{N}$ is $(p+q)$-colorable. In this paper we show that the assumption that $p$ divides $q$ is in fact redundant, and we also prove that $\mathcal{M} \cap \mathcal{N}$ is even $p+q$ list-colorable. The result uses topology and relies on a new parameter yielding a lower bound for the topological connectivity of the intersection of two matroids.

math.CO↗
arXiv2024

Coloring, list coloring, and fractional coloring in intersections of matroids

It is known that in matroids the difference between the chromatic number and the fractional chromatic number is smaller than 1, and that the list chromatic number is equal to the chromatic number. We investigate the gap within these pairs of parameters for hypergraphs that are the intersection of a given number k of matroids. We prove that in such hypergraphs the list chromatic number is at most k times the chromatic number and at most 2k-1 times the maximum chromatic number among the k matroids. We study the relationship between three polytopes associated with k-sets of matroids, and connect them to bounds on the fractional chromatic number of the intersection of the members of the k-set. This also connects to bounds on the matroidal matching and covering number of the intersection of the members of the k-set. The tools used are in part topological.

math.CO↗
arXiv2024

Repelling Random Walks

We present a novel quasi-Monte Carlo mechanism to improve graph-based sampling, coined repelling random walks. By inducing correlations between the trajectories of an interacting ensemble such that their marginal transition probabilities are unmodified, we are able to explore the graph more efficiently, improving the concentration of statistical estimators whilst leaving them unbiased. The mechanism has a trivial drop-in implementation. We showcase the effectiveness of repelling random walks in a range of settings including estimation of graph kernels, the PageRank vector and graphlet concentrations. We provide detailed experimental evaluation and robust theoretical guarantees. To our knowledge, repelling random walks constitute the first rigorously studied quasi-Monte Carlo scheme correlating the directions of walkers on a graph, inviting new research in this exciting nascent domain.

stat.ML↗
arXiv2024

General Graph Random Features

We propose a novel random walk-based algorithm for unbiased estimation of arbitrary functions of a weighted adjacency matrix, coined universal graph random features (u-GRFs). This includes many of the most popular examples of kernels defined on the nodes of a graph. Our algorithm enjoys subquadratic time complexity with respect to the number of nodes, overcoming the notoriously prohibitive cubic scaling of exact graph kernel evaluation. It can also be trivially distributed across machines, permitting learning on much larger networks. At the heart of the algorithm is a modulation function which upweights or downweights the contribution from different random walks depending on their lengths. We show that by parameterising it with a neural network we can obtain u-GRFs that give higher-quality kernel estimates or perform efficient, scalable kernel learning. We provide robust theoretical analysis and support our findings with experiments including pointwise estimation of fixed graph kernels, solving non-homogeneous graph ordinary differential equations, node clustering and kernel regression on triangular meshes.

stat.ML↗
arXiv2024

Bounded diameter tree-decompositions

When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded ``tree-length''. We will show that this is equivalent to being ``boundedly quasi-isometric to a tree'', which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to tie these two areas together. We will prove that there is a tree-decomposition in which each bag has small diameter, if and only if there is a map $ϕ$ from $V(G)$ into the vertex set of a tree $T$, such that for all $u,v\in V(G)$, the distances $d_G(u,v), d_T(ϕ(u),ϕ(v))$ differ by at most a constant. A ``geodesic loaded cycle'' in $G$ is a pair $(C,F)$, where $C$ is a cycle of $G$ and $F\subseteq E(C)$, such that for every pair $u,v$ of vertices of $C$, one of the paths of $C$ between $u,v$ contains at most $d_G(u,v)$ $F$-edges, where $d_G(u,v)$ is the distance between $u,v$ in $G$. We will show that a graph $G$ admits a tree-decomposition in which every bag has small diameter, if and only if $|F|$ is small for every geodesic loaded cycle $(C,F)$. Our proof is an extension of an algorithm to approximate tree-length in finite graphs by Dourisboure and Gavoille. In metric geometry, there is a similar theorem that characterizes when a graph is quasi-isometric to a tree, ``Manning's bottleneck criterion''. The goal of this paper is to tie all these concepts together, and add a few more related ideas. For instance, we prove a conjecture of Rose McCarty, that $G$ admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices $u,v,w$ of $G$, some ball of small radius meets every path joining two of $u,v,w$.

math.CO↗
arXiv2023

2-covers of wide Young diagrams

A Young diagram $Y$ is called wide if every sub-diagram $Z$ formed by a subset of the rows of $Y$ dominates $Z'$, the conjugate of $Z$. A Young diagram $Y$ is called Latin if its squares can be assigned numbers so that for each $i$, the $i$th row is filled injectively with the numbers $1, \ldots ,a_i$, where $a_i$ is the length of $i$th row of $Y$, and every column is also filled injectively. A conjecture of Chow and Taylor, publicized by Chow, Fan, Goemans, and Vondrak is that a wide Young diagram is Latin. We prove a dual version of the conjecture.

math.CO↗
arXiv2023

Graphs with no even holes and no sector wheels are the union of two chordal graphs

Sivaraman conjectured that if $G$ is a graph with no induced even cycle then there exist sets $X_1, X_2 \subseteq V(G)$ satisfying $V(G) = X_1 \cup X_2$ such that the induced graphs $G[X_1]$ and $G[X_2]$ are both chordal. We prove this conjecture in the special case where $G$ contains no sector wheel, namely, a pair $(H, w)$ where $H$ is an induced cycle of $G$ and $w$ is a vertex in $V(G) \setminus V(H)$ such that $N(w) \cap H$ is either $V(H)$ or a path with at least three vertices.

math.CO↗
arXiv2023

Efficient Graph Field Integrators Meet Point Clouds

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

cs.LG↗
arXiv2022

Non-uniform degrees and rainbow versions of the Caccetta-Häggkvist conjecture

The Caccetta-Häggkvist conjecture (denoted below CHC) states that the directed girth (the smallest length of a directed cycle) $dgirth(D)$ of a directed graph $D$ on $n$ vertices is at most $\lceil \frac{n}{δ^+(D)}\rceil$, where $δ^+(D)$ is the minimum out-degree of~$D$. We consider a version involving all out-degrees, not merely the minimum one, and prove that if $D$ does not contain a sink, then $dgirth(D) \le 2 \sum_{v\in V(D)} \frac{1}{deg^+(v)+1}$. In the spirit of a generalization of the CHC to rainbow cycles in \cite{ADH2019}, this suggests the conjecture that given non-empty sets $F_1, \ldots,F_n$ of edges of $K_n$, there exists a rainbow cycle of length at most $2\sum_{1\le i \le n}\frac{1}{|F_i|+1}$. We prove a bit stronger result when $1\le |F_i|\le 2$, thereby strengthening a result of DeVos et. al \cite{DDFGGHMM2021}. We prove a logarithmic bound on the rainbow girth in the case that the sets $F_i$ are triangles.

math.CO↗
arXiv2022

Fractionally balanced hypergraphs and rainbow KKM theorems

A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove lower bounds on the matching number of such hypergraphs. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals.

math.CO↗
arXiv2021

Rainbow paths and large rainbow matchings

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for hypergraphs. For $\{C_3,C_5\}$-free graphs and for disjoint matchings we obtain a lower bound of $\frac{3n}{4}-O(1)$. We also discuss a conjecture on rainbow alternating paths, that if true would yield a lower bound of $n-\sqrt{2n}$. We prove the non-alternating (ordinary paths) version of this conjecture.

math.CO↗
arXiv2021

Tournaments and the Strong Erdős-Hajnal Property

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $ε(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{ε(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament $T$ with no subtorunament isomorphic to $C_5$ there exist disjoint vertex subsets $A$ and $B$, each containing a linear proportion of the vertices of $T$, and such that every vertex of $A$ is adjacent to every vertex of $B$.

math.CO↗