arXiv subjects

Jacob W. Cooper

Publications and source records attributed to Jacob W. Cooper.

8 recordsLinked to original sources

arXiv2022

Matrices of optimal tree-depth and a row-invariant parameterized algorithm for integer programming

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with dual tree-depth d and largest entry D are solvable in time g(d,D)poly(n) for some function g. However, the dual tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth, and thus does not reflect its geometric structure. We prove that the minimum dual tree-depth of a row-equivalent matrix is equal to the branch-depth of the matroid defined by the columns of the matrix. We design a fixed parameter algorithm for computing branch-depth of matroids represented over a finite field and a fixed parameter algorithm for computing a row-equivalent matrix with minimum dual tree-depth. Finally, we use these results to obtain an algorithm for integer programming running in time g(d*,D)poly(n) where d* is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.

cs.DS↗

arXiv2022

Uniform Turán density of cycles

In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open for almost 40 years. In addition to $K_4^{(3)-}$, the only $3$-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77-97] and a specific family with uniform Turán density equal to $1/27$. We develop new tools for embedding hypergraphs in host hypergraphs with positive uniform density and apply them to completely determine the uniform Turán density of a fundamental family of $3$-uniform hypergraphs, namely tight cycles $C_\ell^{(3)}$. The uniform Turán density of $C_\ell^{(3)}$, $\ell\ge 5$, is equal to $4/27$ if $\ell$ is not divisible by three, and is equal to zero otherwise. The case $\ell=5$ resolves a problem suggested by Reiher.

math.CO↗

arXiv2021

Packing and covering directed triangles asymptotically

A well-known conjecture of Tuza asserts that if a graph has at most $t$ pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most $2t$ edges. If true, the factor 2 would be best possible. In the directed setting, also asked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with $t\inΩ(n^2)$ whose smallest such set has $1.5t-o(n^2)$ arcs.

math.CO↗

arXiv2021

Quasirandom Latin squares

We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.

math.CO↗

arXiv2020

Hadwiger meets Cayley

We show that every connected $k$-chromatic graph contains at least $k^{k-2}$ spanning trees.

math.CO↗

arXiv2018

Finitely forcible graph limits are universal

The theory of graph limits represents large graphs by analytic objects called graphons. Graph limits determined by finitely many graph densities, which are represented by finitely forcible graphons, arise in various scenarios, particularly within extremal combinatorics. Lovasz and Szegedy conjectured that all such graphons possess a simple structure, e.g., the space of their typical vertices is always finite dimensional; this was disproved by several ad hoc constructions of complex finitely forcible graphons. We prove that any graphon is a subgraphon of a finitely forcible graphon. This dismisses any hope for a result showing that finitely forcible graphons possess a simple structure, and is surprising when contrasted with the fact that finitely forcible graphons form a meager set in the space of all graphons. In addition, since any finitely forcible graphon represents the unique minimizer of some linear combination of densities of subgraphs, our result also shows that such minimization problems, which conceptually are among the simplest kind within extremal graph theory, may in fact have unique optimal solutions with arbitrarily complex structure.

math.CO↗

arXiv2018

Optimal-size clique transversals in chordal graphs

The following question was raised by Tuza in 1990 and Erdos et al. in 1992: if every edge of an n-vertex chordal graph G is contained in a clique of size at least four, does G have a clique transversal, i.e., a set of vertices meeting all non-trivial maximal cliques, of size at most n/4? We prove that every such graph G has a clique transversal of size at most 2(n-1)/7 if n>=5, which is the best possible bound.

math.CO↗

arXiv2016

Weak regularity and finitely forcible graph limits

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak $\varepsilon$-regular partition with the number of parts bounded by a polynomial in $\varepsilon^{-1}$. We construct a finitely forcible graphon $W$ such that the number of parts in any weak $\varepsilon$-regular partition of $W$ is at least exponential in $\varepsilon^{-2}/2^{5\log^*\varepsilon^{-2}}$. This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.

math.CO↗