Related papers: Long and winding central paths
Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces…
We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central…
The Hammersley problem asks for the maximal number of points in a monotonous path through a Poisson point process. It is exactly solvable and notoriously known to belong to the KPZ universality class, with a cube-root scaling for the…
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients…
We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory, we consider two methods giving such a…
In the previous works, the rational function semifields of abstract tropical curves were characterized. In this paper, we give a contravariant categorical equivalence between the category of abstract tropical curves with morphisms and the…
We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a…
We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power…
We introduce a notion of the \emph{crux} of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalisations of known results about cycles, offering an interesting paradigm of…
An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the…
We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by…
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a…
We show that the winding of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by any standard notion of length, in contrast to the Cauchy distribution arising when allowing arbitrarily…
Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging…
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
We study representations of tropical linear spaces as intersections of tropical hyperplanes of circuits. For several classes of matroids, we describe minimal tropical bases. We also show that every realizable tropical linear space has a…
We prove that the ordered configuration spaces of planar graphs have the highest possible topological complexity generically, as predicted by a conjecture of Farber. Our argument establishes the same generic maximality for all higher…
Finding paths in graphs is a fundamental graph-theoretic task. In this work, we we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident…
The strengthening of linear relaxations and bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit…
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational…