arXiv subjects

May Szedlak

Publications and source records attributed to May Szedlak.

3 recordsLinked to original sources

arXiv2019

Random Sampling with Removal

We study randomized algorithms for constrained optimization, in abstract frameworks that include, in strictly increasing generality: convex programming; LP-type problems; violator spaces; and a setting we introduce, consistent spaces. Such algorithms typically involve a step of finding the optimal solution for a random sample of the constraints. They exploit the condition that, in finite dimension $δ$, this sample optimum violates a provably small expected fraction of the non-sampled constraints, with the fraction decreasing in the sample size~$r$. We extend such algorithms by considering the technique of removal, where a fixed number $k$ of constraints are removed from the sample according to a fixed rule, with the goal of improving the solution quality. This may have the effect of increasing the number of violated non-sampled constraints. We study this increase, and bound it in a variety of general settings. This work is motivated by, and extends, results on removal as proposed for chance-constrained optimization. For many relevant values of $r$, $δ$, and $k$, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of $k$ constraints. For a large range of values of $k$, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases, and not in the generality we consider. Moreover, we show that our results extend from finite to infinite spaces, for chance-constrained optimization.

cs.CG↗

arXiv2017

On the Complexity of Polytopes in $LI(2)$

In this paper we consider polytopes given by systems of $n$ inequalities in $d$ variables, where every inequality has at most two variables with nonzero coefficient. We denote this family by $LI(2)$. We show that despite of the easy algebraic structure, polytopes in $LI(2)$ can have high complexity. We construct a polytope in $LI(2)$, whose number of vertices is almost the number of vertices of the dual cyclic polytope, the difference is a multiplicative factor of depending on $d$ and in particular independent of $n$. Moreover we show that the dual cyclic polytope can not be realized in $LI(2)$.

cs.CC↗

arXiv2016

Redundancies in Linear Systems with two Variables per Inequality

The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs), given by $d$ variables with $n$ inequality constraints. A constraint is called \emph{redundant}, if after its removal, the LP still has the same feasible region. The currently fastest method to detect all redundancies is due to Clarkson: it solves $n$ linear programs, but each of them has at most $s$ constraints, where $s$ is the number of nonredundant constraints. In this paper, we study the special case where every constraint has at most two variables with nonzero coefficients. This family, denoted by $LI(2)$, has some nice properties. Namely, as shown by Aspvall and Shiloach, given a variable $x_i$ and a value $λ$, we can test in time $O(nd)$ whether there is a feasible solution with $x_i = λ$. Hochbaum and Naor present an $O(d^2 n \log n)$ algorithm for solving the feasibility problem in $LI(2)$. Their technique makes use of the Fourier-Motzkin elimination method and the earlier mentioned result by Aspvall and Shiloach. We present a strongly polynomial algorithm that solves redundancy detection in time $O(n d^2 s \log s)$. It uses a modification of Clarkson's algorithm, together with a revised version of Hochbaum and Naor's technique. Finally we show that dimensionality testing can be done with the same running time as solving feasibility.

cs.DS↗