The Niceness of Unique Sink Orientations

Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of \emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least $n^{\Omega(2^n)}$ many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness and discuss some algorithmic properties of the reachmap.Comment: An extended abstract appears in the proceedings of Approx/Random 201

The Niceness of Unique Sink Orientations

Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of reachmaps and niceness of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least n^{Omega(2^n)} many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness

A quantitative Doignon-Bell-Scarf theorem

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n, k), depending only on the dimension n and k, such that if a bounded polyhedron {x : Ax<=b} contains exactly k integer points, then there exists a subset of the rows, of cardinality no more than c(n, k), defining a polyhedron that contains exactly the same k integer points. In this case c(n, 0) = 2^n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant c(n, k) and discuss some consequences, including a Clarkson-style algorithm to find the l-th best solution of an integer program with respect to the ordering induced by the objective function

LEO: Learning Efficient Orderings for Multiobjective Binary Decision Diagrams

Approaches based on Binary decision diagrams (BDDs) have recently achieved state-of-the-art results for multiobjective integer programming problems. The variable ordering used in constructing BDDs can have a significant impact on their size and on the quality of bounds derived from relaxed or restricted BDDs for single-objective optimization problems. We first showcase a similar impact of variable ordering on the Pareto frontier (PF) enumeration time for the multiobjective knapsack problem, suggesting the need for deriving variable ordering methods that improve the scalability of the multiobjective BDD approach. To that end, we derive a novel parameter configuration space based on variable scoring functions which are linear in a small set of interpretable and easy-to-compute variable features. We show how the configuration space can be efficiently explored using black-box optimization, circumventing the curse of dimensionality (in the number of variables and objectives), and finding good orderings that reduce the PF enumeration time. However, black-box optimization approaches incur a computational overhead that outweighs the reduction in time due to good variable ordering. To alleviate this issue, we propose LEO, a supervised learning approach for finding efficient variable orderings that reduce the enumeration time. Experiments on benchmark sets from the knapsack problem with 3-7 objectives and up to 80 variables show that LEO is ~30-300% and ~10-200% faster at PF enumeration than common ordering strategies and algorithm configuration. Our code and instances are available at https://github.com/khalil-research/leo

Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ? 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ? 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem

Routing with Congestion in Acyclic Digraphs

We study the version of the $k$-disjoint paths problem where $k$ demand pairs $(s_1,t_1)$, $\dots$, $(s_k,t_k)$ are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than $c$ paths. We show that on directed acyclic graphs the problem is solvable in time $n^{O(d)}$ if we allow congestion $k-d$ for $k$ paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time $f(k)n^{o(d/\log d)}$ for any computable function $f$