Basic Polyhedral Theory

This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem, faces of polyhedra, projections of polyhedra, integral polyhedra, total dual integrality, and total unimodularity.Comment: 14 page

Computing Optimal Morse Matchings

Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results

Large, Sparse Optimal Matching with Refined Covariate Balance in an Observational Study of the Health Outcomes Produced by New Surgeons

Every newly trained surgeon performs her first unsupervised operation. How do the health outcomes of her patients compare with the patients of experienced surgeons? Using data from 498 hospitals, we compare 1252 pairs comprised of a new surgeon and an experienced surgeon working at the same hospital. We introduce a new form of matching that matches patients of each new surgeon to patients of an otherwise similar experienced surgeon at the same hospital, perfectly balancing 176 surgical procedures and closely balancing a total of 2.9 million categories of patients; additionally, the individual patient pairs are as close as possible. A new goal for matching is introduced, called refined covariate balance, in which a sequence of nested, ever more refined, nominal covariates is balanced as closely as possible, emphasizing the first or coarsest covariate in that sequence. A new algorithm for matching is proposed and the main new results prove that the algorithm finds the closest match in terms of the total within-pair covariate distances among all matches that achieve refined covariate balance. Unlike previous approaches to forcing balance on covariates, the new algorithm creates multiple paths to a match in a network, where paths that introduce imbalances are penalized and hence avoided to the extent possible. The algorithm exploits a sparse network to quickly optimize a match that is about two orders of magnitude larger than is typical in statistical matching problems, thereby permitting much more extensive use of fine and near-fine balance constraints. The match was constructed in a few minutes using a network optimization algorithm implemented in R. An R package called rcbalance implementing the method is available from CRAN

Charting the Algorithmic Complexity of Waypoint Routing

Modern computer networks support interesting new routing models in which traffic flows from a source sto a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network functions (VNFs), to create innovative network services like service chains or segment routing. While the benefits and technological challenges of providing such routing models have been articulated and studied intensively over the last years, less is known about the underlying algorithmic traffic routing problems. The goal of this paper is to provide the network community with an overview of algorithmic techniques for waypoint routing and also inform about limitations due to computational hardness. In particular, we put the waypoint routing problem into perspective with respect to classic graph theoretical problems. For example, we find that while computing a shortest path from a source s to a destination t is simple (e.g., using Dijkstra's algorithm), the problem of finding a shortest route from s to t via a single waypoint already features a deep combinatorial structure.</jats:p

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes. We consider three classes of fitness models. The first is a continuous phenotype space in which individuals are characterized by a large number of continuously varying traits such as size, weight, color, or concentrations of gene products which directly affect fitness. The second is a simple model that explicitly describes genotype-to-phenotype and phenotype-to-fitness maps allowing for neutrality at both phenotype and fitness levels and resulting in a fitness landscape with tunable correlation length. The third is a class of models in which particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have reduced (or zero) fitness. This class of models can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation. We also demonstrate that the discrete NK model shares some signature properties of models with high correlations. Throughout the paper, our focus is on the percolation threshold, on the number, size and structure of connected clusters, and on the number of viable genotypes.Comment: 31 pages, 4 figures, 1 tabl

Scheduling under Linear Constraints

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.Comment: 21 page