Tamon Stephen

On Equitable Resource Allocation Problems: a Lexicographic Minimax Approach

In this expository paper, we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities. Applications for such problems are found in diverse areas, including distribution planning, production planning and scheduling, and emergency services location. Each activity is associated with a performance function, representing, for example, the weighted shortfall of the selected activity level from a specified target. A resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse-off (i.e., larger) performance function value that is associated with a different activity. A lexicographic minimax solution determines this equitable solution; that is, it determines the lexicographically smallest vector whose elements, the performance function values, are sorted in nonincreasing order. The problems reviewed include large-scale allocation problems with multiple knapsack resource constraints, multiperiod allocation problems for storable resources, and problems with substitutable resources. The solution of large-scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures. Indeed, efficient algorithms for many models will be described. We expect that this paper will help practitioners to formulate and solve diverse resource allocation problems, and motivate researchers to explore new models and algorithmic approaches.

Computing knock-out strategies in metabolic networks.

Given a metabolic network in terms of its metabolites and reactions, our goal is to efficiently compute the minimal knock-out sets of reactions required to block a given behavior. We describe an algorithm that improves the computation of these knock-out sets when the elementary modes (minimal functional subsystems) of the network are given. We also describe an algorithm that computes both the knock-out sets and the elementary modes containing the blocked reactions directly from the description of the network and whose worst-case computational complexity is better than the algorithms currently in use for these problems. Computational results are included.

Minimal conflicting sets for the consecutive ones property in ancestral genome reconstruction.

A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has the C1P. Such submatrices have been considered in comparative genomics applications, but very little is known about their combinatorial structure and efficient algorithms to compute them. We first describe an algorithm that detects rows that belong to Minimal Conflicting Sets. This algorithm has a polynomial time complexity when the number of 1s in each row of the considered matrix is bounded by a constant. Next, we show that the problem of computing all Minimal Conflicting Sets can be reduced to the joint generation of all minimal true clauses and maximal false clauses for some monotone boolean function. We use these methods on simulated data related to ancestral genome reconstruction to show that computing Minimal Conflicting Set is useful in discriminating between true positive and false positive ancestral syntenies. We also study a dataset of yeast genomes and address the reliability of an ancestral genome proposal of the Saccharomycetaceae yeasts.

Colourful Simplicial Depth

AbstractInspired by Barany’s Colourful Caratheodory Theorem, we introduce a colourful generalization of Lius simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.

A quadratic lower bound for colourful simplicial depth

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in ℝd is contained in at least ⌊(d+2)2/4⌋ simplices with one vertex from each set.

The colourful feasibility problem

We study a colourful generalization of the linear programming feasibility problem, comparing the algorithms introduced by Barany and Onn with new methods. This is a challenging problem on the borderline of tractability, its complexity is an open question. We perform benchmarking on generic and ill-conditioned problems, as well as recently introduced highly structured problems. We show that some algorithms can lead to cycling or slow convergence and we provide extensive numerical experiments which show that others perform much better than predicted by complexity arguments. We conclude that the most efficient method is a proposed multi-update algorithm.

The distribution of values in the quadratic assignment problem

We obtain a number of results regarding the distribution of values of a quadratic function f on the set of n × n permutation matrices (identified with the symmetric group Sn) around its optimum (minimum or maximum). We estimate the fraction of permutations σ such that f(σ) lies within a given neighborhood of the optimal value of f and relate the optimal value with the average value of f over a neighborhood of the optimal permutation. We describe a natural class of functions (which includes, for example, the objective function in the Traveling Salesman Problem) with a relative abundance of near-optimal permutations. Also, we identify a large class of functions f with the property that permutations close to the optimal permutation in the Hamming metric of Sn tend to produce near optimal values of f (such is, for example, the objective function in the symmetric Traveling Salesman Problem). We show that for general f, just the opposite behavior may take place: an average permutation in the vicinity of the optimal permutation may be much worse than an average permutation in the whole group Sn.

Minimal Conflicting Sets for the Consecutive Ones Property in Ancestral Genome Reconstruction

A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has the C1P. Such submatrices have been considered in comparative genomics applications, but very little is known about their combinatorial structure and efficient algorithms to compute them. We first describe an algorithm that detects rows that belong to Minimal Conflicting Sets. This algorithm has a polynomial time complexity when the number of 1s in each row of the considered matrix is bounded by a constant. Next, we show that the problem of computing all Minimal Conflicting Sets can be reduced to the joint generation of all minimal true clause and maximal false clauses for some monotone boolean function. We use these methods in preliminary experiments on simulated data related to ancestral genome reconstruction.

Embedding a Pair of Graphs in a Surface, and the Width of 4-dimensional Prismatoids

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of d-prismatoids of width larger than d, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases of Q.

More Colourful Simplices

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in ℝd is contained in at least ⌈(d+1)2/2⌉ simplices with one vertex from each set. This improves the known lower bounds for all d≥4.