Birkhoff Polytope Research Articles

Location, Scheduling, Design and Integer Programming

Preface. 1. Location Problems. 2. Scheduling and Design Problems. 3. Solution Approaches. 4. Locally Ideal LP Formulations I. 5. Locally Ideal LP Formulations II. 6. Quadratic Scheduling Problems. 7. Quadratic Assignment Polytopes. 8. Solving Small QAPs. A. FORTRAN Programs for Small SQPs. References. Index.

A GRASP for the biquadratic assignment problem

The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). It is a nonlinear integer programming problem where the objective function is a fourth degree multivariable polynomial and the feasible domain is the assignment polytope. BiQAP problems appear in VLSI synthesis. Due to the difficulty of this problem, only heuristic solution approaches have been proposed. In this paper, we propose a new heuristic for the BiQAP, a greedy randomized adaptive search procedure (GRASP). Computational results on instances described in the literature indicate that this procedure consistently finds better solutions than previously described algorithms.

Approximation Algorithms for Quadratic Programming

We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m=1, we rigorously show that an ∈-minimizer, where error ∈ ∈ (0, 1), can be obtained in polynomial time, meaning that the number of arithmetic operations is a polynomial in n, m, and log(1/∈). For m ≥ 2, we present a polynomial-time (1-\(\frac{1}{{m^2 }}\))-approximation algorithm as well as a semidefinite programming relaxation for this problem. In addition, we present approximation algorithms for solving QP under the box constraints and the assignment polytope constraints.

Adjacent Extreme Points Of A Transportation Polytope

The adjacent extreme points of a Transportation Polytope are characterized by the circuit s contained in the support of its vertices. This result generalizes the Balinski and Russakoff [4] result on the Assignment Polytope

Linear-time separation algorithms for the three-index assignment polytope

Balas and Saltzman identified several classes of facet inducing inequalities for the three-index assignment polytope, and gave O( n 4 ) separation algorithms for two of them. We give O( n 3 ) separation algorithms for these two classes of facets, and also for a third class. Since the three-index assignment problem has n 3 variables, these algorithms are linear-time and their complexity is best possible.

The monotonic diameter of the perfect matching and shortest path polytopes

The monotonic diameter is determined for the perfect matching, assignment, and shortest path polytopes. These results show that the diameter of a polytope and its monotonic diameter may be different. Furthermore, the monotonic Hirsch conjecture is true for these polytopes.

Resource constrained assignment problems

In many applications it is necessary to find a minimum weight assignment that satisfies one or several additional resource constraints. For example, consider the problem of assigning persons to jobs where each assignment utilizes at least two scarce resources and the resource utilization is dependent on the person and the type of task. A practical situation where the above might occur is a slaughter house where the “cutters” are assigned to different cut patterns. In this case the resources are the time, the cost and the productivity measured in terms of quality and amount of the end products. In this paper we study the resource constrained assignment problem and derive several classes of valid inequalities based on the properties of the knapsack and assignment polytopes. We also present an algorithm that uses both the linear programming and the Lagrangean relaxation of the original problem in order to solve the separation problem. Some computational experiments are presented.

The Asymmetric Assignment Problem and Some New Facets of the Traveling Salesman Polytope on a Directed Graph

An assignment (spanning union of node-disjoint dicycles) in a directed graph is called asymmetric if it contains at most one arc of each pair $(i, j)$, $( j,i)$. The asymmetric assignment polytope is the convex hull of the incidence vectors of all asymmetric assignments. A class of facets is described for this polytope defined on a complete digraph, associated with certain odd-length closed alternating trails. The inequalities defining these facets are also facet inducing for the traveling salesman polytope defined on the same digraph. Furthermore, this class of facets is distinct from each of the classes identified earlier.

Facets of the three-index assignment polytope

Given three disjoint n-sets and the family of all weighted triplets that contain exactly one element of each set, the 3-index assignment (or 3-dimensional matching) problem asks for a minimum-weight subcollection of triplets that covers exactly (i.e., partitions) the union of the three sets. Unlike the common (2-index) assignment problem, the 3-index problem is NP-complete. In this paper we examine the facial structure of the 3-index assignment polytope (the convex hull of feasible solutions to the problem) with the aid of the intersection graph of the coefficient matrix of the problem's constraint set. In particular, we describe the cliques of the intersection graph as belonging to three distinct classes, and show that cliques in two of three classes induce inequalities that define facets of our polytope. Furthermore, we give an O(n4) procedure (note that the number of variables is n3) for finding a facet-defining clique-inequality violated by a given noninteger solution to the linear programming relaxation of the 3-index assignment problem, or showing that no such inequality exists. We then describe the odd holes of the intersection graph and identify two classes of facets associated with odd holes that are easy to generate. One class has coefficients of 0 or 1, the other class coefficients of 0, 1 or 2. No odd hole inequality has left-hand side coefficients greater than two.

A note on certain subpolytopes of the assignment polytope associated with circulant graphs

For a nonempty and noncomplete circulant graph G on a prime number n of vertices with adjacency matrix A , consider the polytope DS( G ) of all doubly stochastic matrices X which commute with A . It is shown that DS( G ) is integral iff the spectral structure of G is the same as for the undirected cycleon n vertices. The sufficiency part of our theorem is extended to the case of arbitrary composite vertex numbers n .

Odd cycles and a class of facets of the axial 3-index assignment polytope

Odd cycles and a class of facets of the axial 3-index assignment polytope

On assignment functions

We define a class of functions called assignment functions, one of which is the permanent function. These functions evaluated at matrices of 0's and 1's are shown to be of applied and combinational interest. We define assignment polytopes which generalize the classical assignment polytope and use them to obtain bounds for the assignment function. We propose the determination of the minimum and maximum value of an assignment function on its corresponding assignment polytope and conjecture that the maximum value is 1. A natural conjecture for the minimum value is shown to be false. The minimum and maximum value are determined for special assignment functions.

On the Connectivity of Cayley Digraphs

We prove that the atom of a Cayley diagraph which contains the unity is a subgroup. As an application we obtain a short proof for a theorem of Imrich on the connectivity of the assignment polytope. We prove also that the connectivity of a Cayley digraph defined by a minimal generating set is equal to its indegree. This result generalizes a theorem of Godsil in the undirected case. We construct a class of Cayley digraphs with optimal connectivity. A symmetric subclass of our class was constructed by Boesch and Felger.

A property regarding degenerate pivots for linear assignment networks

AbstractGiven an extreme point of an assignment polytope of size m, there exist 2m−1 mm−2 alternative basis representations according to the choice of the m ‐ 1 degenerate basic variables. This note establishes the basis structure which admits the maximum degenerate pivots and that which admits the minimum degenerate pivots. It is shown that the upper bound is achieved if and only if the basis is a chain graph and that the lower bound is achieved if the degenerate basic arcs all originate at some facility or all terminate at some location.

On the Use of Exact and Heuristic Cutting Plane Methods for the Quadratic Assignment Problem

This paper uses the formulation of the quadratic assignment problem as that of minimizing a concave quadratic function over the assignment polytope. Cutting plane procedures are investigated for solving this problem. A lower bound derived on the number of cuts needed for termination indicates that conventional cutting plane procedures would require a huge computational effort for the exact solution of the quadratic assignment problems. However, several heuristics which are derived from the cutting planes produce optimal or good quality solutions early on in the search process. An illustrative example and computational results are presented.

Edge-Connectivity on the Assignment Polytope

In this paper, a proof is constructed to show tbat assignment polytope of order n is edge N(n) connected where N(n) is the number of adjacent vertices to any vertex on the assignment polytope. This settles the conjecture made by Balinski and Russakeff on the assignment polytope in its weeker form. Furthermore, the set of N(n) edge-disjoint paths need not consist of a path of length greater than four. The result is extended to all polytopes of diameter two and paths can be constructed there of provided the corresponding adjacency rules are known.

A versatile scheme for ranking the extreme points of an assignment polytope

AbstractA cutting plane scheme embedded in an implicit enumeration framework is proposed for ranking the extreme points of linear assignment problems. This method is capable of ranking any desired number of extreme points at each possible objective function value. The technique overcomes storage difficulties by being able to perform the ranking at any particular objective function value independently of other objective values. Computational experience on some test problems is provided.

On removing a vertex from the assignment polytope

Necessary and sufficient conditions are given for a doubly stochastic matrix D to be expressible as a convex combination of permutation matrices distinct from the identity, thus solving a problem posed by L. Mirsky [12].

On the connectivity of cayley graphs

It has been shown by M. E. Watkins that the connectivity of edge transitive finite graphs is greatest possible. The main Theorem of this paper weakens the condition of edge transitivity and is used to show that the connectivity of the graph of the assignment polytope is equal to its degree, thereby proving a conjecture of Balinski and Russakoff.

On the truncated assignment polytope

We investigate the convex polytope Ω m,n ( r ) which is the convex hull of the m × n r -subpermutation matrices. The faces of Ω m,n ( r ) are characterized, and formulae are obtained to compute their dimensions. The faces of Ω m,n ( r ) are themselves convex polytopes, and we determine their facets.