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On the Complexity of Separation: The Three-Index Assignment Problem
Abstract
A fundamental step in any cutting plane algorithm is separation: deciding whether a violated inequality exists within a certain class of inequalities. It is customary to express the complexity of a separation algorithm in n, the number of variables. Here, we argue that the input to a separation algorithm can be expressed in jsup(x)j, where sup(x) denotes the vector containing the positive components of x. This input measure allows one to take sparsity into account. We apply this idea to two known classes of valid inequalities for the three-index assignment problem, and we find separation algorithms with a better complexity than the ones known in literature. We also show empirically the performance of our separation algorithms.
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This paper deals with the cutting-plane approach to the maximum stable set problem. We provide theoretical results regarding the facet-defining property of inequalities obtained by a known project-and-lift-style separation method called edge-projection, and its variants. An implementation of a Branch and Cut algorithm is described, which uses edge-projection and two other separation tools which have been discussed for other problems: local cuts (pioneered by Applegate, Bixby, Chvátal and Cook) and mod-k cuts. We compare the performance of this approach to another one by Rossi and Smiriglio (2001) and discuss the value of the tools we have tested.
The Index Selection Problem (ISP) is a phase of fundamental importance in the physical design of databases, calling for a set of indexes to be built in a database so as to minimize the overall execution time for a given database workload. The problem is a generalization of the well-known Uncapacitated Facility Location Problem (UFLP). In an earlier publication [A. Caprara, J.J. Salazar González, TOP 4 (1996) 135–163], we formulate ISP as a set packing problem, showing that our mathematical model contains all the clique inequalities, and describe a branch-and-cut algorithm based on the separation of odd-hole inequalities. In this paper, we describe an effective exact separation procedure for a suitably-defined family of lifted odd-hole inequalities, obtained by applying a Chvátal-Gomory derivation to the clique inequalities. Our analysis goes in the direction of determining a new class of inequalities over which efficient separation is possible, rather than introducing new classes of (facet-defining) inequalities that later turn out to be difficult to separate. Our separation procedure is embedded within our branch-and-cut algorithm for the exact solution of ISP. Computational results on two different classes of instances are given, showing the effectiveness of the new approach. We also test our algorithm on UFLP instances both taken from the literature and randomly generated.
Valid inequalities for 0-1 knapsack polytopes often prove useful
when tackling hard 0-1 Linear Programming problems. To generate
such inequalities, one needs separation algorithms for them, i.e., rou-
tines for detecting when they are violated. We present new exact
and heuristic separation algorithms for several classes of inequalities,
namely lifted cover, extended cover, weight and lifted pack inequalities.
Moreover, we show how to improve a recent separation algorithm for
the 0-1 knapsack polytope itself. Extensive computational results, on
MIPLIB and OR Library instances, show the strengths and limitations
of the inequalities and algorithms considered.
Winner Determination Problem is the problem of maximizing the benefit when bids can be made on a group of items. in this paper, we consider the set packing formulation of the problem, study its polyhedral structure and then propose a new and tighter formulation. We also present new valid inequalities which are generated by exploiting combinatorial auctions peculiarities. Finally, we implement a branch-and-cut algorithm which shows its efficiency in a big number of instances. (c) 2008 Elsevier B.V. All rights reserved.
We present a strong cutting plane/branch-and-bound algorithm for node packing. The cutting planes are obtained from cliques and lifting odd hole inequalities. Computational results are reported.
We report new results for a time-indexed formulation of nonpreemptive singlemachine scheduling problems. We give complete characterizations of all facet inducing inequalities with integral coefficients and right-hand side 1 or 2 for the convex hull of the set of feasible partial schedules, i.e., schedules in which not all jobs have to be started. Furthermore, we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope, which is the convex hull of the set of feasible complete schedules, i.e., schedules in which all jobs have to be started. To obtain insight in the effectiveness of these classes of facet-inducing inequalities, we develop a branch-and-cut algorithm based on them. We evaluate its performance on the strongly NP-hard single machine scheduling problem of minimizing the weighted sum of the completion times subject to release dates. Key words: scheduling, polyhedral methods, facet inducing inequalities, separation, bra...
INTRODUCTION This chapter deals with approximation algorithms for and applications of multi index assignment problems (MIAPs). MIAPs and relatives of it have a relatively long history both in applications as well as in theoretical results, starting at least in the fifties (see e.g. Motzkin, 1952, Schell, 1955 and Koopmans and Beckmann, 1957). Here we intend to give the reader i) an idea of the range and diversity of practical problems that have been formulated as an MIAP, and ii) an overview on what is known on theoretical aspects of solving instances of MIAPs. In particular, we will discuss complexity and approximability issues for special cases of MIAPs. We feel that investigating special cases of MIAPs is an important topic since real-world instances almost always posses a certain structure that can be exploited when it comes to solving them. Before doing so, let us first describe a somewhat frivolous, quite unrealistic situation that gives rise to an insta
We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope PG of a graph G. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivision W of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facet-inducing for PW . Generalizations arise by allowing subdivision paths to intersect, and by replacing the "hub" of the wheel by a clique. The separation problem for these inequalities can be solved in polynomial time. 1 Introduction Let G = (V; E) be a simple connected graph with jV j = n 2 and jEj = m. A subset of V is called a stable set if it does not contain adjacent vertices of G. Let N be a stable set. The incidence vector of N is x 2 f0; 1g V such that x v = 1 if and only if v 2 N . The stable set polytope of G, denoted by PG , is the convex hull of incidence vectors of stable sets of G. Some well-known valid inequalities for PG ...
This book presents the latest findings on one of the most intensely investigated subjects in computational mathematics--the traveling salesman problem. It sounds simple enough: given a set of cities and the cost of travel between each pair of them, the problem challenges you to find the cheapest route by which to visit all the cities and return home to where you began. Though seemingly modest, this exercise has inspired studies by mathematicians, chemists, and physicists. Teachers use it in the classroom. It has practical applications in genetics, telecommunications, and neuroscience. The authors of this book are the same pioneers who for nearly two decades have led the investigation into the traveling salesman problem. They have derived solutions to almost eighty-six thousand cities, yet a general solution to the problem has yet to be discovered. Here they describe the method and computer code they used to solve a broad range of large-scale problems, and along the way they demonstrate the interplay of applied mathematics with increasingly powerful computing platforms. They also give the fascinating history of the problem--how it developed, and why it continues to intrigue us.
We investigate an integer programming model for multi-dimensional assignment problems. This model enables us to establish the dimension for entire families of assignment polytopes, thus unifying and generalising previous results. In particular, we establish the dimension of the linear assignment polytope as well as that of every axial and planar assignment polytope. Further, for the axial polytopes, we identify a family of clique facets. We also give a necessary condition for the existence of a solution for assignment problems.
We propose a branch and cut algorithm for the Pallet Loading Problem. The 0-1 formulation proposed by Beasley for cutting problems is adapted to the problem, adding new constraints and new procedures for variable reduction. We then take advantage of the relationship between this problem and the maximum independent set problem to use the partial linear description of its associated polyhedron. Finally, we exploit the specific structure of our problem to define the solution graph and to develop efficient separation procedures. We present computational results for the complete sets Cover I (up to 50 boxes) and Cover II (up to 100 boxes).
The multidimensional assignment problem (MAPs) is a higher dimensional version of the standard linear assignment problem. Test problems of known solution are useful in exercising solution methods. A method of generating an axial MAP of controllable size with a known unique solution is presented. Certain characteristics of the generated MAPs that determine realism and difficulty are investigated.
We study two new problems in sequence alignment both from a practical and a theoretical view, using tools from combinatorial optimization to develop branch-and-cut algorithms. The generalized maximum trace formulation captures several forms of multiple sequence alignment problems in a common framework, among them the original formulation of maximum trace. The RNA sequence alignment problem captures the comparison of RNA molecules on the basis of their primary sequence and their secondary structure. Both problems have a characterization in terms of graphs which we reformulate in terms of integer linear programming. We then study the polytopes (or convex hulls of all feasible solutions) associated with the integer linear program for both problems. For each polytope we derive several classes of facet-defining inequalities and show that for some of these classes the corresponding separation problem can be solved in polynomial time. This leads to a polynomial-time algorithm for pairwise sequence alignment that is not based on dynamic programming. Moreover, for multiple sequences the branch-and-cut algorithms for both sequence alignment problems are able to solve to optimality instances that are beyond the range of present dynamic programming approaches.
Given three n-element sequences ai, bi and ci of nonnegative real numbers, the aim is to find two permutations φ and Ψ such that the sum ∑ni = 1aibφ(i)Cψ(i) is minimized (maximized, respectively). We show that the maximization version of this problem can be solved in polynomial time, whereas we present an NP-completeness proof for the minimization version. We identify several special cases of the minimization problem which can be solved in polynomial time, and suggest a local search heuristic for the general case.
Given a graph G, we introduce several classes of valid inequalities, called wheel inequalities, for the stable set polytope of G. Moreover, we show that the corresponding separation problems can be solved in polynomial time. Each wheel configuration generates two wheel inequalities. The most basic wheel configuration is a subdivision of a wheel. More general configurations arise by allowing subdivision paths to intersect, and this generalization is crucial to our solution of the separation problem. A further generalization replaces the centre of the wheel by a clique of fixed size.
E. Balas and M. J. Saltzman [Discrete Appl. Math. 23, No. 3, 201-229 (1989; Zbl 0723.90065); Oper. Res. 39, No. 1, 150-161 (1991; Zbl 0743.90079)] 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.
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.
The (k,s)assignment problem sets a unified framework for studying the facial structure of families of assignment polytopes. Through this framework, we derive classes of clique facets for all axial and planar assignment polytopes. For each of these classes, a polynomial-time separation procedure is described. Furthermore, we provide computational experience illustrating the efficiency of these facet-defining inequalities when applied as cutting planes.
Consider the following classical formulation of the (axial) three-dimensional assignment problem (3DA) (see e.g. Balas and Saltzman (1989)). Given is a complete tripartite graph Kn,n,n
= (I∪J∪K,(I × J) ∪ (I × K) ∪ (J × K)), where I, J, K are disjoint sets of size n, and a cost cijk
for each triangle (i, j, k) ∈ I × J × K. The problem 3DA is to find a subset A of n triangles, A⊆ I × J × K, such that every element of I ∪ J ∪ K occurs in exactly one triangle of A, and the total cost c(A) = ∑(i,j,k)∈Acijk
is minimized. Some recent references to this problem are Balas and Saltzman (1989), Frieze (1974), Frieze and Yadegar (1981), Hansen and Kaufman (1973).
We describe a branch-and-bound algorithm for solving the axial three-index assignment problem. The main features of the algorithm include a Lagrangian relaxation that incorporates a class of facet inequalities and is solved by a modified subgradient procedure to find good lower bounds, a primal heuristic based on the principle of minimizing maximum regret plus a variable depth interchange phase for finding good upper bounds, and a novel branching strategy that exploits problem structure to fix several variables at each node and reduce the size of the total enumeration tree. Computational experience is reported on problems with up to 78 equations and 17,576 variables. The primal heuristics were tested on problems with up to 210 equations and 343,000 variables.
The (axial) three index assignment problem, also known as the threedimensional matching problem, is the problem of assigning one item to one job at one point or interval of time in such a way as to minimize the total cost of the assignment. Until now the most e#cient algorithms explored for solving this problem are based on polyhedral combinatorics. So far, four important facet classes Q, P, B and C have been characterized and O(n 3 ) (linear-time) separation algorithms for five facet subclasses of Q, P and B have been established. The complexity of these separation algorithms is best possible since the number of the variables of three index assignment problem of order n is n 3 . In this paper, we review these progresses and raise some further questions on this topic. 1. INTRODUCTION Consider three disjoint n-sets, I, J , and K, and a weight c ijk associated with each ordered triplet (i, j, k) # I × J × K. The (axial) three index assignment problem, to be denoted AP3...
Algorithms for multi-index assignment problems
- T Dokka
Dokka, T.: Algorithms for multi-index assignment problems. PhD thesis, KU Leuven (2013)
Parametrized grasp heuristics for three-index assignment
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Höfler, B., Fügenschuh, A.: Parametrized grasp heuristics for three-index assignment. EvoCOP Lect.
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