Semidefinite Programming Relaxations for the Quadratic Assignment Problem

Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e., the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods.For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting the special structure of the relaxation. See e.g., Vandenverghe and Boyd (1995) for a similar approach for solving SDP problems arising from control applications.Numerical results are presented which indicate that the described methods yield at least competitive lower bounds.     

A Semidefinite Programming Approach to the Quadratic Knapsack Problem

In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.     

A Tight Semidefinite Relaxation of the MAX CUT Problem

We obtain a tight semidefinite relaxation of the MAX CUT problem which improves several previous SDP relaxation in the literature. Not only is it a strict improvement over the SDP relaxation obtained by adding all the triangle inequalities to the well-known SDP relaxation, but also it satisfy Slater constraint qualification (strict feasibility).     

Restarting after Branching in the SDP Approach to MAX-CUT and Similar Combinatorial Optimization Problems

Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tree should be exploited at the child node in order to speed up the solution of the child. We show how the solution to the parent relaxation can be used as a warm start to construct an appropriate initial dual solution to the child problem. This restart method for SDP branch-and-cut can be regarded as analogous to the use of the dual simplex method in the branch-and-cut method for mixed integer linear programming problems.     

Geometry of Semidefinite Max-Cut Relaxations via Matrix Ranks

Semidefinite programming (SDP) relaxations are proving to be a powerful tool for finding tight bounds for hard discrete optimization problems. This is especially true for one of the easier NP-hard problems, the Max-Cut problem (MC). The well-known SDP relaxation for Max-Cut, here denoted SDP1, can be derived by a first lifting into matrix space and has been shown to be excellent both in theory and in practice.Recently the present authors have derived a new relaxation using a second lifting. This new relaxation, denoted SDP2, is strictly tighter than the relaxation obtained by adding all the triangle inequalities to the well-known relaxation. In this paper we present new results that further describe the remarkable tightness of this new relaxation. Let denote the feasible set of SDP2 for the complete graph with n nodes, let F _n denote the appropriately defined projection of into , the space of real symmetric n × n matrices, and let C _n denote the cut polytope in . Further let be the matrix variable of the SDP2 relaxation and X F _n be its projection. Then for the complete graph on 3 nodes, F ₃ = C ₃ holds. We prove that the rank of the matrix variable of SDP2 completely characterizes the dimension of the face of the cut polytope in which the corresponding matrix X lies. This shows explicitly the connection between the rank of the variable Y of the second lifting and the possible locations of the projected matrix X within C ₃. The results we prove for n = 3 cast further light on how SDP2 captures all the structure of C ₃, and furthermore they are stepping stones for studying the general case n 4. For this case, we show that the characterization of the vertices of the cut polytope via rank Y = 1 extends to all n 4. More interestingly, we show that the characterization of the one-dimensional faces via rank Y = 2 also holds for n 4. Furthermore, we prove that if rank Y = 2 for n 3, then a simple algorithm exhibits the two rank-one matrices (corresponding to cuts) which are the vertices of the one-dimensional face of the cut polytope where X lies.     

An exact semidefinite programming approach for the max-mean dispersion problem

This paper proposes an exact algorithm for the Max-Mean dispersion problem (\(Max-Mean DP\)), an NP-Hard combinatorial optimization problem whose aim is to select the subset of a set such that the average distance between elements is maximized. The problem admits a natural non-convex quadratic fractional formulation from which a semidefinite programming (SDP) relaxation can be derived. This relaxation can be tightened by means of a cutting plane algorithm which iteratively adds the most violated triangular inequalities. The proposed approach embeds the SDP relaxation and the cutting plane algorithm into a branch and bound framework to solve \(Max-Mean DP\) instances to optimality. Computational experiments show that the proposed method is able to solve to optimality in reasonable time instances with up to 100 elements, outperforming other alternative approaches.     

On Approximate Graph Colouring and MAX-k-CUT Algorithms Based on the θ-Function

The problem of colouring a k-colourable graph is well-known to be NP-complete, for k 3. The MAX-k-CUT approach to approximate k-colouring is to assign k colours to all of the vertices in polynomial time such that the fraction of `defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum (1997), using a semidefinite programming (SDP) relaxation which is related to the Lovász -function. In a related work, Karger et al. (1998) devised approximation algorithms for colouring k-colourable graphs exactly in polynomial time with as few colours as possible. They also used an SDP relaxation related to the -function.In this paper we further explore semidefinite programming relaxations where graph colouring is viewed as a satisfiability problem, as considered in De Klerk et al. (2000). We first show that the approximation to the chromatic number suggested in De Klerk et al. (2000) is bounded from above by the Lovász -function. The underlying semidefinite programming relaxation in De Klerk et al. (2000) involves a lifting of the approximation space, which in turn suggests a provably good MAX-k-CUT algorithm. We show that of our algorithm is closely related to that of Frieze and Jerrum; thus we can sharpen their approximation guarantees for MAX-k-CUT for small fixed values of k. For example, if k = 3 we can improve their bound from 0.832718 to 0.836008, and for k = 4 from 0.850301 to 0.857487. We also give a new asymptotic analysis of the Frieze-Jerrum rounding scheme, that provides a unifying proof of the main results of both Frieze and Jerrum (1997) and Karger et al. (1998) for k 0.     

Almost optimal solutions for bin coloring problems

In this paper we study two interesting bin coloring problems: Minimum Bin Coloring Problem (MinBC) and Online Maximum Bin Coloring Problem (OMaxBC), motivated from several applications in networking. For the MinBC problem, we present two near linear time approximation algorithms to achieve almost optimal solutions, i.e., no more than OPT+2 and OPT+1 respectively, where OPT is the optimal solution. For the OMaxBC problem, we first introduce a deterministic 2-competitive greedy algorithm, and then give lower bounds for any deterministic and randomized (against adaptive offline adversary) online algorithms. The lower bounds show that our deterministic algorithm achieves the best possible competitive ratio. The research of this paper was partially supported by an NSF CAREER award CCF-0546509.     

On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

The Kidney Exchange Problem (KEP) is a combinatorial optimization problem and has attracted the attention from the community of integer programming/combinatorial optimisation in the past few years. Defined on a directed graph, the KEP has two variations: one concerns cycles only, and the other, cycles as well as chains on the same graph. We call the former a Cardinality Constrained Multi-cycle Problem (CCMcP) and the latter a Cardinality Constrained Cycles and Chains Problem (CCCCP). The cardinality for cycles is restricted in both CCMcP and CCCCP. As for chains, some studies in the literature considered cardinality restrictions, whereas others did not. The CCMcP can be viewed as an Asymmetric Travelling Salesman Problem that does allow subtours, however these subtours are constrained by cardinality, and that it is not necessary to visit all vertices. In existing literature of the KEP, the cardinality constraint for cycles is usually considered to be small (to the best of our knowledge, no more than six). In a CCCCP, each vertex on the directed graph can be included in at most one cycle or chain, but not both. The CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights, particularly due to their similarities to some travelling salesman- and vehicle routing-family of problems. In this paper, our main focus is to review the existing mathematical programming models and solution methods in the literature, analyse the performance of these models, and identify future research directions. Further, we propose a polynomial-sized and an exponential-sized mixed-integer linear programming model, discuss a number of stronger constraints for cardinality-infeasible-cycle elimination for the latter, and present some preliminary numerical results.     

The bandpass problem: combinatorial optimization and library of problems

A combinatorial optimization problem, called the Bandpass Problem, is introduced. Given a rectangular matrix A of binary elements {0,1} and a positive integer B called the Bandpass Number, a set of B consecutive non-zero elements in any column is called a Bandpass. No two bandpasses in the same column can have common rows. The Bandpass problem consists of finding an optimal permutation of rows of the matrix, which produces the maximum total number of bandpasses having the same given bandpass number in all columns. This combinatorial problem arises in considering the optimal packing of information flows on different wavelengths into groups to obtain the highest available cost reduction in design and operating the optical communication networks using wavelength division multiplexing technology. Integer programming models of two versions of the bandpass problems are developed. For a matrix A with three or more columns the Bandpass problem is proved to be NP-hard. For matrices with two or one column a polynomial algorithm solving the problem to optimality is presented. For the general case fast performing heuristic polynomial algorithms are presented, which provide near optimal solutions, acceptable for applications. High quality of the generated heuristic solutions has been confirmed in the extensive computational experiments. As an NP-hard combinatorial optimization problem with important applications the Bandpass problem offers a challenge for researchers to develop efficient computational solution methods. To encourage the further research a Library of Bandpass Problems has been developed. The Library is open to public and consists of 90 problems of different sizes (numbers of rows, columns and density of non-zero elements of matrix A and bandpass number B), half of them with known optimal solutions and the second half, without.     

Improved approximation algorithms for the max-bisection and the disjoint 2-catalog segmentation problems

We consider the max-bisection problem and the disjoint 2-catalog segmentation problem, two well-known NP-hard combinatorial optimization problems. For the first problem, we apply the semidefinite programming (SDP) relaxation and the RPR² technique of Feige and Langberg (J. Algorithms 60:1–23, 2006) to obtain a performance curve as a function of the ratio of the optimal SDP value over the total weight through finer analysis under the assumption of convexity of the RPR² function. This ratio is shown to be in the range of [0.5,1]. The performance curve implies better approximation performance when this ratio is away from 0.92, corresponding to the lowest point on this curve with the currently best approximation ratio of 0.7031 due to Feige and Langberg (J. Algorithms 60:1–23, 2006). For the second problem, similar technique results in an approximation ratio of 0.7469, improving the previously best known result 0.7317 due to Wu et al. (J. Ind. Manag. Optim. 8:117–126, 2012).     

On threshold BDDs and the optimal variable ordering problem

Many combinatorial optimization problems can be formulated as 0/1 integer programs (0/1 IPs). The investigation of the structure of these problems raises the following tasks: count or enumerate the feasible solutions and find an optimal solution according to a given linear objective function. All these tasks can be accomplished using binary decision diagrams (BDDs), a very popular and effective datastructure in computational logics and hardware verification. We present a novel approach for these tasks which consists of an output-sensitive algorithm for building a BDD for a linear constraint (a so-called threshold BDD) and a parallel AND operation on threshold BDDs. In particular our algorithm is capable of solving knapsack problems, subset sum problems and multidimensional knapsack problems. BDDs are represented as a directed acyclic graph. The size of a BDD is the number of nodes of its graph. It heavily depends on the chosen variable ordering. Finding the optimal variable ordering is an NP-hard problem. We derive a 0/1 IP for finding an optimal variable ordering of a threshold BDD. This 0/1 IP formulation provides the basis for the computation of the variable ordering spectrum of a threshold function. We introduce our new tool azove 2.0 as an enhancement to azove 1.1 which is a tool for counting and enumerating 0/1 points. Computational results on benchmarks from the literature show the strength of our new method.     

A low complexity semidefinite relaxation for large-scale MIMO detection

Many wireless communication problems is based on a convex relaxation of the maximum likelihood problem which further can be cast as binary quadratic programs (BQPs). The two standard relaxation methods that are widely used for solving general BQPs such as spectral methods and semidefinite programming problem (SDP), each have their own advantages and disadvantages. It is widely accepted that small and medium sized SDP problems can be solved efficiently by interior point methods. Albeit, semidefinite relaxation has a tighter bound for large scale problems, but its computational complexity is high. However, Row-by-Row method (RBR) for solving SDPs could be opted for an alternative for large-scale MIMO detection because of low complexity. The present work is a spectral SDP-cut formulation to which the RBR is applied for large-scale MIMO detection. A modified RBR algorithm with tighter bound is presented to specify the efficiency in detecting massive MIMO.     

Exact solution method to solve large scale integer quadratic multidimensional knapsack problems

In this paper we develop a branch-and-bound algorithm for solving a particular integer quadratic multi-knapsack problem. The problem we study is defined as the maximization of a concave separable quadratic objective function over a convex set of linear constraints and bounded integer variables. Our exact solution method is based on the computation of an upper bound and also includes pre-procedure techniques in order to reduce the problem size before starting the branch-and-bound process. We lead a numerical comparison between our method and three other existing algorithms. The approach we propose outperforms other procedures for large-scaled instances (up to 2000 variables and constraints). A extended abstract of this paper appeared in LNCS 4362, pp. 456–464, 2007.     

Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the SteinLib library.     

On Disjunctive Cuts for Combinatorial Optimization

In the successful branch-and-cut approach to combinatorial optimization, linear inequalities are used as cutting planes within a branch-and-bound framework. Although researchers often prefer to use facet-inducing inequalities as cutting planes, good computational results have recently been obtained using disjunctive cuts, which are not guaranteed to be facet-inducing in general.A partial explanation for the success of the disjunctive cuts is given in this paper. It is shown that, for six important combinatorial optimization problems (the clique partitioning, max-cut, acyclic subdigraph, linear ordering, asymmetric travelling salesman and set covering problems), certain facet-inducing inequalities can be obtained by simple disjunctive techniques. New polynomial-time separation algorithms are obtained for these inequalities as a by-product.The disjunctive approach is then compared and contrasted with some other general-purpose frameworks for generating cutting planes and some conclusions are made with respect to the potential and limitations of the disjunctive approach.     

Attacking the Market Split Problem with Lattice Point Enumeration

The market split problem was proposed by Cornuéjols and Dawande as benchmark problem for algorithms solving linear systems with 0/1 variables. Here, we present an algorithm for the more general problem A · x = b with arbitrary lower and upper bound on the variables. The algorithm consists of exhaustive enumeration of all points of a suitable lattice which are contained in a given polyhedron. We present results for the feasibility version as well as for the integer programming version of the market split problem which indicate that the algorithm outperforms the previously published approaches to this problems considerably.     

Solving the Multidimensional Assignment Problem by a Cross-Entropy method

The Multidimensional Assignment Problem (MAP) is a higher dimensional version of the linear assignment problem, where we find tuples of elements from given sets, such that the total cost of the tuples is minimal. The MAP has many recognized applications such as data association, target tracking, and resource planning. While the linear assignment problem is solvable in polynomial time, the MAP is NP-hard. In this work, we develop a new approach based on the Cross-Entropy (CE) methods for solving the MAP. Exploiting the special structure of the MAP, we propose an appropriate family of discrete distributions on the feasible set of the MAP that allow us to design an efficient and scalable CE algorithm. The efficiency and scalability of our method are proved via several tests on large-scale problems with up to 5 dimensions and 20 elements in each dimension, which is equivalent to a 0–1 linear program with 3.2 millions binary variables and 100 constraints.     

Multi-way clustering and biclustering by the Ratio cut and Normalized cut in graphs

In this paper, we consider the multi-way clustering problem based on graph partitioning models by the Ratio cut and Normalized cut. We formulate the problem using new quadratic models. Spectral relaxations, new semidefinite programming relaxations and linearization techniques are used to solve these problems. It has been shown that our proposed methods can obtain improved solutions. We also adapt our proposed techniques to the bipartite graph partitioning problem for biclustering.     

Solution Structure of Some Inverse Combinatorial Optimization Problems

In this paper we consider some inverse combinatorial optimization problems, that is, for a given set of feasible solutions of an optimization problem, we wish to know: under what weight vectors (or capacity vectors) will these feasible solutions become optimal solutions? We analysed shortest path problem, minimum cut problem, minimum spanning tree problem and maximum-weight matching problem, and found that in each of these cases, the solution set of the inverse problem is charaterized by solving another combinatorial optimization problem. The main tool in our approach is Fulkerson's theory of blocking and anti-blocking polyhedra with some necessary revisions.