George L. Nemhauser

Integer and combinatorial optimization

FOUNDATIONS. The Scope of Integer and Combinatorial Optimization. Linear Programming. Graphs and Networks. Polyhedral Theory. Computational Complexity. Polynomial-Time Algorithms for Linear Programming. Integer Lattices. GENERAL INTEGER PROGRAMMING. The Theory of Valid Inequalities. Strong Valid Inequalities and Facets for Structured Integer Programs. Duality and Relaxation. General Algorithms. Special-Purpose Algorithms. Applications of Special- Purpose Algorithms. COMBINATORIAL OPTIMIZATION. Integral Polyhedra. Matching. Matroid and Submodular Function Optimization. References. Indexes.

An analysis of approximations for maximizing submodular set functions--I

LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)≥z(S⋃T)+z(S⋂T) for allS, T inN. Such a function is called submodular. We consider the problem maxS⊂N{a(S):|S|≤K,z(S) submodular}.Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1)/K]K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.

Branch-And-Price: Column Generation for Solving Huge Integer Programs

We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

Vertex packings: Structural properties and algorithms

We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.

The fleet assignment problem: solving a large-scale integer program

Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.

Flight String Models for Aircraft Fleeting and Routing

Given a schedule of flight legs to be flown by an airline, the fleet assignment problem is to determine the minimum cost assignment of flights to aircraft types, called fleets, such that each scheduled flight is assigned to exactly one fleet, and the resulting assignment is feasible to fly given a limited number of aircraft in each fleet. Then the airline must determine a sequence of flights, or routes, to be flown by individual aircraft such that assigned flights are included in exactly one route, and all aircraft can be maintained as necessary. This is referred to as the aircraft routing problem. In this paper, we present a single model and solution approach to solve simultaneously the fleet assignment and aircraft routing problems. Our approach is robust in that it can capture costs associated with aircraft connections and complicating constraints such as maintenance requirements. By setting the number of fleets to one, our approach can be used to solve the aircraft routing problem alone. We show how to extend our model and solution approach to solve aircraft routing problems with additional constraints requiring equal aircraft utilization. With data provided by airlines, we provide computational results for the combined fleet assignment and aircraft routing problems without equal utilization requirements and for aircraft routing problems requiring equal aircraft utilization.

Best Algorithms for Approximating the Maximum of a Submodular Set Function

A real-valued function z whose domain is all of the subsets of N = {1,..., n is said to be submodular if zS + zT ≥ zS ∪ T + zS ∩ T, ∀S, T ⊆ N, and nondecreasing if zS ≤ zT, ∀S ⊂ T ⊆ N. We consider the problem maxS⊂N {zS: |S| ≤ K, z submodular and nondecreasing, zO = 0}. Many combinatorial optimization problems can be posed in this framework. For example, a well-known location problem and the maximization of certain boolean polynomials are in this class. We present a family of algorithms that involve the partial enumeration of all sets of cardinality q and then a greedy selection of the remaining elements, q = 0,..., K-1. For fixed K, the qth member of this family requires Onq+1 computations and is guaranteed to achieve at least

Min-cut clustering

\biggl[1-\biggl\frac{K-q}{K}\biggr\biggl\frac{K-q-1}{K-q}\biggr^{K-q}\biggr]\times100 \quad\mbox {percent of the optimum value}.