We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We present an exact partial enumerative algorithm for the maximum clique problem. The pruning device used is derived from graph colorings. Pruning of the search tree is accomplished not only by the number of colors used to color a tree subproblem but also by using information gained in the process of coloring. This leads to increased pruning which translates into improved computational performance. Experimental results on test problems are presented.

An exact algorithm for the maximum clique problem with accelerated pruning

Job-shop scheduling with blocking and no-wait constraints

Branch-and-bound algorithms

The allocation of surgeries to operating rooms (ORs) is a challenging combinatorial optimization problem. There is also significant uncertainty in the duration of surgical procedures, which further complicates assignment decisions. In this paper, we present stochastic optimization models for the assignment of surgeries to ORs on a given day of surgery. The objective includes a fixed cost of opening ORs and a variable cost of overtime relative to a fixed length-of-day. We describe two types of models. The first is a two-stage stochastic linear program with binary decisions in the first stage and simple recourse in the second stage. The second is its robust counterpart, in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations. We describe the mathematical models, bounds on the optimal solution, and solution methodologies, including an easy-to-implement heuristic. Numerical experiments based on real data from a large health-care provider are used to contrast the results for the two models and illustrate the potential for impact in practice. Based on our numerical experimentation, we find that a fast and easy-to-implement heuristic works fairly well, on average, across many instances. We also find that the robust method performs approximately as well as the heuristic, is much faster than solving the stochastic recourse model, and has the benefit of limiting the worst-case outcome of the recourse problem.

/pdf/optimal-allocation-of-surgery-blocks-to-operating-rooms-7kj6utwulm.pdf

Optimal Allocation of Surgery Blocks to Operating Rooms Under Uncertainty

http://www.optimization-online.org/DB_FILE/2006/04/1373.pdf

A Lagrangian heuristic for satellite range scheduling with resource constraints

We introduce orbital branching, an effective branching method for integer programs containing a great deal of symmetry. The method is based on computing groups of variables that are equivalent with respect to the symmetry remaining in the problem after branching, including symmetry that is not present at the root node. These groups of equivalent variables, called orbits, are used to create a valid partitioning of the feasible region that significantly reduces the effects of symmetry while still allowing a flexible branching rule. We also show how to exploit the symmetries present in the problem to fix variables throughout the branch-and-bound tree. Orbital branching can easily be incorporated into standard integer programming software. Through an empirical study on a test suite of symmetric integer programs, the question as to the most effective orbit on which to base the branching decision is investigated. The resulting method is shown to be quite competitive with a similar method known as isomorphism pruning and significantly better than a state-of-the-art commercial solver on symmetric integer programs.

Orbital branching

http://directory.umm.ac.id/Data%20Elmu/jurnal/O/Operations%20Research%20Letters/Vol28.Issue2.2001/1337.pdf

A branch-and-cut algorithm for the maximum cardinality stable set problem

We introduce orbital branching, an effective branching method for integer programs containing a great deal of symmetry. The method is based on computing groups of variables that are equivalent with respect to the symmetry remaining in the problem after branching, including symmetry which is not present at the root node. These groups of equivalent variables, called orbits, are used to create a valid partitioning of the feasible region which significantly reduces the effects of symmetry while still allowing a flexible branching rule. We also show how to exploit the symmetries present in the problem to fix variables throughout the branch-and-bound tree. Orbital branching can easily be incorporated into standard IP software. Through an empirical study on a test suite of symmetric integer programs, the question as to the most effective orbit on which to base the branching decision is investigated. The resulting method is shown to be quite competitive with a similar method known as isomorphism pruningand significantly better than a state-of-the-art commercial solver on symmetric integer programs.

Stefano Smriglio

Papers

A Lagrangian heuristic for satellite range scheduling with resource constraints

Orbital branching

A branch-and-cut algorithm for the maximum cardinality stable set problem

Orbital Branching

A three-dimensional matching model for perishable production scheduling