The emergence of metaheuristics for solving diÆcult combinatorial optimization problems is one of the most notable achievements of the last two decades in operations research. This paper provides an account of the most recent developments in the eld and identi es some common issues and trends. Examples of applications are also reported for vehicle routing and scheduling problems.

Proof: We show one direction, the one useful for applications. Suppose A is TUM, consider P = {x | x ≥ 0, Ax ≤ b}. Let y = kx∗ be an integral vector where x∗ ∈ P . We prove by induction on k that y = x1 + x2 + . . . .xk for integral vectors x1, x2, . . . , xk in P . Base case for k = 1 is trivial. For k ≥ 2, consider the polyhedron P ′ = {x | 0 ≤ x ≤ y;Ay − kb + b ≤ Ax ≤ b}. P ′ is an integral ...

1. Basic Concepts in Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Connectedness and Optimal Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 3. Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Let f : 2S → R be a submodular set function. We discuss a connection between submodular functions and convexity that was shown by Lovász [3]. Given an arbitrary (not necessarily submodular) set function f : 2S → R, we can view it as assigning values to the integer vectors in the hypercube [0, 1]n where n = |S|. That is, for each U ⊆ S, f(χ(U)) = f(U). We say that a function f̂ : [0, 1]n → R is a...

Throughout this lecture we will use affhull to denote the affine hull, linspace to be the linear space, charcone to denote the characteristic cone and convexhull to be the convex hull. Recall that P = {x | Ax ≤ b} is a polyhedron in Rn where A is a m× n matrix and b is a m× 1 matrix. An inequality aix ≤ bi in Ax ≤ b is an implicit equality if aix = bi ∀x ∈ P . Let I ⊆ {1, 2, . . . ,m} be the in...

Shifted combinatorial optimization is a new nonlinear optimization framework which is a broad extension of standard combinatorial optimization, involving the choice of several feasible solutions at a time. This framework captures well studied and diverse problems ranging from so-called vulnerability problems to sharing and partitioning problems. In particular, every standard combinatorial optim...

One of several major contributions of Edmonds to combinatorial optimization is algorithms and polyhedral theorems for matroid intersection, and more generally polymatroid intersection. From an optimization point of view, the matroid intersection problem is the following: Let M1 = (S, I1) and M2 = (S, I2) be two matroids on the same ground set S. Then I1 ∩ I2 is the collection of all sets that a...