Our ability to solve large, important combinatorial optimization problems has improved dramatically in the decade. The availability of reliable software, extremely fast and inexpensive hardware and high-level languages that make the modeling of complex problems much faster have led to a much greater demand for optimization tools. This paper highlights the major breakthroughs and then describes ...

Here we are given a finite connected undirected graph (V,E) (so V is the set of vertices and E the set of edges) and weights on the edges, i.e., c(e) ∈ R for all e ∈ E. The task is to find a set T ⊆ E such that (V, T ) is a (spanning) tree and ∑ e∈T c(e) is minimum. (Recall that a tree is a connected graph without cycles.) The figure below shows on the left a set V of eight points in the Euclid...

Proof: By double implication: If there is an augmenting path, then M is not maximum: OK. Suppose there is no augmenting path for M but Mopt is maximum. For each node in M Mopt (where is a xor), the degree is either 0, 1 or 2, thus M Mopt consists of disjoint paths or cycles. • The cycles must be even (otherwise M or Mopt is not a matching). • Suppose there is an odd path. Since #Mopt is maximum...

The p-median problem is one of the discrete optimization problem in location theory which aims to satisfy total demand with minimum cost. A high-level algorithmic approach can be specialized to solve optimization problem. In recent years, meta-heuristic methods have been applied to support the solution of Combinatorial Optimization Problems (COP). Collision Bodies Optimization algorithm (CBO) a...