Combinatorial (or discrete) optimization is one of the most active fields in the interface of operations research, computer science, and applied mathΒ ematics. Combinatorial optimization problems arise in various applications, including communications network design, VLSI design, machine vision, airΒ line crew scheduling, corporate planning, computer-aided design and manΒ ufacturing, database query design, cellular telephone frequency assignment, constraint directed reasoning, and computational biology. Furthermore, combinatorial optimization problems occur in many diverse areas such as linear and integer programming, graph theory, artificial intelligence, and number theory. All these problems, when formulated mathematically as the minimization or maximization of a certain function defined on some domain, have a commonality of discreteness. Historically, combinatorial optimization starts with linear programming. Linear programming has an entire range of important applications including production planning and distribution, personnel assignment, finance, allocaΒ tion of economic resources, circuit simulation, and control systems. Leonid Kantorovich and Tjalling Koopmans received the Nobel Prize (1975) for their work on the optimal allocation of resources. Two important discoverΒ ies, the ellipsoid method (1979) and interior point approaches (1984) both provide polynomial time algorithms for linear programming. These algoΒ rithms have had a profound effect in combinatorial optimization. Many polynomial-time solvable combinatorial optimization problems are special cases of linear programming (e.g. matching and maximum flow). In addiΒ tion, linear programming relaxations are often the basis for many approxiΒ mation algorithms for solving NP-hard problems (e.g. dual heuristics).