Chapter 2 POLYHEDRAL COMBINATORICS Robert D. Carr Discrete Mathematics and Algorithms Department, Sandia National Laboratories
[email protected] Goran Konjevod Computer Science and Engineering Department, Arizona State University
[email protected] Abstract Polyhedral combinatorics is a rich mathematical subject motivated by integer and linear programming. While not exhaustive, this survey cov- ers a variety of interesting topics, so let's get right to it! Keywords: combinatorial optimization, integer programming, linear programming, polyhedron, relaxation, separation, duality, compact optimization, pro- jection, lifting, dynamic programming, total dual integrality, integrality gap, approximation algorithms Introduction There exist several excellent books [15, 24, 29, 35, 37] and survey articles [14, 33, 36] on combinatorial optimization and polyhedral com- binatorics. We do not see our article as a competitor to these fine works. Instead, in the first half of our Chapter, we focus on the very founda- tions of polyhedral theory, attempting to present important concepts and problems as early as possible. This material is for the most part classical and we do not trace the original sources. In the second half (Sections 5{7), we present more advanced material in the hope of draw- ing attention to what we believe are some valuable and useful, but less well explored, directions in the theory and applications of polyhedral combinatorics. We also point out a few open questions that we feel have not been given the consideration they deserve. 2-2 Polyhedral Combinatorics 2.1 Combinatorial Optimization and Polyhedral Combinatorics Consider an undirected graph G = (V; E) (i.e. where each edge is a set of 2 vertices (endpoints) in no specified order).