Optimality conditions and complete description of polytopes in combinatorial optimization

TL;DR

This paper develops a comprehensive set of optimality conditions for combinatorial optimization problems, enabling a complete polyhedral description of their feasible regions through linear inequalities.

Contribution

It introduces a general analytical framework for describing the polytopes of combinatorial optimization problems, extending beyond extreme point convex hulls.

Findings

Provides conditions for optimality based on cones of vectors

Characterizes facet-inducing inequalities for polytopes

Enables complete polyhedral descriptions of combinatorial problems

Abstract

A combinatorial optimization problem (COP) has a finite groundset $E(\left|E\right|=N$), a weight vector $c=\left(c^e:e\in E\right)$ and a family $T\in E$ of feasible subsets with objective to find $t\in T$ with maximal weight: ${max}\{\sum_{e\in t}c^e$: $t\in T\}$. Polyhedral combinatorics reformulates combinatorial optimization as linear program: $T$ is mapped into the set $X\in R^N$ of 0/1 incidence vectors and $c\in R^N$ is maximized over the convex hull of $X$: ${max}\{cx: x\in conv(X)\}$. In theory, complementary slackness conditions for the induced linear program provide optimality conditions for the COP. However, in general case, optimality conditions for combinatorial optimization have not been formulated analytically as for many problems complete description of the induced polytopes is available only as a convex hull of extreme points rather than a system of linear inequalities. Here, we formulate optimality conditions for a COP in general case: $x_k\in X\ \ $is optimal if and only if $c\in cone\left(H_k\right)\ $ where $H_k$ $=\{h\in V:\ {hx}_k\ \ge hx$ for all $x\in X\}$ and $V$ is the set of all -1/0/1 valued vectors in $R^N$. This provides basis to get, in theory, a complete description of a combinatorial polytope induced by any COP: all facet inducing inequalities for $conv(X)$ can be written as $hx\le l$ where $h\in V$ and $l$ is integer. A vector $h\in V$ induces a nonredundent facet if and only if $h\in H_k^o$ $\in H_k$ for at least one $x_k\in X\ $(where $H_k^o$$=\{h\in H_k:h\notin cone(H_k\setminus \{h\})\}$ ) and $l=\ $ $x_kh$.