On the Cycle Polytope of a Directed Graph and Its Relaxations

Abstract

This paper continues the investigation of the cycle polytope of a directed graph begun by Balas and Oosten [2]. Given a digraph G = (N,A) and the collection C of its simple directed cycles, the cycle polytope defined on G is P C ≔ conv {X C:C∈C}, where χ C is the incidence vector of C. According to the integer programming formulation given in [2], P C is the convex hull of points x∈ℝ satisfying $$ x(\delta ^ + (i)) - x(\delta ^ - (i)) = 0{\text{ }}for{\text{ all }}i \in N $$ ((1)) , $$ x(\delta ^ + (i)) \leqslant 1{\text{ }}for{\text{ all }}i \in N $$ ((2)) , $$ \begin{array}{*{20}c} { - x(S,N\backslash S) + x(\delta ^ + (i)) + x(\delta ^ + (j)) \leqslant 1{\text{ }}for{\text{ all }}S \subseteq N,2 \leqslant |S| \leqslant n - 2,} \\ {i \in S,j \in N\backslash S} \\ \end{array} $$ ((3)) , $$ \sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^n {x_{\pi (i)\pi (j)} } \geqslant 1{\text{ for all permutations }}\pi {\text{ of }}N} $$ ((4)) , $$ x_{ij} \in \{ 0,1\} {\text{ }}for all (i,j) \in A $$ ((5)) KeywordsDirect GraphLinear OrderingValid InequalityLinear Programming RelaxationIncidence VectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.