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))

Keywords

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Research supported by the National Science Foundation through grant #DMI-0098427 and by the Office of Naval Research through contract N00014-97-1-0196.