Results 1 
3 of
3
OneThirdIntegrality in the MaxCut Problem
"... Given a graph G = (V; E), the metric polytope S(G) is defined by the inequalities x(F ) \Gamma x(C n F ) jF j \Gamma 1 for F ` C; jF j odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the maxcut problem. The graph G is called 1 d integral if all ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Given a graph G = (V; E), the metric polytope S(G) is defined by the inequalities x(F ) \Gamma x(C n F ) jF j \Gamma 1 for F ` C; jF j odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the maxcut problem. The graph G is called 1 d integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d integral graphs is closed under minors, and we present several minimal forbidden minors for 1 3 integrality. In particular, we characterize the 1 3 integral graphs on 7 nodes. We study several operations preserving 1 d integrality, in particular, the ksum operation for 0 k 3. We prove that series parallel graphs are characterized by the following stronger property. All vertices of the polytope S(G) " fx j ` x ug are 1 3 integral for every choice of 1 3 integral bounds `, u on the edges of G.
Packing Circuits in Matroids
, 2007
"... The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: For any nonnegative integral weight function w defined on E(M), Maximum {k: M has k circuits (repetition allowed) such that each element e of M is used at most 2w(e) times by these circuits} = Mi ..."
Abstract
 Add to MetaCart
(Show Context)
The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: For any nonnegative integral weight function w defined on E(M), Maximum {k: M has k circuits (repetition allowed) such that each element e of M is used at most 2w(e) times by these circuits} = Minimum { � x∈X w(x) : X is a collection of elements (repetition allowed) of M such that every circuit in M meets X at least twice}. Our characterization contains a complete solution to a research problem on 2edgeconnected subgraph polyhedra posed by Cornuéjols, Fonlupt, and Naddef in 1985, which was independently solved by Vandenbussche and Nemhauser in [11].
A Characterization of BoxMengerian Matroid Ports
"... Let M be a matroid on E ∪ {`}, where ` 6 ∈ E is a distinguished element of M. The `port of M is the set P = {P: P ⊆ E with P ∪ {`} a circuit of M}. Let A be the PE incidence matrix. Let U2,4 be the uniform matroid on four elements of rank two, F7 be the Fano matroid, F 7 be the dual of F7, and F+7 ..."
Abstract
 Add to MetaCart
(Show Context)
Let M be a matroid on E ∪ {`}, where ` 6 ∈ E is a distinguished element of M. The `port of M is the set P = {P: P ⊆ E with P ∪ {`} a circuit of M}. Let A be the PE incidence matrix. Let U2,4 be the uniform matroid on four elements of rank two, F7 be the Fano matroid, F 7 be the dual of F7, and F+7 be the unique series extension of F7. In this paper, we prove that the system Ax ≥ 1, x ≥ 0 is boxtotally dual integral (boxTDI) if and only if M has no U2,4minor using `, no F 7minor using `, and no F+7minor using ` as a series element. Our characterization yields a number of interesting results in combinatorial optimization.