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## The optimal path-matching problem (1997)

Venue: | COMBINATORICA |

Citations: | 24 - 2 self |

### Citations

1891 |
Theory of linear and integer programming.
- Schrijver
- 1987
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Citation Context ...0 The independent path-matching polytope We now prove Theorem 1.7 as a consequence of Corollary 3.2. For the special case of the Matching Polyhedron Theorem, the proof reduces to one due to Schrijver =-=[18]-=-. Proof of Theorem 1.7: It is clear that inequalities (9){(15) are valid for conv(K ). Now suppose that x 2 R E satises all of these inequalities. Create a copy ~v of each v 2 V , and for S V , d... |

712 |
Matching Theory
- Lovasz, Plummer
- 1986
(Show Context)
Citation Context ... x( (S)) jS \ Rj (T 1 S T 1 [R) (10) x( (S)) jS \ Rj (T 2 S T 2 [R) (11) x( (S)) jSj 1 (S R; jSj odd) (12) x((A)) r 1 (A) (A T 1 ) (13) x((A)) r 2 (A) (A T 2 ) (14) x 0: =-=(15)-=- Theorem 1.7 is proved from Theorem 1.4 in Section 3. It is easy to derive from it the polyhedral theorems of Edmonds on matchings and common independent sets. We also call attention to the path-match... |

481 |
The ellipsoid method and its consequences in combinatorial optimization,”
- Grotschel, Lovasz, et al.
- 1981
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Citation Context ...o prove the algorithmic results stated above from Theorem 1.4. Proof of Theorems 1.2 and 1.3 from Theorem 1.4: By the equivalence of optimization and separation|see Grotschel, Lovasz, and Schrijver =-=[13]-=-|it is possible to optimize an arbitrary linear function over conv(K(G;M 1 ;M 2 )) in polynomial time if and only if it is possible to solve the separation problem for the same polytope in polynomial ... |

408 |
Maximum matching and a polyhedron with 0, 1 vertices,
- Edmonds
- 1963
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Citation Context ...ubset of edges such that each vertex of G is incident to exactly one edge of the subset. Tutte [19] gave a necessary and sucient condition for the existence of a perfect matching. Later Edmonds [6], =-=[7]-=- gave polynomial-time algorithms to decide whether a given graph has a perfect matching, and (given a weighting of the edges) tosnd a perfect matching of maximum weight. He also gave 1 a polyhedral de... |

346 | Submodular functions, matroids, and certain polyhedra, Combinatorial optimization—Eureka, you shrink
- Edmonds
- 2003
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Citation Context ... [5] proved the total dual integrality of the system of inequalities. Given matroids M 1 ;M 2 on the same set T , a common basis of M 1 ;M 2 is a subset of T that is a basis in both matroids. Edmonds =-=[9]-=- gave a necessary and sucient condition for the existence of a common basis, and polynomial-time algorithms to determine whether there exists a common basis and tosnd a common basis of maximum weight... |

174 |
The factorization of linear graphs,
- Tutte
- 1947
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Citation Context ... certain matrix of indeterminates. 1 Introduction Given a graph G = (V;E); a perfect matching of G is a subset of edges such that each vertex of G is incident to exactly one edge of the subset. Tutte =-=[19]-=- gave a necessary and sucient condition for the existence of a perfect matching. Later Edmonds [6], [7] gave polynomial-time algorithms to decide whether a given graph has a perfect matching, and (gi... |

111 |
Odd minimum cut-sets and b-matchings
- Padberg, Rao
- 1982
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Citation Context ...es (2) can be checked by solving a minimum-cut problem. Inequalities (3) require a more sophisticated use of minimum-cut methods, but these can also be checked in polynomial time; see Padberg and Rao =-=[16]-=-. Next, x̂ satises inequalities (4) and (6) if and only if the vector y 2 R T 1 dened by y v = x̂((v)) is in the convex hull of incidence vectors of bases of M 1 . Polynomial-time algorithms for th... |

41 |
A primal algorithm for optimum matching
- Cunningham, Marsh
- 1978
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Citation Context ...e also gave 1 a polyhedral description of the perfect matchings of G, by characterizing their convex hull as the solution set of a certain system of linear inequalities. Finally, Cunningham and Marsh =-=[5]-=- proved the total dual integrality of the system of inequalities. Given matroids M 1 ;M 2 on the same set T , a common basis of M 1 ;M 2 is a subset of T that is a basis in both matroids. Edmonds [9] ... |

38 |
Systems of distinct representatives and linear
- Edmonds
- 1967
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Citation Context ..., both of size k, we want to determine whether the submatrix A[I; J ] is nonsingular, that is, whether its determinant is nonzero (as a polynomial), or more generally, to determine its rank. (Edmonds =-=[8]-=- seems to have been thesrst to emphasize such algorithmic questions. For example, he proposed the problem ofsnding a polynomial-time algorithm to compute the rank of a matrix whose entries are multiva... |

37 |
Testing membership in matroid polyhedra
- CUNNINGHAM
- 1984
(Show Context)
Citation Context ...he vector y 2 R T 1 dened by y v = x̂((v)) is in the convex hull of incidence vectors of bases of M 1 . Polynomial-time algorithms for the separation problem for this polytope are given in [13] and =-=[2]-=-. The inequalities involving M 2 can be handled similarly. This completes the proof. Note that the algorithms that result from these proofs use the ellipsoid method, and are not practical. Independent... |

27 | The perfectly matchable subgraph polytope of a bipartite graph. Networks - Balas, Pulleyblank - 1983 |

20 |
private communication
- Lovász
- 1983
(Show Context)
Citation Context ...orem 2.2 with the min-max theorem Theorem 1.9 to obtain a formula for the rank of A[I; J ]. However, this formula is not really new. It can be proved directly using a linear-algebra method of Lovasz =-=[14]-=-. His proof, which can be found in [11], predates our polyhedral proof, also in [11], and our generalization, whichsrst appeared in [4]. Theorem 2.4 The rank of A[I; J ] is equal to the minimum over a... |

18 |
Transversals and matroid
- Edmonds, Fulkerson
- 1965
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Citation Context ...efore, we 9 may abbreviate conv(fsK : K 2 K g) to conv(K ).) Theorem 1.7 conv(K (G;M 1 ;M 2 )) is the set of all x 2 R E satisfying: x((v)) 2 (v 2 R) (9) x( (S)) jS \ Rj (T 1 S T 1 [R) =-=(10)-=- x( (S)) jS \ Rj (T 2 S T 2 [R) (11) x( (S)) jSj 1 (S R; jSj odd) (12) x((A)) r 1 (A) (A T 1 ) (13) x((A)) r 2 (A) (A T 2 ) (14) x 0: (15) Theorem 1.7 is proved from Theorem 1.... |

18 | Tutte, The factorizations of linear graphs - T - 1947 |

10 |
Short proofs on the matching polyhedron
- Schrijver
- 1983
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Citation Context ...gral polyhedron. This is false. Our proof of Theorem 1.4 follows a technique that was used previously in proofs of Edmonds' description of the perfect matching polyhedron (Theorem 1.5); see Schrijver =-=[17]-=- or Green-Krotki [12]. The proof uses Theorem 1.5, but could easily be modied to avoid doing so. Proof of Theorem 1.4: Let P (G;M 1 ;M 2 ) R E (or simply P ) denote the polyhedron dened by the in... |

4 |
A Separation Algorithm for the Matchable Set Polytope
- Cunningham, Green-Krótki
- 1994
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Citation Context ...ation problem is based on the ellipsoid method, and so is not combinatorial, and is not strongly polynomial. There does exist a polynomial-time combinatorial algorithm for the separation problem; see =-=[3]-=-. However, that algorithm uses scaling, and is not strongly polynomial. Here we describe a strongly polynomial algorithm, based on the results of this paper. In fact, it was this problem that original... |

2 |
Matching Polyhedra
- Green-Krotki
- 1980
(Show Context)
Citation Context ...is false. Our proof of Theorem 1.4 follows a technique that was used previously in proofs of Edmonds' description of the perfect matching polyhedron (Theorem 1.5); see Schrijver [17] or Green-Krotki =-=[12]-=-. The proof uses Theorem 1.5, but could easily be modied to avoid doing so. Proof of Theorem 1.4: Let P (G;M 1 ;M 2 ) R E (or simply P ) denote the polyhedron dened by the inequalities (1){(8). Cl... |

2 | Matching Theory - LovJsz, PLUMMER - 1986 |

2 |
The perfectly matchable subgraph polyhedron of an arbitrary graph, Combinatorica 9:495{516
- Balas, Pulleyblank
- 1989
(Show Context)
Citation Context ...s of the edges of some matching. The matchable set polyhedron Q(G) of a graph G is the convex hull of incidence vectors of matchable sets of G. This polyhedron was introduced by Balas and Pulleyblank =-=[1]-=-, who gave a nice description by linear inequalities. (We will not need that description here.) There are several ways to obtain a polynomial-time algorithm for the separation problem for Q(G). The ea... |

1 | SCHR1JVER: Theory of Linear and Integer Programming - unknown authors - 1986 |

1 |
Unimodular Matrices, doctoral thesis
- Geelen, Matroids
- 1995
(Show Context)
Citation Context ...he graph obtained from G 0 by deleting the vertices not in I [ J and the edges having both ends in InJ and those having both ends in JnI. The following result can be proved by elementary methods; see =-=[11]-=- or [4]. Theorem 2.2 The rank of A[I; J ] is the maximum ofsK (E) over pathmatchings K with respect to G; InJ; JnI. 13 From Theorems 2.2 and 1.2 we get immediately the following consequence. Corollary... |