## Algebraic structures and algorithms for matching and matroid problems

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Citations: | 10 - 2 self |

### BibTeX

@MISC{Harvey_algebraicstructures,

author = {Nicholas J. A. Harvey},

title = {Algebraic structures and algorithms for matching and matroid problems},

year = {}

}

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### Abstract

We present new algebraic approaches for several wellknown combinatorial problems, including non-bipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For non-bipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal.

### Citations

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(Show Context)
Citation Context ... basic path-matching problems and a randomized algorithm for constructing basic path-matchings in Õ(nω ) time, where n is the number of vertices and ω < 2.38 is the exponent for matrix multiplication =-=[7]-=-. Our approach involves two assumptions. First, as is common, we assume that the matroids associated with the basic path-matching problem are linear. Additionally, we make the mild technical assumptio... |

704 |
Linear Algebra and Its Applications
- Strang
- 1976
(Show Context)
Citation Context ...� � I 0 · B I � , � � I − Y 0 . 0 I = X(i) ∗, i:n = X(n) ∗, i . Now consider It is well-known that B∗,i is precisely the (lower half of the) column involved in the i th elimination (see, e.g., Strang =-=[32]-=-). Thus B = X (n) = X ⊗ Y . The lemma follows by observing that X = B · (I − Y ). � We can therefore compute NSW = Ũ ⊗(− ˜ C · ˜ V ) in O(nω ) time by the sequential update lemma. The columns from NSW... |

552 |
Combinatorial Optimization: Polyhedra and Efficiency
- Schrijver
- 2003
(Show Context)
Citation Context ...mization [24]. Matroid intersection is another fundamental problem; its min-max characterization and efficient algorithms have had far-reaching consequences in both combinatorics and computer science =-=[31]-=-. Basic path-matchings, due to Cunningham and Geelen [11, 12], are an elegant generalization of these two classic problems. Cunningham and Geelen showed that maximum-weight basic path-matchings can be... |

302 | Maximum matchings and a polyhedron with (0, 1)-vertices - Edmonds - 1965 |

241 |
Submodular functions, matroids, and certain polyhedra
- Edmonds
- 1970
(Show Context)
Citation Context ...-factors in odd-cycle symmetric graphs. Matroid intersection algorithms have a more storied history. Polynomial time algorithms for matroid intersection were developed in the 1970s by various authors =-=[13, 14, 22]-=-. The efficiency of these early algorithms was typically measured relative to an oracle for testing independence. Cunningham [9] later developed an efficient algorithm for intersection of linear matro... |

222 | Combinatorial Optimization
- Cook, Cunningham, et al.
- 1998
(Show Context)
Citation Context ... intersection algorithm resembles a well-known proof of the matroid intersection theorem, for which it is remarked that the “proof gives no hint of how to find the [matroid intersection] efficiently” =-=[6, p289]-=-. Now suppose that we consider only algorithms using naive matrix multiplication. Under this restriction, the best known algorithm for linear matroid intersection is Cunningham’s, which uses O(nr 2 lo... |

147 | Algebraic Combinatorics - Godsil - 1993 |

142 | An O( √ |V| ·|E|) algorithm for finding maximum matching in general graphs - Micali, Vazirani - 1980 |

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80 |
Matching Theory. Akadémiai Kiadó
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- 1986
(Show Context)
Citation Context ...lem of Mucha and Sankowski (FOCS 2004). 1 Introduction Non-bipartite matching is a fundamental problem that has played a pivotal role in the development of graph theory and combinatorial optimization =-=[24]-=-. Matroid intersection is another fundamental problem; its min-max characterization and efficient algorithms have had far-reaching consequences in both combinatorics and computer science [31]. Basic p... |

60 |
On determinants, matchings and random algorithms
- Lovász
- 1979
(Show Context)
Citation Context ...ms for matroid intersection and bipartite matroid matching are all based on augmenting paths; indeed, so are most algorithms for these problems [1, 9, 14, 15, 16, 17, 18, 22, 34]. In contrast, Lovász =-=[23]-=- introduced a randomized algebraic approach for matroid matchings; this approach was employed and extended by Barvinok [3] and Camerini et al. [5]. This paper, building on results of Geelen [20], exte... |

57 |
Maximum matchings via Gaussian elimination
- Mucha, Sankowski
- 2004
(Show Context)
Citation Context ...aic framework to basic path-matching problems. Algebraic approaches have recently been used by Mucha and Sankowski to obtain efficient algorithms for matching problems in bipartite and general graphs =-=[25]-=-. For bipartite graphs, they consider the Edmonds matrix and apply a simple but elegant variant of the Hopcroft-Bunch Gaussian elimination algorithm [4]. For general graphs, their algorithm is much mo... |

52 |
Triangular factorization and inversion by fast matrix multiplication
- Bunch, Hopcroft
- 1974
(Show Context)
Citation Context ...ching problems in bipartite and general graphs [25]. For bipartite graphs, they consider the Edmonds matrix and apply a simple but elegant variant of the Hopcroft-Bunch Gaussian elimination algorithm =-=[4]-=-. For general graphs, their algorithm is much more complicated: it maintains the canonical partition of the graph using sophisticated data structures for testing dynamic connectivity. Sankowski [30] a... |

46 |
Matrices and Matroids for Systems Analysis
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- 2000
(Show Context)
Citation Context ...n det M = � J⊂{1,...,n}, |J|=|I| det M[I,J] · detM[ Ī, ¯ J] · (−1) � i∈I i+� j∈J j . (2) An analogous statement holds for a set of rows I by taking the transpose of A. Proof. See Aitken [2] or Murota =-=[27]-=-. � 3sA matroid is a tuple M = (V, I, B,r) where V is the ground set, I ⊆ 2 V is the collection of independent sets, B ⊆ I is the collection of bases, and r is the rank function. The rank of the matro... |

42 | A theory of alternating paths and blossoms for proving correctness of the O( √ VE) general graph maximum matching algorithm - Vazirani - 1994 |

37 |
Matroid intersection
- Edmonds
- 1979
(Show Context)
Citation Context ...-factors in odd-cycle symmetric graphs. Matroid intersection algorithms have a more storied history. Polynomial time algorithms for matroid intersection were developed in the 1970s by various authors =-=[13, 14, 22]-=-. The efficiency of these early algorithms was typically measured relative to an oracle for testing independence. Cunningham [9] later developed an efficient algorithm for intersection of linear matro... |

28 |
Submodular Functions and Optimization, volume 58 of Annals of Discrete Mathematics. Elsevier, second edition, 2005. Documents de Travail du Centre d'Economie de la Sorbonne - 2011.59 M. Grabisch. The core of games on ordered structures and graphs. 4OR: A
- Fujishige
(Show Context)
Citation Context ...aphs are not solvable in polynomial time (in the oracle model) [24]. On the other hand, bipartite matroid matching problems are much more tractable, and have close connections to matroid intersection =-=[16]-=-. It was shown by Edmonds [13, Theorem 81] that bipartite matroid matching reduces to matroid intersection, and therefore can be solved efficiently. Efficient algorithms for bipartite matroid matching... |

25 | Approximating capacited routing and delivery problem - Chalasani, Motwani - 1999 |

24 | GEELEN: The optimal path-matching problem
- CUNNINGHAM, F
- 1996
(Show Context)
Citation Context ...l problem; its min-max characterization and efficient algorithms have had far-reaching consequences in both combinatorics and computer science [31]. Basic path-matchings, due to Cunningham and Geelen =-=[11, 12]-=-, are an elegant generalization of these two classic problems. Cunningham and Geelen showed that maximum-weight basic path-matchings can be found in polynomial time via the ellipsoid method [12]. Late... |

22 |
A weighted matroid intersection algorithm
- Frank
- 1981
(Show Context)
Citation Context ...plicit in Tomizawa-Iri [34]. The aforementioned algorithms for matroid intersection and bipartite matroid matching are all based on augmenting paths; indeed, so are most algorithms for these problems =-=[1, 9, 14, 15, 16, 17, 18, 22, 34]-=-. In contrast, Lovász [23] introduced a randomized algebraic approach for matroid matchings; this approach was employed and extended by Barvinok [3] and Camerini et al. [5]. This paper, building on re... |

22 |
Matroid intersection algorithms
- Lawler
- 1975
(Show Context)
Citation Context ...-factors in odd-cycle symmetric graphs. Matroid intersection algorithms have a more storied history. Polynomial time algorithms for matroid intersection were developed in the 1970s by various authors =-=[13, 14, 22]-=-. The efficiency of these early algorithms was typically measured relative to an oracle for testing independence. Cunningham [9] later developed an efficient algorithm for intersection of linear matro... |

22 | Deterministic network coding by matrix completion - Harvey, Karger, et al. - 2005 |

21 |
Maximum matchings in general graphs through randomization
- Rabin, Vazirani
- 1984
(Show Context)
Citation Context ...Z[A,B], by the Hopcroft-Bunch [4] algorithm. Lovász [23] showed that |A| is twice the cardinality of a maximum matching. Furthermore, it is known that Z[A,A] also has full rank (by Frobenius’ theorem =-=[29]-=- or the Grassman-Plücker identity [27, p434]). This implies that G[A] has a perfect matching, so we may apply the algorithm of Section 3 to G[A]. Since Z(M) is skew-symmetric, the matrix N = Z(M) −1 i... |

20 | Dynamic Transitive Closure via Dynamic Matrix Inverse - Sankowski |

16 | Randomized Parallel Algorithms for Matroid Union and Intersection, With Applications to Arboresences and Edge-Disjoint Spanning Trees - NARAYANAN, SARAN, et al. - 1994 |

15 |
New algorithms for linear k-matroid intersection and matroid k-parity problems
- Barvinok
- 1995
(Show Context)
Citation Context ...s for these problems [1, 9, 14, 15, 16, 17, 18, 22, 34]. In contrast, Lovász [23] introduced a randomized algebraic approach for matroid matchings; this approach was employed and extended by Barvinok =-=[3]-=- and Camerini et al. [5]. This paper, building on results of Geelen [20], extends the algebraic framework to basic path-matching problems. Algebraic approaches have recently been used by Mucha and San... |

15 | Random pseudo-polynomial algorithms for exact matroid problems
- Camerini, Galbiati, et al.
- 1992
(Show Context)
Citation Context ... 9, 14, 15, 16, 17, 18, 22, 34]. In contrast, Lovász [23] introduced a randomized algebraic approach for matroid matchings; this approach was employed and extended by Barvinok [3] and Camerini et al. =-=[5]-=-. This paper, building on results of Geelen [20], extends the algebraic framework to basic path-matching problems. Algebraic approaches have recently been used by Mucha and Sankowski to obtain efficie... |

12 | On the history of combinatorial optimization (till - Schrijver - 1960 |

11 |
Improved bounds for matroid partition and intersection
- Cunningham
- 1986
(Show Context)
Citation Context ...id intersection were developed in the 1970s by various authors [13, 14, 22]. The efficiency of these early algorithms was typically measured relative to an oracle for testing independence. Cunningham =-=[9]-=- later developed an efficient algorithm for intersection of linear matroids (those that can ∗ Supported by a Natural Sciences and Engineering Research Council of Canada PGS Scholarship. 1sbe represent... |

11 |
Valuated matroid intersection, II: algorithms
- Murota
- 1996
(Show Context)
Citation Context ... digraphs known as “weakly-symmetric”. Cunningham and Geelen [10] devised a combinatorial algorithm to compute a maximum even-factor in such graphs. Combining this with a matroid (or valuated matroid =-=[26]-=-) intersection algorithm, they obtain an algorithm to compute a maximum (or maximum-weight) basic even-factor. Pap [28] considers even-factors in a class of digraphs known as “odd-cycle symmetric”, an... |

11 | Randomized Õ(M(|V |)) Algorithms for Problems in Matching Theory - Cheriyan - 1997 |

9 |
Efficient theoretic and practical algorithms for linear matroid intersection problems
- Gabow, Xu
- 1996
(Show Context)
Citation Context ...st class of matroids with efficient representations. Cunningham’s algorithm requires O(nr 2 log r) time, where n denotes the size of the ground set and r bounds the rank of the matroids. Gabow and Xu =-=[17, 18]-=- obtained an improved bound of O(nr 1.62 ) through the use of fast matrix multiplication and quite technical arguments. However, their bound does not seem to be a natural one; Gabow and Xu explicitly ... |

8 | Bounded degree minimum spanning trees - Goemans |

6 |
Vertex-disjoint directed paths and even circuits. Manuscript
- Cunningham, Geelen
(Show Context)
Citation Context ...atchings can be found in polynomial time via the ellipsoid method [12]. Later, path-matchings were generalized to evenfactors in a class of digraphs known as “weakly-symmetric”. Cunningham and Geelen =-=[10]-=- devised a combinatorial algorithm to compute a maximum even-factor in such graphs. Combining this with a matroid (or valuated matroid [26]) intersection algorithm, they obtain an algorithm to compute... |

5 |
A combinatorial algorithm to find a maximum even
- Pap
- 2005
(Show Context)
Citation Context ...m even-factor in such graphs. Combining this with a matroid (or valuated matroid [26]) intersection algorithm, they obtain an algorithm to compute a maximum (or maximum-weight) basic even-factor. Pap =-=[28]-=- considers even-factors in a class of digraphs known as “odd-cycle symmetric”, and gives a combinatorial algorithm to find a maximum even-factor in such graphs. Finally, Takazawa [33] gives a strongly... |

4 | Matching theory for combinatorial geometries
- Aigner, Dowling
- 1971
(Show Context)
Citation Context ...eorem 81] that bipartite matroid matching reduces to matroid intersection, and therefore can be solved efficiently. Efficient algorithms for bipartite matroid matching were implicit in Aigner-Dowling =-=[1]-=- and explicit in Tomizawa-Iri [34]. The aforementioned algorithms for matroid intersection and bipartite matroid matching are all based on augmenting paths; indeed, so are most algorithms for these pr... |

4 |
Efficient algorithms for independent assignments on graphic and linear matroids
- Gabow, Xu
- 1989
(Show Context)
Citation Context ...st class of matroids with efficient representations. Cunningham’s algorithm requires O(nr 2 log r) time, where n denotes the size of the ground set and r bounds the rank of the matroids. Gabow and Xu =-=[17, 18]-=- obtained an improved bound of O(nr 1.62 ) through the use of fast matrix multiplication and quite technical arguments. However, their bound does not seem to be a natural one; Gabow and Xu explicitly ... |

4 |
Matching theory. Lecture notes from the Euler Institute for Discrete Mathematics and its Applications
- Geelen
- 2001
(Show Context)
Citation Context ...¯J 2 Q J 1 Q ¯ J 1 X del(J,J) By an argument similar to Theorem 3, it follows that the cardinality of a maximum intersection in M1/J and M2/J is rankZ(J)+|J|−|V |. This result was mentioned by Geelen =-=[21]-=- (and attributed to Murota [27]) for the special case that J = ∅. The characterization of allowed elements for matroid intersection problems is as follows. We identify the elements of V with the rows ... |

4 |
Processor efficient parallel matching
- Sankowski
- 2005
(Show Context)
Citation Context ...hm [4]. For general graphs, their algorithm is much more complicated: it maintains the canonical partition of the graph using sophisticated data structures for testing dynamic connectivity. Sankowski =-=[30]-=- also developed an RNC 5 algorithm for constructing perfect matchings that uses only Õ(nω ) processors, yielding another sequential algorithm that uses only Õ(n ω ) time. However, this algorithm is ba... |

4 | A weighted even factor algorithm
- Takazawa
- 2008
(Show Context)
Citation Context ...even-factor. Pap [28] considers even-factors in a class of digraphs known as “odd-cycle symmetric”, and gives a combinatorial algorithm to find a maximum even-factor in such graphs. Finally, Takazawa =-=[33]-=- gives a strongly-polynomial algorithm to compute maximum-weight even-factors in odd-cycle symmetric graphs. Matroid intersection algorithms have a more storied history. Polynomial time algorithms for... |

3 |
Determinants and Matrices. Interscience
- Aitken
- 1956
(Show Context)
Citation Context ...{1,...,n}. Then det M = � J⊂{1,...,n}, |J|=|I| det M[I,J] · detM[ Ī, ¯ J] · (−1) � i∈I i+� j∈J j . (2) An analogous statement holds for a set of rows I by taking the transpose of A. Proof. See Aitken =-=[2]-=- or Murota [27]. � 3sA matroid is a tuple M = (V, I, B,r) where V is the ground set, I ⊆ 2 V is the collection of independent sets, B ⊆ I is the collection of bases, and r is the rank function. The ra... |

3 |
An algorithm for determining the rank of a triple matrix product AXB with application to the problem of discerning the existence of the unique solution in a network
- Tomizawa, Iri
- 1974
(Show Context)
Citation Context ...matching reduces to matroid intersection, and therefore can be solved efficiently. Efficient algorithms for bipartite matroid matching were implicit in Aigner-Dowling [1] and explicit in Tomizawa-Iri =-=[34]-=-. The aforementioned algorithms for matroid intersection and bipartite matroid matching are all based on augmenting paths; indeed, so are most algorithms for these problems [1, 9, 14, 15, 16, 17, 18, ... |

2 | The Theory of Matrices, volume 1 - Gantmakher - 1960 |

2 | Applications of matroid theory - Iri - 1983 |