Algebraic Algorithms for B-Matching, Shortest Undirected Paths, and F-Factors

@article{Gabow2013AlgebraicAF,
  title={Algebraic Algorithms for B-Matching, Shortest Undirected Paths, and F-Factors},
  author={Harold N. Gabow and Piotr Sankowski},
  journal={2013 IEEE 54th Annual Symposium on Foundations of Computer Science},
  year={2013},
  pages={137-146}
}
  • H. Gabow, P. Sankowski
  • Published 24 April 2013
  • Mathematics, Computer Science
  • 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
Let G = (V, E) be a graph with f : V → Z+ a function assigning degree bounds to vertices. We present the first efficient algebraic algorithm to find an f-factor. The time is O(f(V )ω). More generally for graphs with integral edge weights of maximum absolute value W we find a maximum weight f-factor in time Õ(Wf(V )ω). (The algorithms are correct with high probability and can be made Las Vegas.) We also present three specializations of these algorithms: For maximum weight perfect f-matching the… 

Figures and Tables from this paper

Data Structures for Weighted Matching and Extensions to b-matching and f-factors
  • H. Gabow
  • Computer Science
    ACM Trans. Algorithms
  • 2018
TLDR
The weighted matching problem on general graphs can be solved in time O(n(m + n log n)) for n and m the number of vertices and edges, respectively, which was previously known only for bipartite graphs.
Faster all-pairs shortest paths via circuit complexity
TLDR
A new randomized method for computing the min-plus product of two n × n matrices is presented, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights.
A Simpler Scaling Algorithm for Weighted Matching in General Graphs
We present a new scaling approach for the maximum weight perfect matching problem in general graphs, with running time O((m + n log n) √ n log(nN)), where n,m,N denote the number of vertices, number
New Algorithms for Maximum Weight Matching and a Decomposition Theorem
We revisit the classical maximum weight matching problem in general graphs with nonnegative integral edge weights. We present an algorithm that operates by decomposing the problem into W unweighted
Scaling Algorithms for Weighted Matching in General Graphs
TLDR
This work presents a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs that runs in O(m√nlog(nN)) time, which matches the running time of the best cardinality matching algorithms on sparse graphs.
Negative-Weight shortest paths and unit capacity minimum cost flow in Õ(m[superscript 10/7] log W) Time
TLDR
This paper shows that each one of these four combinatorial optimization problems on weighted graphs can be solved in Õ(m logW ) time, where W is the absolute maximum weight of an edge in the graph, which gives the first in over 25 years polynomial improvement in their sparse-graph time complexity.
Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in Õ (m10/7 log W) Time (Extended Abstract)
TLDR
This paper studies a set of combinatorial optimization problems on weighted graphs, and shows that each of these four problems can be solved in O(m10/7 log W) time, providing the first polynomial improvement in their sparse-graph time complexity in over 25 years.
A Scaling Algorithm for Weighted f-Factors in General Graphs
TLDR
The running time of the algorithm is independent of f(V), and consequently it first breaks the $\Omega(mn)$ barrier for large $f(V)$ even for the unweighted $f$-factor problem in general graphs.
NC Algorithms for Weighted Planar Perfect Matching and Related Problems
TLDR
A new relatively simple but versatile framework is developed that handles the combinatorial structure of matchings directly and needs to only know weights of appropriately defined matchings from algebraic subroutines.
Approximate Generalized Matching: $f$-Factors and $f$-Edge Covers
TLDR
An efficient method for maintaining {\em relaxed complementary slackness} in generalized matching problems and approximation-preserving reductions between the $f-factor and $f$-edge cover problems are included.
...
1
2
...

References

SHOWING 1-10 OF 48 REFERENCES
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
TLDR
A general framework for solving problems on such graphs using matrix multiplication based on the Baur-Strassen Theorem and Strojohann's determinant algorithm is introduced and an Õ(Wnω) time algorithm for minimum weight perfect matching is presented.
Undirected distances and the postman-structure of graphs
Matching algorithms.
TLDR
A bipartite graph with sets of vertices A, B has a perfect matching iff |A| = |B| and (∀U ⊆ A)|N (U)| ≥ |U |.
Maximum matchings via Gaussian elimination
  • M. Mucha, P. Sankowski
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
TLDR
The results resolve a long-standing open question of whether Lovasz's randomized technique of testing graphs for perfect matching in time O(n/sup w/) can be extended to an algorithm that actually constructs a perfect matching.
Maximum Matchings in General Graphs Through Randomization
Potentials in Undirected Graphs and Planar Multiflows
TLDR
The goal of the present paper is to extrapolate from this $\pm 1$-weighted bipartite special case the arbitrarily weighted general min-path-max-potential theorem and to show some algorithmic consequences related to planar multiflows, the Chinese postman problem, the weighted and unweighted matching structure, etc.
Maximum matching and a polyhedron with 0,1-vertices
TLDR
The emphasis in this paper is on relating the matching problem to the theory of continuous linear programming, and the algorithm described does not involve any "blind-alley programming" -which, essentially, amounts to testing a great many combinations.
Maximum weight bipartite matching in matrix multiplication time
  • P. Sankowski
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 2009
An Efficient Implementation of Edmonds' Algorithm for Maximum Matching on Graphs
TLDR
This paper presents an efficient implementation of Edmonds' algorithm for finding a maximum matching based on a system of labels that encodes the structure of alternating paths.
...
1
2
3
4
5
...