The b-Matching Problem in Distance-Hereditary Graphs and Beyond
We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((k log^2{k})*(m+n) * log{n})-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.
maximum-cardinality matching
b-matching
FPT in P
split decomposition
distance-hereditary graphs
Mathematics of computing~Graph theory
Theory of computation~Design and analysis of algorithms
30:1-30:13
Regular Paper
This work was supported by the Institutional research programme PN 1819 "Advanced IT resources to support digital transformation processes in the economy and society - RESINFO-TD" (2018), project PN 1819-01-01 "Modeling, simulation, optimization of complex systems and decision support in new areas of IT&C research", funded by the Ministry of Research and Innovation, Romania.
https://arxiv.org/abs/1804.09393
Guillaume
Ducoffe
Guillaume Ducoffe
ICI – National Institute for Research and Development in Informatics, Bucharest, Romania , The Research Institute of the University of Bucharest ICUB, Bucharest, Romania
Alexandru
Popa
Alexandru Popa
University of Bucharest, Bucharest, Romania , ICI – National Institute for Research and Development in Informatics, Bucharest, Romania
10.4230/LIPIcs.ISAAC.2018.30
H.-J. Bandelt and H. Mulder. Distance-hereditary graphs. J. of Combinatorial Theory, Series B, 41(2):182-208, 1986.
C. Berge. Two theorems in graph theory. Proceedings of the National Academy of Sciences, 43(9):842-844, 1957.
J. A. Bondy and U. S. R. Murty. Graph theory. Grad. Texts in Math., 2008.
P. Charbit, F. De Montgolfier, and M. Raffinot. Linear time split decomposition revisited. SIAM J. on Discrete Mathematics, 26(2):499-514, 2012.
D. Coudert, G. Ducoffe, and A. Popa. Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. In SODA'18, pages 2765-2784. SIAM, 2018.
W. Cunningham. Decomposition of directed graphs. SIAM Journal on Algebraic Discrete Methods, 3(2):214-228, 1982.
F. Dragan. On greedy matching ordering and greedy matchable graphs. In WG'97, volume 1335 of LNCS, pages 184-198. Springer, 1997.
F. Dragan and F. Nicolai. LexBFS-orderings of distance-hereditary graphs with application to the diametral pair problem. Discrete Applied Mathematics, 98(3):191-207, 2000.
G. Ducoffe and A. Popa. A quasi linear-time b-Matching algorithm on distance-hereditary graphs and bounded split-width graphs. Technical Report arXiv:1804.09393, arXiv, 2018.
G. Ducoffe and A. Popa. The use of a pruned modular decomposition for Maximum Matching algorithms on some graph classes. In ISAAC, 2018. To appear.
J. Edmonds. Paths, trees, and flowers. Canadian J. of mathematics, 17(3):449-467, 1965.
J. Edmonds and E. Johnson. Matching: A well-solved class of integer linear programs. In Combinatorial structures and their applications. Citeseer, 1970.
F. Fomin, D. Lokshtanov, M. Pilipczuk, S. Saurabh, and M. Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. In SODA'17, pages 1419-1432. SIAM, 2017.
H. Gabow. An Efficient Reduction Technique for Degree-Constrained Subgraph and Bidirected Network Flow Problems. In STOC'83, pages 448-456. ACM, 1983.
H. Gabow. Data Structures for Weighted Matching and Extensions to b-matching and f-factors. arXiv, 2016. URL: http://arxiv.org/abs/1611.07541.
http://arxiv.org/abs/1611.07541
A. C. Giannopoulou, G. B. Mertzios, and R. Niedermeier. Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. Theoretical Computer Science, 689:67-95, 2017.
M. Golumbic and U. Rotics. On the clique-width of some perfect graph classes. International J. of Foundations of Computer Science, 11(03):423-443, 2000.
J. Hopcroft and R. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on computing, 2(4):225-231, 1973.
Y. Iwata, T. Ogasawara, and N. Ohsaka. On the Power of Tree-Depth for Fully Polynomial FPT Algorithms. In STACS'18, 2018.
S. Kratsch and F. Nelles. Efficient and adaptive parameterized algorithms on modular decompositions. In ESA'18. LIPIcs, 2018. To appear.
G. Mertzios, A. Nichterlein, and R. Niedermeier. The Power of Linear-Time Data Reduction for Maximum Matching. In MFCS'17, pages 46:1-46:14, 2017.
S. Micali and V. Vazirani. An O(√V E) Algorithm for Finding Maximum Matching in General Graphs. In FOCS'80, pages 17-27. IEEE, 1980.
M. Rao. Solving some NP-complete problems using split decomposition. Discrete Applied Mathematics, 156(14):2768-2780, 2008.
L. Roditty and V. Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In STOC'13, pages 515-524. ACM, 2013.
W. Tutte. A short proof of the factor theorem for finite graphs. Canad. J. Math, 6(1954):347-352, 1954.
M.-S. Yu and C.-H. Yang. An O(n)-time algorithm for maximum matching on cographs. Information processing letters, 47(2):89-93, 1993.
Guillaume Ducoffe and Alexandru Popa
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode