research-article

Perfect Phylogenies via Branchings in Acyclic Digraphs and a Generalization of Dilworth’s Theorem

Authors:

Ademir Hujdurović,

Martin Milanić,

Alexandru I. TomescuAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 2

Article No.: 20, Pages 1 - 26

https://doi.org/10.1145/3182178

Published: 16 April 2018 Publication History

Abstract

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, 2018) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurović et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs.

We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including (1) a strengthening of the heuristic by Hujdurović et al. via a new min-max result in digraphs generalizing Dilworth’s theorem, which may be of independent interest; (2) APX-hardness results for both problems; (3) approximation algorithms; and (4) exponential-time algorithms solving the two problems to optimality faster than the naïve brute-force approach. Our work relates to several well-studied notions in combinatorial optimization: chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.

References

[1]

A. V. Aho, M. R. Garey, and J. D. Ullman. 1972. The transitive reduction of a directed graph. SIAM J. Comput. 1, 2 (1972), 131--137.

Digital Library

[2]

P. Alimonti and V. Kann. 2000. Some APX-completeness results for cubic graphs. Theoret. Comput. Sci. 237, 1--2 (2000), 123--134.

Digital Library

[3]

J. Araujo, N. Nisse, and S. Pérennes. 2014. Weighted coloring in trees. SIAM J. Discrete Math. 28, 4 (2014), 2029--2041.

[4]

G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. 1999. Complexity and Approximation. Springer-Verlag, Berlin. xx+524 pages.

[5]

J. Cheriyan, T. Jordán, and R. Ravi. 1999. On 2-coverings and 2-packings of laminar families. In Proceedings of the 7th Annual European Symposium on Algorithms (ESA’99). Lecture Notes in Comput. Sci., Vol. 1643. Springer, Berlin, 510--520.

Digital Library

[6]

D. de Werra, M. Demange, B. Escoffier, J. Monnot, and V. Th. Paschos. 2009. Weighted coloring on planar, bipartite and split graphs: Complexity and approximation. Discrete Appl. Math. 157, 4 (2009), 819--832.

Digital Library

[7]

R. P. Dilworth. 1950. A decomposition theorem for partially ordered sets. Ann. Math. (2) 51 (1950), 161--166.

[8]

B. Escoffier, J. Monnot, and V. Th. Paschos. 2006. Weighted coloring: Further complexity and approximability results. Inform. Process. Lett. 97, 3 (2006), 98--103.

Digital Library

[9]

G. F. Estabrook, C. S. Johnson Jr., and F. R. McMorris. 1975. An idealized concept of the true cladistic character. Math. Biosci. 23 (1975), 263--272.

[10]

T. Feder and R. Motwani. 1995. Clique partitions, graph compression and speeding-up algorithms. J. Comput. Syst. Sci. 51, 2 (1995), 261--272. 0022-0000

Digital Library

[11]

A. Frank. 1999. Finding minimum weighted generators of a path system. In Contemporary Trends in Discrete Mathematics (Štiřín Castle, 1997). DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 49. Amer. Math. Soc., Providence, RI, 129--138.

[12]

D. R. Fulkerson. 1956. Note on Dilworth’s decomposition theorem for partially ordered sets. Proc. Amer. Math. Soc. 7 (1956), 701--702.

[13]

H. N. Gabow and K. S. Manu. 1998. Packing algorithms for arborescences (and spanning trees) in capacitated graphs. Math. Program. 82, 1--2, Ser. B (1998), 83--109.

[14]

M. C. Golumbic. 2004. Algorithmic Graph Theory and Perfect Graphs (second ed.). Elsevier Science B.V., Amsterdam.

Digital Library

[15]

D. J. Guan and X. Zhu. 1997. A coloring problem for weighted graphs. Inform. Process. Lett. 61, 2 (1997), 77--81.

Digital Library

[16]

D. Gusfield. 1991. Efficient algorithms for inferring evolutionary trees. Networks 21, 1 (1991), 19--28.

[17]

D. Gusfield. 1997. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. Cambridge University Press.

Digital Library

[18]

I. Hajirasouliha and B. J. Raphael. 2014. Reconstructing mutational history in multiply sampled tumors using perfect phylogeny mixtures. In Proceedings of the 14th International Workshop on Algorithms in Bioinformatics (WABI’14). Lecture Notes in Comput. Sci., Vol. 8701. Springer, Heidelberg, 354--367.

[19]

A. Hujdurović, E. Husić, M. Milanič, R. Rizzi, and A. I. Tomescu. 2017. The minimum conflict-free row split problem revisited. In Proceedings of the 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG’17). Revised Selected Papers. 303--315.

[20]

A. Hujdurović, U. Kačar, M. Milanič, B. Ries, and A. I. Tomescu. 2018. Complexity and algorithms for finding a perfect phylogeny from mixed tumor samples. IEEE Transactions on Computational Biology and Bioinformatics 15 (2018) 96–108. For an extended abstract, see In Proceedings of the 15th International Workshop on Algorithms in Bioinformatics (WABI’15). Lecture Notes in Computer Science 9289 (2015) 80--92.

Digital Library

[21]

O. H. Ibarra and S. Moran. 1981. Deterministic and probabilistic algorithms for maximum bipartite matching via fast matrix multiplication. Inform. Process. Lett. 13, 1 (1981), 12--15. 0020-0190

[22]

K. Jain. 2001. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21, 1 (2001), 39--60.

[23]

F. Le Gall. 2014. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC’14). ACM, New York, 296--303.

Digital Library

[24]

V. Mäkinen, D. Belazzougui, F. Cunial, and A. I. Tomescu. 2015. Genome-Scale Algorithm Design. Cambridge University Press.

[25]

L. S. Moonen and F. C. R. Spieksma. 2008. Partitioning a weighted partial order. J. Comb. Optim. 15, 4 (2008), 342--356.

[26]

S. C. Ntafos and S. L. Hakimi. 1979. On path cover problems in digraphs and applications to program testing. IEEE Trans. Software Engrg. 5, 5 (1979), 520--529.

Digital Library

[27]

M. Sakashita, K. Makino, and S. Fujishige. 2008. Minimizing a monotone concave function with laminar covering constraints. Discrete Appl. Math. 156, 11 (2008), 2004--2019.

Digital Library

[28]

A. Schrijver. 2003. Combinatorial Optimization. Polyhedra and Efficiency. Algorithms and Combinatorics, Vol. 24. Springer-Verlag, Berlin.

[29]

H. Shum and L. E. Trotter Jr. 1996. Cardinality-restricted chains and antichains in partially ordered sets. Discrete Appl. Math. 65, 1--3 (1996), 421--439.

Digital Library

Cited By

Sheu WWang B(2023)Parameterized Complexity for Finding a Perfect Phylogeny from Mixed Tumor SamplesSIAM Journal on Discrete Mathematics10.1137/21M144926937:3(2049-2071)Online publication date: 14-Sep-2023
https://doi.org/10.1137/21M1449269
Chen ZLee CLai H(2022)Speedup the optimization of maximal closure of a node-weighted directed acyclic graphOPSEARCH10.1007/s12597-022-00595-z59:4(1413-1437)Online publication date: 8-Jul-2022
https://doi.org/10.1007/s12597-022-00595-z
Wu YZhu Y(2020)Weighted Rooted Trees: Fat or Tall?Computer Science – Theory and Applications10.1007/978-3-030-50026-9_30(406-418)Online publication date: 22-Jun-2020
https://doi.org/10.1007/978-3-030-50026-9_30
Show More Cited By

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Perfect Phylogenies via Branchings in Acyclic Digraphs and a Generalization of Dilworth’s Theorem

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 2

April 2018

339 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3196491

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

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Publication History

Published: 16 April 2018

Accepted: 01 January 2018

Received: 01 September 2017

Published in TALG Volume 14, Issue 2

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Cited By

Sheu WWang B(2023)Parameterized Complexity for Finding a Perfect Phylogeny from Mixed Tumor SamplesSIAM Journal on Discrete Mathematics10.1137/21M144926937:3(2049-2071)Online publication date: 14-Sep-2023
https://doi.org/10.1137/21M1449269
Chen ZLee CLai H(2022)Speedup the optimization of maximal closure of a node-weighted directed acyclic graphOPSEARCH10.1007/s12597-022-00595-z59:4(1413-1437)Online publication date: 8-Jul-2022
https://doi.org/10.1007/s12597-022-00595-z
Wu YZhu Y(2020)Weighted Rooted Trees: Fat or Tall?Computer Science – Theory and Applications10.1007/978-3-030-50026-9_30(406-418)Online publication date: 22-Jun-2020
https://doi.org/10.1007/978-3-030-50026-9_30
Husić ELi XHujdurović AMehine MRizzi RMäkinen VMilanič MTomescu A(2018)MIPUP: minimum perfect unmixed phylogenies for multi-sampled tumors via branchings and ILPBioinformatics10.1093/bioinformatics/bty68335:5(769-777)Online publication date: 8-Aug-2018
https://doi.org/10.1093/bioinformatics/bty683

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