Carla Groenland
Abstract
We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of \(O(t\sqrt {\log t})\). This is the first algorithm to achieve an f(t)-approximation for some function f.
Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA’18): there exists a universal constant c such that every graph with pathwidth Ω(kc) has treewidth at least k or contains a subdivision of a complete binary tree of height k.
Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t′ in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC’05) for treewidth.
- [1] . 1987. Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8, 2 (1987), 277–284.Google Scholar
Digital Library
- [2] . 2002. Two short proofs concerning tree-decompositions. Combinatorics, Probability and Computing 11, 6 (2002), 541–547.Google ScholarDigital Library
- [3] . 1991. Quickly excluding a forest. Journal of Combinatorial Theory, Series B 52, 2 (1991), 274–283.Google ScholarDigital Library
- [4] . 1992. A tourist guide through treewidth. Acta Cybernetica 11, 1–2 (1992), 1–21.Google Scholar
- [5] . 1996. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 6 (1996), 1305–1317.Google ScholarDigital Library
- [6] . 2012. Fixed-parameter tractability of treewidth and pathwidth. In The Multivariate Algorithmic Revolution and Beyond: Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday. Springer, Berlin, 196–227.Google Scholar
- [7] . 2002. Approximation of pathwidth of outerplanar graphs. Journal of Algorithms 43, 2 (2002), 190–200.Google ScholarDigital Library
- [8] . 1996. Efficient and constructive algorithms for the pathwidth and treewidth of Graphs. Journal of Algorithms 21, 2 (1996), 358–402.Google ScholarDigital Library
- [9] . 1996. A simple linear-time algorithm for finding path-decompositions of small width. Information Processing Letters 57, 4 (1996), 197–203.Google ScholarDigital Library
- [10] . 2016. Polynomial bounds for the grid-minor theorem. J. ACM 63, 5 (2016), Article No. 40.Google ScholarDigital Library
- [11] . 2021. Towards tight(er) bounds for the Excluded Grid Theorem. Journal of Combinatorial Theory, Series B 146 (2021), 219–265.Google ScholarCross Ref
- [12] . 2021. Improved bounds for the excluded-minor approximation of treedepth. SIAM Journal on Discrete Mathematics 35, 2 (2021), 934–947.Google ScholarDigital Library
- [13] . 1993. Open problems. In Graph Structure Theory: Proceedings of a Joint Summer Research Conference on Graph Minors, and (Eds.).
Contemporary Mathematics , Vol. 147. American Mathematical Society, Providence, 677–688.Google ScholarCross Ref - [14] . 2010. Graph Theory (4th ed.).
Graduate Texts in Mathematics , Vol. 173. Springer, Berlin.Google ScholarCross Ref - [15] . 2008. Improved approximation algorithms for minimum weight vertex separators. SIAM Journal on Computing 38, 2 (2008), 629–657.Google ScholarDigital Library
- [16] . 2006. A 3-approximation for the pathwidth of Halin graphs. Journal of Discrete Algorithms 4, 4 (2006), 499–510.Google ScholarDigital Library
- [17] . 2006. Connected treewidth and connected graph searching. In LATIN 2006: Theoretical Informatics, , , and (Eds.).
Lecture Notes in Computer Science , Vol. 3887. Springer, Berlin, 479–490.Google Scholar - [18] . 1993. On the pathwidth of chordal graphs. Discrete Applied Mathematics 45, 3 (1993), 233–248.Google ScholarDigital Library
- [19] . 1994. Treewidth of cocomparability graphs and a new order-theoretic parameter. Order 11, 1 (1994), 47–60.Google ScholarCross Ref
- [20] . 2019. Graph minors and tree decompositions. Bachelor’s thesis. School of Mathematics, Monash University, Melbourne.Google Scholar
- [21] . 1972. Three results for trees, using mathematical induction. Journal of Research of the National Bureau of Standards, Series B: Mathematical Sciences 76B, 1–2 (1972), 39–43.Google Scholar
- [22] . 2022. A polynomial excluded-minor approximation of treedepth. Journal of the European Mathematical Society 24, 4 (2022), 1449–1470.Google ScholarCross Ref
- [23] . 1993. Computing treewidth and minimum fill-in: All you need are the minimal separators. In Algorithms—ESA’93, (Ed.).
Lecture Notes in Computer Science , Vol. 726. Springer, Berlin, 260–271.Google Scholar - [24] . 1995. Dominoes. In Graph-Theoretic Concepts in Computer Science, , , and (Eds.).
Lecture Notes in Computer Science , Vol. 903. Springer, Berlin, 106–120.Google Scholar - [25] . 2015. Circumference and pathwidth of highly connected graphs. Journal of Graph Theory 79, 3 (2015), 222–232.Google ScholarDigital Library
- [26] . 1988. Min Cut is NP-complete for edge weighted trees. Theoretical Computer Science 58, 1 (1988), 209–229.Google ScholarDigital Library
- [27] . 1986. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B 41, 1 (1986), 92–114.Google ScholarDigital Library
- [28] . 1989. Die Baumweite von Graphen als ein Maß für die Kompliziertheit algorithmischer Probleme. Ph.D. Dissertation. Akademie der Wissenschaften der DDR, Berlin.Google Scholar
- [29] . 2013. Personal communication. (2013).Google Scholar
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- Approximating Pathwidth for Graphs of Small Treewidth
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