Pliability and Approximating Max-CSPs
Abstract
We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general.
Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.
References
[1]
Noga Alon, Wenceslas Fernandez de la Vega, Ravi Kannan, and Marek Karpinski. 2003. Random sampling and approximation of MAX-CSPs. J. Comput. Syst. Sci. 67, 2 (2003), 212–243. DOI:
[2]
Noga Alon, Paul D. Seymour, and Robin Thomas. 1990. A separator theorem for graphs with an excluded minor and its applications. In 22nd Annual ACM Symposium on Theory of Computing (STOC’90). ACM, 293–299. DOI:
[3]
Gunnar Andersson and Lars Engebretsen. 2002. Property testers for dense constraint satisfaction programs on finite domains. Rand. Struct. Algor. 21, 1 (2002), 14–32. DOI:
[4]
Sanjeev Arora, David R. Karger, and Marek Karpinski. 1999. Polynomial time approximation schemes for dense instances of NP-hard problems. J. Comput. Syst. Sci. 58, 1 (1999), 193–210. DOI:
[5]
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. 1998. Proof verification and the hardness of approximation problems. J. ACM 45, 3 (1998), 501–555. DOI:
[6]
Brenda S. Baker. 1994. Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 1 (1994), 153–180. DOI:
[7]
Itai Benjamini, Oded Schramm, and Asaf Shapira. 2008. Every minor-closed property of sparse graphs is testable. In 40th Annual ACM Symposium on Theory of Computing (STOC’08). ACM, 393–402. DOI:
[8]
Eli Berger, Zdeněk Dvořák, and Sergey Norin. 2018. Treewidth of grid subsets. Combinatorica 38, 6 (2018), 1337–1352. DOI:
[9]
Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., and Pasin Manurangsi. 2018. Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH. Technical Report. arXiv:1803.09717.
[10]
Jacek Brodzki, Graham A. Niblo, Jan Spakula, Rufus Willett, and Nick Wright. 2013. Uniform local amenability. J. Noncommut. Geom. 7, 2 (2013), 583–604. DOI:
[11]
Clément Carbonnel, Miguel Romero, and Stanislav Živný. 2022. The complexity of general-valued constraint satisfaction problems seen from the other side. SIAM J. Comput. 51, 1 (2022), 19–69. DOI:
[12]
Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. 2020. From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more. SIAM J. Comput. 49, 4 (2020), 772–810. DOI:
[13]
Moses Charikar, MohammadTaghi Hajiaghayi, and Howard J. Karloff. 2011. Improved approximation algorithms for label cover problems. Algorithmica 61, 1 (2011), 190–206. DOI:
[14]
Artur Czumaj, Asaf Shapira, and Christian Sohler. 2009. Testing hereditary properties of nonexpanding bounded-degree graphs. SIAM J. Comput. 38, 6 (2009), 2499–2510. DOI:
[15]
David P. Dailey. 1980. Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete. Discr. Math. 30, 3 (1980), 289–293. DOI:
[16]
Víctor Dalmau, Phokion G. Kolaitis, and Moshe Y. Vardi. 2002. Constraint satisfaction, bounded treewidth, and finite-variable logics. In 8th International Conference on Principles and Practice of Constraint Programming (CP’02) (LNCS), Vol. 2470. Springer, 310–326. DOI:
[17]
Anuj Dawar, Martin Grohe, Stephan Kreutzer, and Nicole Schweikardt. 2006. Approximation schemes for first-order definable optimisation problems. In 21st IEEE Symposium on Logic in Computer Science (LICS’06). IEEE Computer Society, 411–420. DOI:
[18]
Wenceslas Fernandez de la Vega. 1996. MAX-CUT has a randomized approximation scheme in dense graphs. Rand. Struct. Algor. 8, 3 (1996), 187–198.
[19]
Wenceslas Fernandez de la Vega and Marek Karpinski. 2002. A Polynomial Time Approximation Scheme for Subdense MAX-CUT. Technical Report 044. Retrieved from http://eccc.hpi-web.de/eccc-reports/2002/TR02-044/index.html
[20]
Wenceslas Fernandez de la Vega and Marek Karpinski. 2006. Approximation Complexity of Nondense Instances of MAX-CUT. Technical Report 101. Retrieved from http://eccc.hpi-web.de/eccc-reports/2006/TR06-101/index.html
[21]
Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. 2005. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM 52, 6 (2005), 866–893. DOI:
[22]
Erik D. Demaine and Mohammad Taghi Hajiaghayi. 2005. Bidimensionality: New connections between FPT algorithms and PTASs. In 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’05). SIAM, 590–601. Retrieved from http://dl.acm.org/citation.cfm?id=1070432.1070514
[23]
Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Ken-ichi Kawarabayashi. 2005. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05). IEEE Computer Society, 637–646. DOI:
[24]
Matt DeVos, Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, Bruce Reed, Paul Seymour, and Dirk Vertigan. 2004. Excluding any graph as a minor allows a low tree-width 2-coloring. J. Comb. Theor. B 91, 1 (2004), 25–41. DOI:
[25]
Reinhard Diestel. 2010. Graph Theory (4th ed.). Springer.
[26]
Irit Dinur. 2016. Mildly Exponential Reduction from Gap 3SAT to Polynomial-gap Label-cover. Technical Report 128. Retrieved from https://eccc.weizmann.ac.il/report/2016/128/
[27]
Irit Dinur, Eldar Fischer, Guy Kindler, Ran Raz, and Shmuel Safra. 2011. PCP characterizations of NP: Toward a polynomially-small error-probability. Comput. Complex. 20, 3 (2011), 413–504. DOI:
[28]
Irit Dinur and Pasin Manurangsi. 2018. ETH-hardness of approximating 2-CSPs and directed Steiner network. In 9th Innovations in Theoretical Computer Science Conference (ITCS’18) (Leibniz International Proceedings in Informatics (LIPIcs)), Vol. 94. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 36:1–36:20. DOI:
[29]
Zdeněk Dvořák. 2016. Sublinear separators, fragility and subexponential expansion. Eur. J. Comb. 52, Part A (2016), 103–119. DOI:
[30]
Zdeněk Dvořák. 2020. Baker game and polynomial-time approximation schemes. In 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). SIAM, 2227–2240. DOI:
[31]
Zdeněk Dvořák, Tony Huynh, Gwenael Joret, Chun-Hung Liu, and David R. Wood. 2021. Notes on Graph Product Structure Theory. Springer International Publishing, 513–533. DOI:
[32]
Zdeněk Dvořák and Jean-Sébastien Sereni. 2020. On fractional fragility rates of graph classes. Electron. J. Comb. 27, 4 (2020), 4. DOI:
[33]
Zdeněk Dvořák. 2018. Thin graph classes and polynomial-time approximation schemes. In 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’18). 1685–1701. DOI:
[34]
Zdenek Dvořák and Abhiruk Lahiri. 2021. Approximation schemes for bounded distance problems on fractionally treewidth-fragile graphs. In 29th Annual European Symposium on Algorithms (ESA’21) (LIPIcs), Vol. 204. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 40:1–40:10. DOI:
[35]
Zdeněk Dvořák and Sergey Norin. 2016. Strongly sublinear separators and polynomial expansion. SIAM J. Discr. Math. 30, 2 (2016), 1095–1101. DOI:
[36]
Keith Edwards and Graham Farr. 2012. Graph Fragmentability. Cambridge University Press, 203–218.
[37]
Keith Edwards and Colin McDiarmid. 1994. New upper bounds on harmonious colorings. J. Graph Theor. 18, 3 (1994), 257–267. DOI:
[38]
Gábor Elek. 2008. L2-spectral invariants and convergent sequences of finite graphs. J. Funct. Anal. 10, 254 (2008), 2667–2689. DOI:
[39]
Gábor Elek. 2021. Uniform local amenability implies Property A. Am. Math. Soc. 149, 6 (2021), 2573–2577. DOI:
[40]
David Eppstein. 2000. Diameter and treewidth in minor-closed graph families. Algorithmica 27, 3–4 (2000), 275–291. DOI:
[41]
Alan M. Frieze and Ravi Kannan. 1996. The regularity lemma and approximation schemes for dense problems. In 37th Annual Symposium on Foundations of Computer Science (FOCS’96). IEEE Computer Society, 12–20. DOI:
[42]
Michel X. Goemans and David P. Williamson. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 6 (1995), 1115–1145. DOI:
[43]
Oded Goldreich. 2017. Introduction to Property Testing. Cambridge University Press. Retrieved from http://www.wisdom.weizmann.ac.il/oded/pt-intro.html
[44]
Oded Goldreich, Shafi Goldwasser, and Dana Ron. 1998. Property testing and its connection to learning and approximation. J. ACM 45, 4 (1998), 653–750. DOI:
[45]
Alexander Grigoriev and Hans L. Bodlaender. 2007. Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49, 1 (01 9 2007), 1–11. DOI:
[46]
Martin Grohe. 2003. Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23, 4 (2003), 613–632. DOI:
[47]
Martin Grohe. 2007. The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54, 1 (2007), 1–24. DOI:
[48]
Martin Grohe, Thomas Schwentick, and Luc Segoufin. 2001. When is the evaluation of conjunctive queries tractable? In 33rd Annual ACM Symposium on Theory of Computing (STOC’01). ACM, 657–666. DOI:
[49]
Venkatesan Guruswami and Sanjeev Khanna. 2004. On the hardness of 4-coloring a 3-colorable graph. SIAM J. Discr. Math. 18, 1 (2004), 30–40. DOI:
[50]
F. Hadlock. 1975. Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput. 4, 3 (1975), 221–225. DOI:
[51]
Avinatan Hassidim, Jonathan A. Kelner, Huy N. Nguyen, and Krzysztof Onak. 2009. Local graph partitions for approximation and testing. In 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09). 22–31. DOI:
[52]
Pavol Hell and Jaroslav Nešetřil. 2004. Graphs and Homomorphisms. Oxford University Press.
[53]
Dorit S. Hochbaum and Wolfgang Maass. 1985. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 1 (1 1985), 130–136. DOI:
[54]
Harry B. Hunt III, Riko Jacob, Madhav V. Marathe, and Richard E. Stearns. 1998. Towards syntactic characterizations of approximation schemes via predicate and graph decompositions. In IEEE Symposium on Foundations of Computer Science (FOCS’98). Retrieved from https://www.osti.gov/biblio/677169
[55]
Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, S. S. Ravi, Daniel J. Rosenkrantz, and Richard E. Stearns. 1998. NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algor. 26, 2 (1998), 238–274. DOI:
[56]
Richard M. Karp. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations. 85–103. Retrieved from http://www.cs.berkeley.edu/%7Eluca/cs172/karp.pdf
[57]
Sanjeev Khanna and Rajeev Motwani. 1996. Towards a syntactic characterization of PTAS. In 28th Annual ACM Symposium on the Theory of Computing (STOC’96). 329–337. DOI:
[58]
Subhash Khot. 2002. On the power of unique 2-prover 1-round games. In 34th Annual ACM Symposium on Theory of Computing (STOC’02). ACM, 767–775. DOI:
[59]
Alexandr V. Kostochka. 1984. Lower bound of the Hadwiger number of graphs by their average degree. Combinatorica 4, 4 (1984), 307–316. DOI:
[60]
Robert Krauthgamer and James R. Lee. 2007. The intrinsic dimensionality of graphs. Combinatorica 27, 5 (2007), 551–585. DOI:
[61]
Akash Kumar, C. Seshadhri, and Andrew Stolman. 2019. Random walks and forbidden minors II: A poly(\(d\epsilon ^{-1}\))-query tester for minor-closed properties of bounded degree graphs. In 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC’19). ACM, 559–567. DOI:
[62]
Michael Langberg, Yuval Rabani, and Chaitanya Swamy. 2006. Approximation algorithms for graph homomorphism problems. In Conference on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM’06) (Lecture Notes in Computer Science), Vol. 4110. Springer, 176–187. DOI:
[63]
Richard J. Lipton and Robert Endre. Tarjan. 1979. A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 2 (1979), 177–189. DOI:
[64]
Richard J. Lipton and Robert Endre Tarjan. 1980. Applications of a planar separator theorem. SIAM J. Comput. 9, 3 (1980), 615–627. DOI:
[65]
Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. 2020. Parameterized complexity and approximability of directed odd cycle transversal. In 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). SIAM, 2181–2200. DOI:
[66]
László Lovász. 2012. Large Networks and Graph Limits. Colloquium Publications, Vol. 60. American Mathematical Society. Retrieved from http://web.cs.elte.hu/lovasz/bookxx/hombook-almost.final.pdf
[67]
László Lovász and Balázs Szegedy. 2007. Szemerédi’s lemma for the analyst. Geom. Funct. Anal. 17, 1 (2007), 252–270. DOI:
[68]
Pasin Manurangsi and Dana Moshkovitz. 2015. Approximating dense max 2-CSPs. In Conference on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM’15). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 396–415. DOI:
[69]
Pasin Manurangsi and Prasad Raghavendra. 2017. A birthday repetition theorem and complexity of approximating dense CSPs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP’17) (LIPIcs), Vol. 80. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 78:1–78:15. DOI:
[70]
Madhav V. Marathe, Harry B. Hunt III, and Richard E. Stearns. 1997. Level-treewidth property, exact algorithms and approximation schemes. In ACM Symposium on Theory of Computing (STOC’97). Retrieved from https://www.osti.gov/biblio/471394
[71]
Balázs F. Mezei, Marcin Wrochna, and Stanislav Živný. 2023. PTAS for sparse general-valued CSPs. ACM Trans. Algor. 19, 2 (2023), 14:1–14:31. DOI:
[72]
Guy Moshkovitz and Asaf Shapira. 2018. Decomposing a graph into expanding subgraphs. Rand. Struct. Algor. 52, 1 (2018), 158–178. DOI:
[73]
Jaroslav Nešetřil and Patrice Ossona de Mendez. 2012. Property Testing, Hyperfiniteness and Separators. Springer, 363–379. DOI:
[74]
Jaroslav Nešetřil and Patrice Ossona de Mendez. 2006. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb. 27, 6 (2006), 1022–1041. DOI:
[75]
Jaroslav Nešetřil and Patrice Ossona de Mendez. 2008. Grad and classes with bounded expansion II. Algorithmic aspects. Eur. J. Comb. 29, 3 (2008), 777–791. DOI:
[76]
Ilan Newman and Christian Sohler. 2013. Every property of hyperfinite graphs is testable. SIAM J. Comput. 42, 3 (2013), 1095–1112. DOI:
[77]
Christos H. Papadimitriou and Mihalis Yannakakis. 1991. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 3 (1991), 425–440. DOI:
[78]
Prasad Raghavendra. 2008. Optimal algorithms and inapproximability results for every CSP? In 40th Annual ACM Symposium on Theory of Computing (STOC’08). 245–254. DOI:
[79]
Vojtěch Rödl and Mathias Schacht. 2007. Regular partitions of hypergraphs: Regularity lemmas. Comb. Probab. Comput. 16, 6 (2007), 833–885. DOI:
[80]
Miguel Romero, Marcin Wrochna, and Stanislav Živný. 2021. Treewidth-pliability and PTAS for Max-CSPs. In Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’21). SIAM, 473–483. DOI:
[81]
Sartaj Sahni and Teofilo F. Gonzalez. 1976. P-complete approximation problems. J. ACM 23, 3 (1976), 555–565. DOI:
[82]
Alexander Schrijver. 1986. Theory of Linear and Integer Programming. John Wiley & Sons, Inc.
[83]
Hanif D. Sherali and Warren P. Adams. 1990. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discr. Math. 3, 3 (1990), 411–430. DOI:
[84]
Joel Spencer. 1990. Counting extensions. J. Comb. Theor., Ser. A 55, 2 (1990), 247–255. DOI:
[85]
Endre Szeméredi. 1978. Regular partitions of graphs. In Colloq. Inter. CNRS. 399–401. Retrieved from https://mathscinet.ams.org/mathscinet/relay-station?mr=0540024
[86]
Johan Thapper and Stanislav Živný. 2016. The complexity of finite-valued CSPs. J. ACM 63, 4 (2016). DOI:
[87]
Carsten Thomassen. 1983. Girth in graphs. J. Comb. Theor., Ser. B 35, 2 (1983), 129–141. DOI:
[88]
Jan van den Heuvel, Patrice Ossona de Mendez, Daniel Quiroz, Roman Rabinovich, and Sebastian Siebertz. 2017. On the generalised colouring numbers of graphs that exclude a fixed minor. Eur. J. Comb. 66 (2017), 129–144. DOI:
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- Pliability and Approximating Max-CSPs
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December 2023
314 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/3633310
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- Venkatesan Guruswami
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Publication History
Published: 30 November 2023
Online AM: 06 October 2023
Accepted: 26 September 2023
Revised: 16 September 2023
Received: 12 December 2020
Published in JACM Volume 70, Issue 6
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