Abstract
We continue the development of matroid-based techniques for kernelization, initiated by the present authors [47]. We significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to Lovász [51] and Marx [53]. As a first result, we show how representative sets can be used to derive a polynomial kernel for the elusive ALMOST 2-SAT problem (where the task is to remove at most k clauses to make a 2-CNF formula satisfiable), solving a major open problem in kernelization. This result also yields a new O(√log OPT)-approximation for the problem, improving on the O(√log n)-approximation of Agarwal et al. [3] and an implicit O(log OPT)-approximation due to Even et al. [24].
We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, that is, vertices that can be made undeletable without affecting the answer to the problem. This gives the first significant progress towards a polynomial kernel for the MULTIWAY CUT problem; in particular, we get a kernel of O(ks+1) vertices for MULTIWAY CUT instances with at most s terminals. Both these kernelization results have significant spin-off effects, producing the first polynomial kernels for a range of related problems.
More generally, the irrelevant vertex results have implications for covering min cuts in graphs. For a directed graph G=(V,E) and sets S, T ⊆ V, let r be the size of a minimum (S,T)-vertex cut (which may intersect S and T). We can find a set Z ⊆ V of size O(|S| . |T| . r) that contains a minimum (A,B)-vertex cut for every A ⊆ S, B ⊆ T. Similarly, for an undirected graph G=(V,E), a set of terminals X ⊆ V, and a constant s, we can find a set Z⊆ V of size O(|X|s+1) that contains a minimum multiway cut for every partition of X into at most s pairwise disjoint subsets. Both results are polynomial time. We expect this to have further applications; in particular, we get direct, reduction rule-based kernelizations for all problems above, in contrast to the indirect compression-based kernel previously given for ODD CYCLE TRANSVERSAL [47].
All our results are randomized, with failure probabilities that can be made exponentially small in n, due to needing a representation of a matroid to apply the representative sets tool.
- Faisal N. Abu-Khzam. 2010. A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76, 7 (2010), 524--531.Google ScholarDigital Library
- Leonard M. Adleman. 1978. Two theorems on random polynomial time. In 19th Annual Symposium on Foundations of Computer Science (Ann Arbor, Michigan, 16-18 October 1978). IEEE Computer Society, 75--83. DOI:https://doi.org/10.1109/SFCS.1978.37Google ScholarDigital Library
- Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. 2005. approximation algorithms for Min UnCut, Min 2CNF deletion, and directed cut problems. In STOC, Harold N. Gabow and Ronald Fagin (Eds.). ACM, 573--581.Google Scholar
- Sepehr Assadi, Sanjeev Khanna, Yang Li, and Val Tannen. 2015. Dynamic sketching for graph optimization problems with applications to cut-preserving sketches. In 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS'15), (December 16-18, 2015, Bangalore, India (LIPIcs)), Prahladh Harsha and G. Ramalingam (Eds.), Vol. 45. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 52--68. DOI:https://doi.org/10.4230/LIPIcs.FSTTCS.2015.52Google Scholar
- M. Balinski. 1970. On the maximum matching, minimum covering. In Proceedings of the Symposium on Mathematical Programming. Princeton University Press, 301--312.Google Scholar
- Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. 2009. On problems without polynomial kernels. J. Comput. Syst. Sci. 75, 8 (2009), 423--434.Google ScholarDigital Library
- Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. 2016. (Meta) Kernelization. J. ACM 63, 5 (2016), 44:1–44:69. http://dl.acm.org/citation.cfm?id=2973749Google ScholarDigital Library
- Hans L. Bodlaender, Fedor V. Fomin, and Saket Saurabh. 2010. Open Problems, WORKER 2010. Available at http://fpt.wikidot.com/open-problems.Google Scholar
- Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. 2011. Cross-composition: A new technique for kernelization lower bounds. In STACS (LIPIcs), Thomas Schwentick and Christoph Dürr (Eds.), Vol. 9. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 165--176.Google Scholar
- Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. 2011. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412, 35 (2011), 4570--4578.Google ScholarDigital Library
- Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michal Pilipczuk. 2016. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput. 45, 4 (2016), 1171--1229. DOI:https://doi.org/10.1137/15M1032077Google ScholarDigital Library
- Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis A. Goddyn, Michael Lohman, and Paul D. Seymour. 2006. Packing non-zero a-paths in group-labelled graphs. Combinatorica 26, 5 (2006), 521--532.Google ScholarDigital Library
- Julia Chuzhoy. 2012. On vertex sparsifiers with Steiner nodes. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, Howard J. Karloff and Toniann Pitassi (Eds.). ACM, 673--688. DOI:https://doi.org/10.1145/2213977.2214039Google ScholarDigital Library
- Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer. DOI:https://doi.org/10.1007/978-3-319-21275-3Google Scholar
- Marek Cygan, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Magnus Wahlström. 2014. Clique cover and graph separation: New incompressibility results. TOCT 6, 2 (2014), 6:1–6:19. DOI:https://doi.org/10.1145/2594439Google ScholarDigital Library
- Marek Cygan, Marcin Pilipczuk, and Michal Pilipczuk. 2016. On group feedback vertex set parameterized by the size of the cutset. Algorithmica 74, 2 (2016), 630--642. DOI:https://doi.org/10.1007/s00453-014-9966-5Google ScholarDigital Library
- Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. 2013. On multiway cut parameterized above lower bounds. TOCT 5, 1 (2013), 3. DOI:https://doi.org/10.1145/2462896.2462899Google ScholarDigital Library
- Holger Dell and Dániel Marx. 2012. Kernelization of packing problems. In 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'12), (Kyoto, Japan, January 17-19, 2012), Yuval Rabani (Ed.). SIAM, 68--81.Google ScholarCross Ref
- Holger Dell and Dieter van Melkebeek. 2014. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61, 4 (2014), 23. DOI:https://doi.org/10.1145/2629620Google ScholarDigital Library
- Bistra N. Dilkina, Carla P. Gomes, and Ashish Sabharwal. 2007. Tradeoffs in the complexity of backdoor detection. In CP (Lecture Notes in Computer Science), Christian Bessiere (Ed.), Vol. 4741. Springer, 256--270.Google Scholar
- Rodney G. Downey and Michael R. Fellows. 2013. Fundamentals of Parameterized Complexity. Springer. DOI:https://doi.org/10.1007/978-1-4471-5559-1Google ScholarDigital Library
- Andrew Drucker. 2015. New limits to classical and quantum instance compression. SIAM J. Comput. 44, 5 (2015), 1443--1479. DOI:https://doi.org/10.1137/130927115Google Scholar
- Paul Erdős and Richard Rado. 1960. Intersection theorems for systems of sets. Journal of the London Mathematical Society s1-35, 1 (1960), 85--90. DOI:https://doi.org/10.1112/jlms/s1-35.1.85 arXiv:http://jlms.oxfordjournals.org/content/s1-35/1/85.full.pdf+html.Google ScholarCross Ref
- Guy Even, Joseph Naor, Satish Rao, and Baruch Schieber. 2000. Divide-and-conquer approximation algorithms via spreading metrics. J. ACM 47, 4 (2000), 585--616. DOI:https://doi.org/10.1145/347476.347478Google ScholarDigital Library
- Stefan Fafianie, Stefan Kratsch, and Vuong Anh Quyen. 2016. Preprocessing under uncertainty. In Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), Nicolas Ollinger and Heribert Vollmer (Eds.). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 43:1–43:14. DOI:https://doi.org/10.4230/LIPIcs.STACS.2016.33Google Scholar
- Stephen A. Fenner, Rohit Gurjar, and Thomas Thierauf. 2016. Bipartite perfect matching is in quasi-NC. In 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC'16), (Cambridge, MA, June 18-21, 2016), Daniel Wichs and Yishay Mansour (Eds.). ACM, 754--763. DOI:https://doi.org/10.1145/2897518.2897564Google ScholarDigital Library
- Jörg Flum and Martin Grohe. 2006. Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). Springer.Google ScholarDigital Library
- Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. 2016. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63, 4 (2016), 29:1–29:60. DOI:https://doi.org/10.1145/2886094Google ScholarDigital Library
- Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. 2010. Bidimensionality and kernels. In 21st Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA 2010), (Austin, Texas, January 17-19, 2010). SIAM, 503--510. DOI:https://doi.org/10.1137/1.9781611973075.43Google Scholar
- Fedor V. Fomin, Saket Saurabh, and Yngve Villanger. 2013. A polynomial kernel for proper interval vertex deletion. SIAM J. Discrete Math. 27, 4 (2013), 1964--1976.Google ScholarCross Ref
- Lance Fortnow and Rahul Santhanam. 2011. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77, 1 (2011), 91--106.Google ScholarDigital Library
- Shivam Garg and Geevarghese Philip. 2016. Raising the bar for vertex cover: Fixed-parameter tractability above a higher guarantee. In 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'16), (Arlington, VA, January 10-12, 2016), Robert Krauthgamer (Ed.). SIAM, 1152--1166. DOI:https://doi.org/10.1137/1.9781611974331.ch80Google ScholarCross Ref
- Georg Gottlob and Stefan Szeider. 2008. Fixed-parameter algorithms for artificial intelligence, constraint satisfaction and database problems. Comput. J. 51, 3 (2008), 303--325.Google ScholarDigital Library
- Sylvain Guillemot. 2011. FPT algorithms for path-transversal and cycle-transversal problems. Disc. Optim. 8, 1 (2011), 61--71.Google ScholarDigital Library
- Jiong Guo, Hannes Moser, and Rolf Niedermeier. 2009. Iterative compression for exactly solving NP-hard minimization problems. In Algorithmics of Large and Complex Networks (Lecture Notes in Computer Science), Jürgen Lerner, Dorothea Wagner, and Katharina Anna Zweig (Eds.), vol. 5515. Springer, 65--80.Google Scholar
- Rohit Gurjar and Thomas Thierauf. 2017. Linear matroid intersection is in quasi-NC. In 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC'17), (Montreal, QC, Canada, June 19-23, 2017), Hamed Hatami, Pierre McKenzie, and Valerie King (Eds.). ACM, 821--830. DOI:https://doi.org/10.1145/3055399.3055440Google ScholarDigital Library
- Gregory Z. Gutin, M. S. Ramanujan, Felix Reidl, and Magnus Wahlström. 2019. Path-contractions, edge deletions and connectivity preservation. J. Comput. Syst. Sci. 101 (2019), 1--20. DOI:https://doi.org/10.1016/j.jcss.2018.10.001Google ScholarCross Ref
- Danny Harnik and Moni Naor. 2010. On the compressibility of instances and cryptographic applications. SIAM J. Comput. 39, 5 (2010), 1667--1713.Google ScholarDigital Library
- Danny Hermelin and Xi Wu. 2012. Weak compositions and their applications to polynomial lower bounds for kernelization. In 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'12), (Kyoto, Japan, January 17-19, 2012), Yuval Rabani (Ed.). SIAM, 104--113.Google ScholarCross Ref
- Eva-Maria C. Hols and Stefan Kratsch. 2016. A randomized polynomial kernel for subset feedback vertex set. In Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), Nicolas Ollinger and Heribert Vollmer (Eds.). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 43:1–43:14. http://drops.dagstuhl.de/opus/volltexte/2016/5744.Google Scholar
- Yoichi Iwata, Magnus Wahlström, and Yuichi Yoshida. 2016. Half-integrality, LP-branching, and FPT algorithms. SIAM J. Comput. 45, 4 (2016), 1377--1411. DOI:https://doi.org/10.1137/140962838Google ScholarCross Ref
- Bart M. P. Jansen and Hans L. Bodlaender. 2013. Vertex cover kernelization revisited - Upper and lower bounds for a refined parameter. Theory Comput. Syst. 53, 2 (2013), 263--299. DOI:https://doi.org/10.1007/s00224-012-9393-4Google ScholarCross Ref
- Subhash Khot. 2002. On the power of unique 2-prover 1-round games. In IEEE Conference on Computational Complexity. 25.Google ScholarDigital Library
- Stephan Kottler, Michael Kaufmann, and Carsten Sinz. 2008. Computation of renameable Horn backdoors. In SAT (Lecture Notes in Computer Science), Hans Kleine Büning and Xishun Zhao (Eds.), Vol. 4996. Springer, 154--160.Google Scholar
- Stefan Kratsch. 2018. A randomized polynomial kernelization for vertex cover with a smaller parameter. SIAM J. Discrete Math. 32, 3 (2018), 1806--1839. DOI:https://doi.org/10.1137/16M1104585Google ScholarCross Ref
- Stefan Kratsch and Magnus Wahlström. 2012. Representative sets and irrelevant vertices: New tools for kernelization. In FOCS. IEEE Computer Society, 450--459.Google Scholar
- Stefan Kratsch and Magnus Wahlström. 2014. Compression via matroids: A randomized polynomial kernel for odd cycle transversal. ACM Transactions on Algorithms 10, 4 (2014), 20. DOI:https://doi.org/10.1145/2635810Google ScholarDigital Library
- Frank Thomson Leighton and Ankur Moitra. 2010. Extensions and limits to vertex sparsification. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, Leonard J. Schulman (Ed.). ACM, 47--56. DOI:https://doi.org/10.1145/1806689.1806698Google ScholarDigital Library
- Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. 2018. Deterministic truncation of linear matroids. ACM Trans. Algorithms 14, 2 (2018), 14:1–14:20. DOI:https://doi.org/10.1145/3170444Google ScholarDigital Library
- Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. 2014. Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11, 2 (2014), 15:1–15:31. DOI:https://doi.org/10.1145/2566616Google ScholarDigital Library
- László Lovász. 1977. Flats in matroids and geometric graphs. In Proc. Sixth British Combinatorial Conf. (Combinatorial Surveys). 45--86.Google Scholar
- Dániel Marx. 2006. Parameterized graph separation problems. Theor. Comput. Sci. 351, 3 (2006), 394--406.Google ScholarDigital Library
- Dániel Marx. 2009. A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410, 44 (2009), 4471--4479.Google ScholarDigital Library
- Dániel Marx and Igor Razgon. 2014. Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput. 43, 2 (2014), 355--388. DOI:https://doi.org/10.1137/110855247Google ScholarCross Ref
- Dániel Marx and Ildikó Schlotter. 2011. Stable assignment with couples: Parameterized complexity and local search. Disc, Optim 8, 1 (2011), 25--40.Google ScholarDigital Library
- Syed Mohammad Meesum, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. 2019. Rank vertex cover as a natural problem for algebraic compression. SIAM J. Discrete Math. 33, 3 (2019), 1277--1296. https://doi.org/10.1137/17M1154370Google ScholarCross Ref
- Karl Menger. 1927. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae 10, 1 (1927), 96--115.Google ScholarCross Ref
- Sounaka Mishra, Venkatesh Raman, Saket Saurabh, Somnath Sikdar, and C. Subramanian. 2011. The complexity of König subgraph problems and above-guarantee vertex cover. Algorithmica 61 (2011), 857--881. Issue 4. http://dx.doi.org/10.1007/s00453-010-9412-2 10.1007/s00453-010-9412-2.Google ScholarDigital Library
- Pranabendu Misra, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. 2017. Linear representation of transversal matroids and gammoids parameterized by rank. In - 23rd International Conference on Computing and Combinatorics (COCOON'17), (Hong Kong, China, August 3-5, 2017), (Lecture Notes in Computer Science), Yixin Cao and Jianer Chen (Eds.), vol. 10392. Springer, 420--432. DOI:https://doi.org/10.1007/978-3-319-62389-4_35Google Scholar
- Ankur Moitra. 2009. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS'09), (October 25-27, 2009, Atlanta, Ga.). IEEE Computer Society, 3--12. DOI:https://doi.org/10.1109/FOCS.2009.28Google ScholarDigital Library
- N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. 2012. LP can be a cure for parameterized problems. In STACS (LIPIcs), Christoph Dürr and Thomas Wilke (Eds.), vol. 14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 338--349.Google Scholar
- George L. Nemhauser and Leslie E. Trotter Jr.1975. Vertex packings: Structural properties and algorithms.Math. Program. 8 (1975), 232--248. DOI:https://doi.org/10.1007/BF01580444Google ScholarDigital Library
- James Oxley. 2011. Matroid Theory. Oxford University Press.Google Scholar
- Hazel Perfect. 1968. Applications of Menger’s graph theorem. J. Math. Anal. Appl. 22 (1968), 96--111.Google ScholarCross Ref
- Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. 2011. Paths, flowers and vertex cover. In ESA (Lecture Notes in Computer Science), Camil Demetrescu and Magnús M. Halldórsson (Eds.), vol. 6942. Springer, 382--393.Google Scholar
- Igor Razgon and Barry O’Sullivan. 2009. Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75, 8 (2009), 435--450.Google ScholarDigital Library
- Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. 2004. Finding odd cycle transversals. Oper. Res. Lett. 32, 4 (2004), 299--301.Google ScholarDigital Library
- Neil Robertson and Paul D. Seymour. 1995. Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B 63, 1 (1995), 65--110. DOI:https://doi.org/10.1006/jctb.1995.1006Google ScholarDigital Library
- Alexander Schrijver. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Springer.Google Scholar
- Stéphan Thomassé. 2010. A 4k kernel for feedback vertex set. ACM Trans. Alg. 6, 2 (2010).Google Scholar
- Magnus Wahlström. 2013. Abusing the Tutte matrix: An algebraic instance compression for the K-set-cycle problem. In 30th International Symposium on Theoretical Aspects of Computer Science, (STACS 2013), (February 27 - March 2, 2013, Kiel, Germany (LIPIcs)), Natacha Portier and Thomas Wilke (Eds.), vol. 20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 341--352. DOI:https://doi.org/10.4230/LIPIcs.STACS.2013.341Google Scholar
- Sergei Winitzki. 2010. Linear Algebra via Exterior Products. https://sites.google.com/site/winitzki/linalg.Google Scholar
Index Terms
- Representative Sets and Irrelevant Vertices: New Tools for Kernelization
Recommendations
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given undirected graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result, Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4kkmn) time ...
Representative Sets and Irrelevant Vertices: New Tools for Kernelization
FOCS '12: Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer ScienceThe existence of a polynomial kernel for Odd Cycle Transversal was a notorious open problem in parameterized complexity. Recently, this was settled by the present authors (Kratsch and Wahlstr\"om, SODA 2012), with a randomized polynomial kernel for the ...
Parameterized algorithms for locating-dominating sets
AbstractA locating-dominating set D of a graph G is a dominating set of G where each vertex not in D has a unique neighborhood in D, and the Locating-Dominating Set problem asks if G contains such a dominating set of bounded size. This problem is known to ...
Comments