Abstract
Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition’s width. Following the work of Allender et al. [5], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [18], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we extend the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition’s depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.
- Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. 2015. Tight hardness results for LCS and other sequence similarity measures. In Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS’15). IEEE Computer Society, 59--78. Google Scholar
Digital Library
- Dmitri Akatov. 2010. Exploiting parallelism in decomposition methods for constraint satisfaction. Ph.D. Dissertation. University of Oxford.Google Scholar
- Dmitri Akatov and Georg Gottlob. 2010. Balanced queries: Divide and conquer. In Proceedings of the 35th International Symposium on Mathematical Foundations of Computer Science (MFCS’10) (Lecture Notes in Computer Science), Vol. 6281. Springer, 42--54. Google Scholar
Digital Library
- Jochen Alber, Frederic Dorn, and Rolf Niedermeier. 2005. Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs. Discrete Applied Mathematics 145, 2 (2005), 219--231. Google Scholar
Digital Library
- Eric Allender, Shiteng Chen, Tiancheng Lou, Periklis A. Papakonstantinou, and Bangsheng Tang. 2014. Width-parametrized SAT: Time--space tradeoffs. Theor. Comput. 10 (2014), 297--339.Google Scholar
Cross Ref
- Stefan Arnborg, Jens Lagergren, and Detlef Seese. 1991. Easy problems for tree-decomposable graphs. J. Algorithms 12, 2 (1991), 308--340. Google Scholar
Digital Library
- Sanjeev Arora and Boaz Barak. 2009. Computational Complexity - A Modern Approach. Cambridge University Press. Google Scholar
Digital Library
- Per Austrin, Petteri Kaski, Mikko Koivisto, and Jussi Määttä. 2013. Space-time tradeoffs for subset sum: An improved worst case algorithm. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP’13), Part I (Lecture Notes in Computer Science), Vol. 7965. Springer, 45--56. Google Scholar
Digital Library
- Brenda S. Baker. 1994. Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 1 (1994), 153--180. Google Scholar
Digital Library
- Greg Barnes, Jonathan F. Buss, Walter L. Ruzzo, and Baruch Schieber. 1998. A sublinear space, polynomial time algorithm for directed s-t connectivity. SIAM J. Comput. 27, 5 (1998), 1273--1282. Google Scholar
Digital Library
- Marina Barsky, Ulrike Stege, Alex Thomo, and Chris Upton. 2007. Shortest path approaches for the longest common subsequence of a set of strings. In Proceedings of the 7th IEEE International Conference on Bioinformatics and Bioengineering (BIBE’07). IEEE Computer Society, 327--333.Google Scholar
Cross Ref
- Nadja Betzler, Rolf Niedermeier, and Johannes Uhlmann. 2006. Tree decompositions of graphs: Saving memory in dynamic programming. Discrete Optim. 3, 3 (2006), 220--229. Google Scholar
Digital Library
- Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. 2010. Narrow sieves for parameterized paths and packings. CoRR abs/1007.1161 (2010). http://arxiv.org/abs/1007.1161Google Scholar
- Hans L. Bodlaender. 2005. Discovering treewidth. In Proceedings of the 31st Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM’05) (Lecture Notes in Computer Science), Vol. 3381. Springer, 1--16. Google Scholar
Digital Library
- Hans L. Bodlaender, Jitender S. Deogun, Klaus Jansen, Ton Kloks, Dieter Kratsch, Haiko Müller, and Zsolt Tuza. 1998. Rankings of graphs. SIAM J. Discrete Math. 11, 1 (1998), 168--181. Google Scholar
Digital Library
- Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Harold T. Wareham. 1995. The parameterized complexity of sequence alignment and consensus. Theor. Comput. Sci. 147, 182 (1995), 31--54. Google Scholar
Digital Library
- Liming Cai, Jianer Chen, Rodney G. Downey, and Michael R. Fellows. 1995. On the structure of parameterized problems in NP. Inf. Comput. 123, 1 (1995), 38--49. Google Scholar
Digital Library
- Liming Cai and David W. Juedes. 2003. On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67, 4 (2003), 789--807. Google Scholar
Digital Library
- Diptarka Chakraborty and Raghunath Tewari. 2015. An O(n^ε) space and polynomial time algorithm for reachability in directed layered planar graphs. In Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC’15) (Lecture Notes in Computer Science), Vol. 9472. Springer, 614--624.Google Scholar
- Hubie Chen and Moritz Müller. 2014. One hierarchy spawns another: Graph deconstructions and the complexity classification of conjunctive queries. In Proceedings of the Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (CSL-LICS’14). ACM, 32:1--32:10. Google Scholar
Digital Library
- Andrew Chiu, George I. Davida, and Bruce E. Litow. 2001. Division in logspace-uniform NC1. Theoret. Informatics and Appl. 35, 3 (2001), 259--275.Google Scholar
Cross Ref
- Bruno Courcelle. 1990. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 1 (1990), 12--75. Google Scholar
Digital Library
- Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer. Google Scholar
Digital Library
- Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. 2005. Subexponential parameterized algorithms on bounded-genus graphs and -minor-free graphs. J. ACM 52, 6 (2005), 866--893. Google Scholar
Digital Library
- Rodney G. Downey and Michael R. Fellows. 1999. Parameterized Complexity. Springer.Google Scholar
Digital Library
- Rodney G. Downey and Michael R. Fellows. 2013. Fundamentals of Parameterized Complexity. Springer. Google Scholar
Digital Library
- Jeff Edmonds, Chung Keung Poon, and Dimitris Achlioptas. 1999. Tight lower bounds for st-connectivity on the NNJAG model. SIAM J. Comput. 28, 6 (1999), 2257--2284. Google Scholar
Digital Library
- Michael Elberfeld, Martin Grohe, and Till Tantau. 2012. Where first-order and monadic second-order logic coincide. In Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science (LICS’12). IEEE Computer Society, 265--274. Google Scholar
Digital Library
- Michael Elberfeld, Andreas Jakoby, and Till Tantau. 2010. Logspace versions of the theorems of bodlaender and courcelle. Electronic Colloquium on Computational Complexity (ECCC) 17 (2010), 62. Extended abstract included in the proceedings of FOCS 2010.Google Scholar
- Michael Elberfeld, Andreas Jakoby, and Till Tantau. 2012. Algorithmic meta theorems for circuit classes of constant and logarithmic depth. In Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS’12), Vol. 14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 66--77.Google Scholar
- Michael Elberfeld, Christoph Stockhusen, and Till Tantau. 2015. On the space and circuit complexity of parameterized problems: Classes and completeness. Algorithmica 71, 3 (2015), 661--701. Google Scholar
Digital Library
- J. Flum and M. Grohe. 2006. Parameterized Complexity Theory (1 ed.). Springer. Google Scholar
Digital Library
- Fedor V. Fomin, Petteri Kaski, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. 2015. Parameterized single-exponential time polynomial space algorithm for steiner tree. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP’15), Part I (Lecture Notes in Computer Science), Vol. 9134. Springer, 494--505.Google Scholar
Cross Ref
- Martin Fürer and Huiwen Yu. 2014. Space saving by dynamic algebraization. In Proceedings of the 9th International Computer Science Symposium in Russia (CSR’14) (Lecture Notes in Computer Science), Vol. 8476. Springer, 375--388.Google Scholar
Cross Ref
- M. R. Garey and David S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman. Google Scholar
Digital Library
- M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. 1976. Some simplified NP-complete graph problems. Theor. Comput. Sci. 1, 3 (1976), 237--267.Google Scholar
Cross Ref
- Georg Gottlob, Nicola Leone, and Francesco Scarcello. 2001. The complexity of acyclic conjunctive queries. J. ACM 48, 3 (2001), 431--498. Google Scholar
Digital Library
- Sylvain Guillemot. 2011. Parameterized complexity and approximability of the longest compatible sequence problem. Discrete Optim. 8, 1 (2011), 50--60. Google Scholar
Digital Library
- Meir Katchalski, William McCuaig, and Suzanne M. Seager. 1995. Ordered colourings. Discrete Math. 142, 1-3 (1995), 141--154. Google Scholar
Digital Library
- Ton Kloks. 1994. Treewidth, Computations and Approximations. Lecture Notes in Computer Science, Vol. 842. Springer.Google Scholar
- Arie M. C. A. Koster, Stan P. M. van Hoesel, and Antoon W. J. Kolen. 2002. Solving partial constraint satisfaction problems with tree decomposition. Networks 40, 3 (2002), 170--180.Google Scholar
Cross Ref
- Alexander Langer, Felix Reidl, Peter Rossmanith, and Somnath Sikdar. 2014. Practical algorithms for MSO model-checking on tree-decomposable graphs. Comput. Sci. Rev. 13--14 (2014), 39--74. Google Scholar
Digital Library
- Richard J. Lipton. 2010. Savitch’s theorem. In The P=NP Question and Gödel’s Lost Letter. Springer, 135--138.Google Scholar
- Richard J. Lipton and Robert Endre Tarjan. 1980. Applications of a planar separator theorem. SIAM J. Comput. 9, 3 (1980), 615--627.Google Scholar
Digital Library
- Daniel Lokshtanov, Matthias Mnich, and Saket Saurabh. 2011. Planar k-path in subexponential time and polynomial space. In Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science (WG’11), Revised Papers (Lecture Notes in Computer Science), Vol. 6986. Springer, 262--270. Google Scholar
Digital Library
- Daniel Lokshtanov and Jesper Nederlof. 2010. Saving space by algebraization. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC’10). ACM, 321--330. Google Scholar
Digital Library
- Pinyan Lu, Jialin Zhang, Chung Keung Poon, and Jin-yi Cai. 2005. Simulating undirected st-connectivity algorithms on uniform JAGs and NNJAGs. In Proceedings of the 16th International Symposium on Algorithms and Computation (ISAAC’05) (Lecture Notes in Computer Science), Vol. 3827. Springer, 767--776. Google Scholar
Digital Library
- Burkhard Monien and Ivan Hal Sudborough. 1985. Bandwidth constrained NP-complete problems. Theor. Comput. Sci. 41 (1985), 141--167. Google Scholar
Digital Library
- Jesper Nederlof. 2013. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica 65, 4 (2013), 868--884. Google Scholar
Digital Library
- Jaroslav Nešetřil and Patrice Ossona de Mendez. 2006. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb. 27, 6 (2006), 1022--1041. Google Scholar
Digital Library
- J. Nešetřil and P. Ossona de Mendez. 2012. Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics, Vol. 28. Springer. Google Scholar
Cross Ref
- Noam Nisan. 1994. RL <= SC. Computational Complexity 4 (1994), 1--11. Google Scholar
Digital Library
- Periklis A. Papakonstantinou. 2009. A note on width-parameterized SAT: An exact machine-model characterization. Inf. Process. Lett. 110, 1 (2009), 8--12. Google Scholar
Digital Library
- Periklis A. Papakonstantinou. 2014. The depth irreducibility hypothesis. In Proceedings of the Electronic Colloquium on Computational Complexity (ECCC) 21 (2014), 124. http://eccc.hpi-web.de/report/2014/124Google Scholar
- Krzysztof Pietrzak. 2003. On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci. 67, 4 (2003), 757--771. Google Scholar
Digital Library
- Alex Pothen. 1988. The complexity of optimal elimination trees. (1988). Technical Report CS 88-16, Pennsylvania State University.Google Scholar
- Neil Robertson and Paul D. Seymour. 1986. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms 7, 3 (1986), 309--322.Google Scholar
Cross Ref
- J. Barkley Rosser and Lowell Schoenfeld. 1962. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 1 (03 1962), 64--94.Google Scholar
- Francesca Rossi, Peter van Beek, and Toby Walsh (Eds.). 2006. Handbook of Constraint Programming. Foundations of Artificial Intelligence, Vol. 2. Elsevier. Google Scholar
Digital Library
- Walter L. Ruzzo. 1980. Tree-size bounded alternation. J. Comput. Syst. Sci. 21, 2 (1980), 218--235.Google Scholar
Cross Ref
- Walter J. Savitch. 1970. Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci. 4, 2 (1970), 177--192. Google Scholar
Digital Library
- V. Vinay and V. Chandru. 1990. The expressibility of nondeterministic auxiliary stack automata and its relation to treesize bounded alternating auxiliary pushdown automata. In Proceedings of the 10th Conference on Foundations of Software Technology and Theoretical Computer Science (Lecture Notes in Computer Science), Vol. 472. Springer, 104--114. Google Scholar
Digital Library
- Avi Wigderson. 1992. The complexity of graph connectivity. In Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science (MFCS’92) (Lecture Notes in Computer Science), Vol. 629. Springer, 112--132. Google Scholar
Digital Library
Index Terms
On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs
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