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Multivariate Complexity Theory

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Abstract

Multivariate complexity analysis and algorithm design techniques have developed over many decades, starting from a number of early research themes in Computer Science. The basic insight is that in many situations, one or more secondary measurements of problem instances or computational objectives, beyond the overall input size, govern a problem’s computational complexity.

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Notes

  1. 1.

    A proper \( k \)-coloring of a graph is an assignment of at most \( k \) colors to its vertices such that vertices of the same color form an independent set.

References

  • K. Abrahamson, J. Ellis, M. Fellows and M. Mata (1989). On the complexity of fixed-parameter problems. FOCS 1989, 210–215.

    Google Scholar 

  • S. Arnborg, A. Proskurowski, D. Seese (1990). Monadic Second Order Logic, Tree Automata and Forbidden Minors. CSL 1990,1–16.

    Google Scholar 

  • C. Bessiere, E. Hebrard, B. Hnich, Z. Kiziltan, C.-G. Quimper, and T. Walsh (2008). The parameterized complexity of global constraints. AAAI 2008, 235–240.

    Google Scholar 

  • N. Betzler, M. R. Fellows, J. Guo, R. Niedermeier, and F. A. Rosamond (2009). Fixed-parameter algorithms for Kemeny rankings. Theoretical Computer Science 410(45), 4554–4570.

    Article  MATH  MathSciNet  Google Scholar 

  • H. L. Bodlaender (1988). Dynamic programming on graphs with bounded treewidth. ICALP 1988, 105–118.

    Google Scholar 

  • H. L. Bodlaender: A linear time algorithm for finding tree-decompositions of small treewidth. STOC 1993: 226–234.

    Google Scholar 

  • H. L. Bodlaender, R. G. Downey, M. R. Fellows, and D. Hermelin (2009). On problems without polynomial kernels. Journal of Computer and System Sciences 75(8), 423–434.

    Article  MATH  MathSciNet  Google Scholar 

  • L. Cai, J. Chen, R. G. Downey, and M. R. Fellows (1997). The parameterized complexity of short computation and factorization. Archive for Mathematical Logic 36(4/5), 321–337.

    Article  MATH  MathSciNet  Google Scholar 

  • L. Cai, X. Huang, C. Liu, F. Rosamond, and Y. Song (2008). Parameterized complexity and biopolymer sequence comparison. The Computer Journal 51(3), 270–291.

    Article  Google Scholar 

  • L. Cai and D. Juedes (2003). On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences 67(4), 789–807.

    Article  MATH  MathSciNet  Google Scholar 

  • M-C. Cai, X. Deng (2003). Approximation and computation of arbitrage in frictional foreign exchange market (extended abstract). Electronic Notes in Theoretical Computer Science 78, 293–302.

    Article  Google Scholar 

  • J. Cheetham, F. Dehne, A. Rau-Chaplin, U. Stege, and P. J. Taillon (2003). Solving large FPT problems on coarse grained parallel machines. Journal of Computer and System Sciences 67(4), 691–706.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Chen, B. Chor, M. Fellows, X. Huang, D. W. Juedes, I. Kanj, and G. Xia (2005). Tight lower bounds for certain parameterized NP-hard problems. Information and Computation 201(2), 216–231.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Chen, X. Huang, I. Kanj, and G. Xia (2004). Linear FPT reductions and computational lower bounds. STOC 2004, 212–221.

    Google Scholar 

  • J. Chen, I. A. Kanj, and G. Xia (2006). Improved parameterized upper bounds for Vertex Cover. MFCS 2006, 238–249.

    Google Scholar 

  • J. Chen, Y. Liu, S. Lu, B. O’Sullivan, and I. Razgon (2008). A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM 55(5).

    Google Scholar 

  • J. Chen and J. Meng (2008). On parameterized intractability: hardness and completeness. The Computer Journal 51(1), 39–59.

    Article  Google Scholar 

  • B. Chor, M. R. Fellows, and D. W. Juedes (2004). Linear kernels in linear time, or how to save k colors in O(n) steps. WG 2004, 257–269.

    Google Scholar 

  • The Computer Journal (2008). Two special issues of surveys of various aspects of parameterized complexity and algorithmics (Guest Editors: M. Fellows, R. Downey and M. Langston). The Computer Journal, Volume 51: Number 1 and Number 3.

    Google Scholar 

  • B. Courcelle (1990) The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs Inf. Comput. 85(1): 12–75.

    Article  MATH  MathSciNet  Google Scholar 

  • H. Dell and D. van Melkebeek (2010). Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. STOC 2010, 251–260.

    Google Scholar 

  • E. Demaine (2001). Playing games with algorithms: algorithmic combinatorial game theory. MFCS 2001, 18–32.

    Google Scholar 

  • E. Demaine and M. T. Hajiaghayi (2008). The bidimensionality theory and its algorithmic applications. The Computer Journal 51(3), 292–302.

    Article  Google Scholar 

  • Discrete Optimization (2011). Special issue on parameterized complexity of discrete optimization (Guest Editors: M. R. Fellows, F. V. Fomin, and G. Gutin) 8(1).

    Google Scholar 

  • R. G. Downey and M. R. Fellows (1995a). Fixed-parameter tractability and completeness I: basic results. SIAM Journal on Computing 24, 873–921.

    Article  MATH  MathSciNet  Google Scholar 

  • R. G. Downey and M. R. Fellows (1995b). Fixed-parameter tractability and completeness II: on completeness for W[1]. Theoretical Computer Science 141, 109–131.

    Article  MATH  MathSciNet  Google Scholar 

  • R. G. Downey and M. R. Fellows (1998). Parameterized Complexity. Springer-Verlag.

    Google Scholar 

  • R. G. Downey, M. R. Fellows, B. Kapron, M. T. Hallett, and H. T. Wareham (1994). Parameterized complexity of some problems in logic and linguistics. LFCS 1994, 89–100.

    Google Scholar 

  • R. G. Downey, M. R. Fellows, and C. McCartin (2006). Parameterized approximation algorithms. IWPEC 2006, 121–129.

    Google Scholar 

  • R. G. Downey, M. Fellows, and U. Taylor (1997). The parameterized complexity of relational database queries and an improved characterization of W[1]. DMTCS 1996, 194–213.

    Google Scholar 

  • M. Fellows (2002). Parameterized complexity: the main ideas and connections to practical computing. In: Experimental Algorithmics, Springer-Verlag, 51–77.

    Google Scholar 

  • M. R. Fellows (2003). Blow-ups, win/win’s and crown rules: some new directions in FPT. WG 2003, 1–12.

    Google Scholar 

  • M. R. Fellows (2009). Towards fully multivariate algorithmics: some new results and directions in parameter ecology. IWOCA 2009, 2–10.

    Google Scholar 

  • M. R. Fellows, F. V. Fomin, D. Lokshtanov, F. A. Rosamond, S. Saurabh, S. Szeider, and C. Thomassen (2011). On the complexity of some colorful problems parameterized by treewidth. Information and Computation 209(2), 143–153.

    Article  MATH  MathSciNet  Google Scholar 

  • M. R. Fellows, D. Hermelin, F. A. Rosamond, and S. Vialette (2009a). Theoretical Computer Science 410(1), 53–61.

    Article  MATH  MathSciNet  Google Scholar 

  • M. R. Fellows and N. Koblitz (1993). Fixed-parameter complexity and cryptography. AAECC 1993, 121–131.

    Google Scholar 

  • M. R. Fellows and M. Langston (1987). Nonconstructive advances in polynomial time complexity. Information Processing Letters 26, 157–162.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Fellows, M. A. Langston. (1989a) On Search, Decision and the Efficiency of Polynomial-Time Algorithms (Extended Abstract) STOC 1989, 501–512.

    Google Scholar 

  • M. Fellows, M.A. Langston(1989b) An Analogue of the Myhill-Nerode Theorem and Its Use in Computing Finite-Basis Characterizations (Extended Abstract) FOCS 1989: 520–525.

    Google Scholar 

  • M. R. Fellows and F. Rosamond (2007). The complexity ecology of parameters: an illustration using bounded max leaf number. CiE 2007, 268–277.

    Google Scholar 

  • M. R. Fellows, F. A. Rosamond, F. V. Fomin, D. Lokshtanov, S. Saurabh, and Y. Villanger (2009b). Local search: is brute-force avoidable? IJCAI 2009, 486–491.

    Google Scholar 

  • H. Fernau, T. Hagerup, N. Nishimura, P. Ragde and K. Reinhardt (2003). On the parameterized complexity of the generalized rush hour puzzle. CCCG 2003, 6–9.

    Google Scholar 

  • F. V. Fomin, D. Lokshtanov, V. Raman, and S. Saurabh (2010). Fast local search algorithm for Weighted Feedback Arc Set in Tournaments. AAAI 2010.

    Google Scholar 

  • J. Flum and M. Grohe (2006). Parameterized Complexity Theory, Springer-Verlag.

    Google Scholar 

  • S. Gaspers, D. Kratsch, and M. Liedloff. On independent sets and bicliques in graphs. Algorithmica, to appear.

    Google Scholar 

  • S. Gaspers and S. Szeider (2011). Kernels for global constraints. IJCAI 2011, to appear.

    Google Scholar 

  • G. Gottlob and S. Szeider (2008). Fixed-parameter algorithms for Artificial Intelligence, Constraint Satisfaction and Database Problems. The Computer Journal 51(3), 303–325.

    Article  Google Scholar 

  • M. Grohe (2002). The parameterized complexity of database queries. PODS 2002, 82–92.

    Google Scholar 

  • J. Guo, H. Moser, and R. Niedermeier (2009). Iterative compression for exactly solving NP-hard minimization problems. Algorithmics of Large and Complex Networks 2009, 65–80.

    Google Scholar 

  • J. Guo and R. Niedermeier (2007). Invitation to data reduction and problem kernelization. SIGACT News, March 2007, 31–45.

    Google Scholar 

  • S. Hartung and R. Niedermeier (2010). Incremental list coloring of graphs, parameterized by conservation. TAMC 2010, 258–270.

    Google Scholar 

  • S. Helwig, F. Hüffner, I. Rössling, and M. Weinard (2010). Selected design issues. Algorithm Engineering 2010, 58–126.

    Google Scholar 

  • F. Henglein and H. G. Mairson (1991). The complexity of type inference for higher-order typed lambda calculi. POPL 1991, 119–130.

    Google Scholar 

  • F. Hüffner (2009). Algorithm engineering for optimal graph bipartization. Journal of Graph Algorithms and Applications 13(2), 77–98.

    MATH  MathSciNet  Google Scholar 

  • F. Hüffner, R. Niedermeier, and S. Wernicke (2008). Techniques for practical fixed-parameter algorithms. The Computer Journal, 51(1):7–25.

    Article  Google Scholar 

  • Journal of Computer and System Sciences (2003). Parameterized computation and complexity. (Guest Editors: Jianer Chen, Michael R. Fellows ), 67(4).

    Google Scholar 

  • S. Khot and O. Regev (2008). Vertex cover might be hard to approximate to within 2 − ε. Journal of Computer and System Sciences 74(3), 335–349.

    Article  MATH  MathSciNet  Google Scholar 

  • C. Komusiewicz, R. Niedermeier, and J. Uhlmann (2009). Deconstructing intractability a case study for interval constrained coloring. CPM 2009, 207–220.

    Google Scholar 

  • M. A. Langston, F. N. Abu-Khzam, R. L. Collins, M. R. Fellows, W. H. Suters, and C. T. Symons (2004). Kernelization algorithms for the Vertex Cover problem: theory and experiments. ALENEX 2004, 62–69.

    Google Scholar 

  • M. A. Langston, F. N. Abu-Khzam, and P. Shanbhag (2003). Scalable parallel algorithms for difficult combinatorial problems: a case study in optimization. PDCS 2003, 649–654.

    Google Scholar 

  • O. Lichtenstein and A. Pneuli (1985). Checking that finite-state concurrent programs satisfy their linear specification. POPL 1985, 97–107.

    Google Scholar 

  • M. Mahajan and V. Raman (1999). Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms 31(2), 335–354.

    Article  MATH  MathSciNet  Google Scholar 

  • D. Marx (2008a). Parameterized complexity and approximation algorithms. The Computer Journal 51(1), 60–78.

    Article  Google Scholar 

  • D. Marx (2008b). Searching the k-change neighborhood for TSP is W[1]-hard. Operations Research Letters 36(1), 31–36.

    Google Scholar 

  • K. Mehlhorn (1984). Data Structures and Efficient Algorithms, Volume 2: Graph Algorithms and NP-Completeness, Springer.

    Google Scholar 

  • N. Misra, V. Raman, and S. Saurabh (2011). Lower bounds on kernelization. Discrete Optimization 8(1), 110–128.

    Article  MathSciNet  Google Scholar 

  • G. L. Nemhauser and L. E. Trotter (1975). Vertex packings: structural properties and algorithms. Mathematical Programming 8, 232–248.

    Article  MATH  MathSciNet  Google Scholar 

  • R. Niedermeier (2006). Invitation to Fixed Parameter Algorithms. Oxford University Press.

    Google Scholar 

  • R. Niedermeier (2010). Reflections on multivariate algorithmics and problem parameterization. STACS 2010, 17–32.

    Google Scholar 

  • S. Ordyniak, D. Paulusma, and S. Szeider (2010). Satisfiability of acyclic and almost acyclic CNF formulas. FSTTCS 2010, 84–95.

    Google Scholar 

  • S.-I. Oum (2005). Approximating rank-width and cliquewidth quickly. WG 2005, 49–58.

    Google Scholar 

  • C. Papadimitriou and M. Yannakakis (1997). On the complexity of database queries. POPL 1997, 12–19.

    Google Scholar 

  • R. Rizzi, P. Mahata, L. Mathieson, and P. Moscato (2010). Hierarchical clustering using the arithmetic-harmonic cut: complexity and experiments. PLoS ONE 5(12): e14067.

    Article  Google Scholar 

  • N. Robertson and P. D. Seymour (1983) Graph minors. I. Excluding a forest. J. Comb. Theory, Ser. B 35(1), 39–61.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Samer and S. Szeider (2010). Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76(2), 103–114.

    Google Scholar 

  • A. Scott (2010). The parameterized complexity of finding short winning strategies in combinatorial games. Ph.D. dissertation, University of Victoria.

    Google Scholar 

  • T. Shrot, Y. Aumann, and S. Kraus (2009). Easy and hard coalition resource game formation problems: a parameterized complexity analysis. AAMAS (1) 2009, 433–440.

    Google Scholar 

  • C. Sloper and J. A. Telle (2008). An overview of techniques for designing parameterized algorithms. The Computer Journal 51(1), 122–136.

    Article  Google Scholar 

  • O. Suchy (2011). Parameterized complexity: nonstandard parameterizations of graph problems. Ph.D. dissertation, Charles University in Prague (Czech Republic).

    Google Scholar 

  • S. Tazari, M. Müller-Hannemann (2009). Dealing with Large Hidden Constants: Engineering a Planar Steiner Tree PTAS. ALENEX 2009: 120–131.

    Google Scholar 

  • I. van Rooij and T. Wareham (2008). Parameterized complexity in cognitive modeling: foundations, applications, and opportunities. The Computer Journal 51(3), 385–404.

    Google Scholar 

  • T. Walsh (2010). Parameterized complexity results in symmetry breaking. In: Proceedings of the Fifth International Symposium of Parameterized and Exact Computation 2010 (Chennai, India), Springer, LNCS (6478) 4–13.

    Google Scholar 

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Authors and Affiliations

  1. School of Engineering and Information Technology, Charles Darwin University, Casuarina, Australia

    Michael R. Fellows & Frances Rosamond

  2. Institute of Information Systems, Vienna University of Technology, Vienna, Austria

    Serge Gaspers

Authors
  1. Michael R. Fellows
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  2. Serge Gaspers
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  3. Frances Rosamond
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Correspondence to Michael R. Fellows .

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Editors and Affiliations

  1. , Mathematics Department, University of Southern California, Los Angeles, 90089, California, USA

    Edward K. Blum

  2. , Department of Computer Science, Columbia University, New York, 10027, New York, USA

    Alfred V. Aho

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Fellows, M.R., Gaspers, S., Rosamond, F. (2011). Multivariate Complexity Theory. In: Blum, E., Aho, A. (eds) Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1168-0_13

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  • DOI: https://doi.org/10.1007/978-1-4614-1168-0_13

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