The theory of parameterized computation and complexity is a recently developed subarea in theoretical computer science. The theory is aimed at practically solving a large number of computational problems that are theoretically intractable. The theory is based on the observation that many intractable computational problems in practice are associated with a parameter that varies within a small or moderate range. Therefore, by taking the advantages of the small parameters, many theoretically intractable problems can be solved effectively and practically. On the other hand, the theory of parameterized computation and complexity has also offered powerful techniques that enable us to derive strong computational lower bounds for many computational problems, thus explaining why certain theoretically tractable problems cannot be solved effectively and practically. The theory of parameterized computation and complexity has found wide applications in areas such as database systems, programming languages, networks, VLSI design, parallel and distributed computing, computational biology, and robotics.
This survey gives an overview on the fundamentals, algorithms, techniques, and applications developed in the research of parameterized computation and complexity. We will also report the most recent advances and excitements, and discuss further research directions in the area.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Garey M R, Johnson D S. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, 1979.
Ausiello G, Crescenzi P, Gambosi G, Kann V, Marchetti-Spaccamela A, Protasi M. Complexity and Approximation — Combinatorial Optimization Problems and Their Approximability Properties. Springer, 1999.
Motwani R, Raghavan P. Randomized Algorithms. Cambridge University Press, New York, 1995.
Michalewicz Z, Fogel D B. How to Solve It: Modern Heuristics. Springer, Berlin, 2000.
Roth-Korostensky C. Algorithms for Building Multiple Sequence Alignments and Evolutionary Trees [Dissertation]. No. 13550, ETH Zürich, 2000.
Stege U. Resolving Conflicts from Problems in Computational Biology [Dissertation]. No. 13364, ETH Zürich, 2000.
Cai L, Juedes D, Kanj I A. Inapproximability of non NP-hard optimization problems. Theoretical Computer Science, 2002, 289: 553–571.
Cheetham J, Dehne F, Rau-Chaplin A, Stege U, Taillon P J. Solving large FPT problems on coarse-granined parallel machines. J. Computer and System Sciences, 2003, 67: 691–701.
Lichtenstein O, Pnueli A. Finite state concurrent programs satisfy their linear specification. In Proc. 12th ACM Symposium on the Principles of Programming Languages, 1985, pp. 97–107.
Henglein F, Mairson H G. The complexity of type inference for higher-order typed lambda calculi. Journal of Functional Programming, 1994, 4: 435–477.
Chen J, Kanj I A. Constrained minimum vertex cover in bipartite graphs: Complexity and parameterized algorithms. Journal of Computer and System Sciences, 2003, 67: 833–847.
Nesetril J, Poljak S. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 1985, 26: 415–419.
Coppersmith D, Winograd S. Matrix multiplication via arithmetic progression. J. Symbolic Logic, 1990, 9: 251–280.
Papadimitriou C H, Yannakakis M. On the complexity of database queries. Journal of Computer and System Sciences, 1999, 58: 407–427.
Chen J, Chor B, Fellows M, Huang X, Juedes D, Kanj I, Xia G. Tight lower bounds for certain parameterized NP-hard problems. In Proc. 19th Annual IEEE Conference on Computational Complexity (CCC 2004), 2004, pp. 150–160.
Anthony M, Biggs N. Computational Learning Theory. Cambridge University Press, Cambridge, UK, 1992.
Papadimitriou C H, Yannakakis M. On limited nondeterminism and the complexity of VC dimension. Journal of Computer and System Sciences, 1996, 53: 161–170.
Downey R, Fellows M. Parameterized Complexity. Springer-Verlag, 1999.
Pevzner P A, Sze S-H. Combinatorial approaches to finding subtle signals in DNA sequences. In Proc. 8th International Conf. Intelligent Systems for Molecular Biology (ISMB’00), 2000, pp. 269–278.
Sze S H, Lu S, Chen J. Integrating sample-driven and pattern-driven approaches in Motif finding, Lecture Notes in Computer Science 3240, 2004, pp. 438–449.
Cai L, Chen J, Downey R, Fellows M. On the structure of parameterized problems in NP. Information and Computation, 1995, 123: 38–49.
Chen Y, Flum J. Machine characterizations of the classes of the W-hierarchy. Lecture Notes in Computer Science 2803 (CSL’03), 2003, pp. 114–127.
Flum J, Grohe M. Describing parameterized complexity classes. Lecture Notes in Computer Science 2285 (STACS’02), 2002, pp. 359–371.
Chen J. Simpler computation and deeper theory: On development of efficient parameterized algorithms. International Workshop on Parameterized Complexity, Chennai, India, 2000.
Downey R, Fellows M, Stege U. Parameterized complexity: A framework for systematically confronting computational intractability. In Contemporary Trends in Discrete Mathematics, Graham R, Kratochvil J, Nesetril J, Roberts F (eds.), AMS-DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1999, 49: 49–99.
Fellows M. Parameterized complexity: The main ideas and some research frontiers. Lecture Notes in Computer Science 2223 (ISAAC’01), 2001, pp. 291–307.
Cormen T, Leiserson C, Rivest R, Stein C. Introduction to Algorithms. McGraw-Hill, Boston, 2001.
Lovasz L, Plummer M. Matching Theory. North-Holland, Amsterdam, 1986.
Nemhauser G, Trotter L. Vertex packing: Structural properties and algorithms. Math. Programming, 1975, 8: 232–248.
Chen J, Kanj I A, Jia W. Vertex cover: Further observations and further improvements. J. Algorithms, 2001, 41: 280–301.
Fellows M. Blow-ups, win/win’s, and crown rules: Some new directions in FPT. Lecture Notes in Computer Science (WG’03), 2003, pp. 1–12.
Chor B, Fellows M, Juedes D. An efficient FPT algorithm for saving k colors. Manuscript, 2003.
Alber J, Fellows M, Niedermeier R. Polynomial time data reduction for dominating set. J. ACM, 2004, 11: 363–384.
Robson J M. Algorithms for maximum independent sets. J. Algorithms, 1986, 7: 425–440.
Tarjan R E, Trojanowski A E. Finding a maximum independent set. SIAM J. Comput., 1977, 6: 537–546.
Woeginger G. Exact algorithms for NP-hard problems: A survey. Lecture Notes in Computer Science 2570, 2001, pp. 185–207.
Niedermeier R, Rossmanith P. A general method to speed up fixed-parameter tractable algorithms. Inform. Process. Lett., 2000, 73: 125–129.
Balasubramanian R, Fellows M R, Raman V. An improved fixed parameter algorithm for vertex cover. Inform. Process. Lett., 1998, 65: 163–168.
Chen J, Liu L, Jia W. Improvement on vertex cover for low-degree graphs. Networks, 2000, 35: 253–259.
Niedermeier R, Rossmanith P. Upper bounds for vertex cover further improved. Lecture Notes in Computer Science 1563 (STACS’99), 1999, pp. 561–570.
Baker B S. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 1994, 41: 153–180.
Lipton R, Tarjan R. Application of a graph separator theorem. SIAM Journal on Computing, 1980, 9: 615–627.
Chen J, Huang X, Kanj I, Xia G. Polynomial time approximation schemes and parameterized complexity. In Proc. 29th International Symposium on Mathematical Foundations of Computer Science (MFCS 2004), Lecture Notes in Computer Science 3153, 2004, pp. 500–512.
Bodlaender H L. A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 1996, 25: 1305–1317.
Eppstein D. Diameter and treewidth in minor-closed graph families. Algorithmica, 2000, 27: 275–291.
Arnborg S. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey. BIT, 1985, 25: 2–23.
Alber J, Bodlaender H L, Fernau H, Kloks T, Niedermeier R. Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica, 2002, 33: 461–493.
Fomin F V, Thilikos D M. Dominating sets in planar graphs: Branch-width and exponential speed-up. In Proc. 14th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA’03), 2003, pp. 168–177.
Kanj I, Perkovic L. Improved parameterized algorithms for planar dominating set. Lecture Notes in Computer Science 2420 (MFCS’02), 2002, pp. 399–410.
Alon N, Yuster R, Zwick U. Color-coding. Journal of the ACM, 1995, 42: 844–856.
Chen J, Friesen D, Kanj I, Jia W. Using nondeterminism to design efficient deterministic algorithms. Algorithmica, 2004, 40: 83–97.
Jia W, Zhang C, Chen J. An efficient parameterized algorithm for m-set packing. Journal of Algorithms, 2004, 50: 106–117.
Dehne F, Fellows M, Rosamond F. An FPT algorithm for set splitting. Lecture Notes in Computer Science 2880 (WG’03), 2003, pp. 180–191.
Niedermeier R. Invitation to Fixed-Parameter Algorithms [Thesis]. Universitat Tubingen, 2002.
Robertson N, Seymour P D. Graph Minors — A Survey. In Surveys in Combinatorics 1985, Anderson I (ed.), Cambridge Univ. Press, Cambridge, 1985, pp. 153–171.
Robertson N, Seymour P D. Graph minors VIII. A Kuratowski theorem for general surfaces. J. Combin. Theory Ser. B, 1990, 48: 255–288.
Robertson N, Seymour P D. Graph minors XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 1995, 63: 65–110.
Fellows M, Langston M. Nonconstructive tools for proving polynomial-time decidability. J. ACM, 1988, 35: 727–739.
DIMACS Workshop on Faster Exact Solutions for NP-Hard Problems. Princeton, Feb. 23–24, 2000.
Papadimitriou C H, Yannakakis M. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 1991, 43: 425–440.
Impagliazzo R, Paturi R. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 2001, 63: 512–530.
Abrahamson K, Downey R, Fellows M. Fixed-parameter tractability and completeness IV: On completeness of W[P] and PSPACE analogs. Ann. Pure Appl. Logic, 1995, 73: 235–276.
Downey R, Estivill-Castro V, Fellows M, Prieto-Rodriguez E, Rosamond F. Cutting up is hard to do: The parameterized complexity of k-cut and related problems. Electronic Notes in Theoretical Computer Science, 2003, 78: 205–218.
Bodlaender H, Downey R, Fellows M, Hallett M, Wareham H. Parameterized complexity analysis in computational biology. Computer Applications in Biosciences, 1995, 11: 49–57.
Cesati M. Compendium of parameterized problems (2004 version). Department of Computer Science, Systems, and Industrial Engineering, University of Rome “Tor Vergata”, Italy. http://bravo.ce.uniroma2.it/home/cesati/research/compendium.ps.
Deng X, Li G, Li Z, Ma B, Wang L. Genetic design of drugs without side-effects. SIAM J. Comput., 2003, 32: 1073–1090.
Gramm J, Guo J, Niedermeier R. On exact and approximation algorithms for distinguishing substring selection. Lecture Notes in Computer Science 2751 (FCT’03), 2003, pp. 195–209.
Chen J, Huang X, Kanj I A, Xia G. Linear FPT-reductions and computational lower bounds. In Proc. 36th ACM Symp. Theory of Computing (STOC 2004), 2004, pp. 212–221.
Goldschmidt O, Hochbaum D. Polynomial algorithm for the k-cut problem. In Proc. 29th Ann. Symp. Foundations of Computer Science (FOCS’88), 1988, pp. 444–451.
Bodlaender H, Fellows M, Hallett M. Beyond NP-completeness for problems of bounded width: Hardness for the W-hierarchy. In Proc. 26th Ann. ACM Symp. Theory of Computing (STOC’94), 1994, pp. 449–458.
Robson J M. Finding a maximum independent set in time O(2n/4)? LaBRI, Universite Bordeaux I, 1251-01, 2001.
Cai L, Juedes D. On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences, 2003, 67: 789–807.
Cai L, Chen J. On fixed parameter tractability and approximability of NP optimization problems. Journal of Computer and System Sciences, 1997, 54: 465–474.
Ausiello G, Marchetti-spaccamela A, Protasi M. Toward a unified approach for the classification of NP-complete optimization problems. Theoretical Computer Science, 1980, 12: 83–96.
Paz A, Moran S. Non deterministic polynomial optimization problems and their approximations. Theoretical Computer Science, 1981, 15: 251–277.
Woeginger G. When does a dynamic programming formulation guarantee the existence of an FPTAS? In Proc. 10th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA’99), 2001, pp. 820–829.
Kolaitis P, Thakur M. Approximation properties of NP minimization classes. Journal of Computer and System Sciences, 1995, 50: 391–411.
Arora S, Lund C, Motwani R, Sudan M, Szegedy M. Proof verification and hardness of approximation problems. Journal of the ACM, 1998, 45: 501–555.
Downey R. Parameterized complexity for the skeptic. In Proc. 18th IEEE Conference on Computational Complexity (CCC’03), 2003, pp. 147–169.
Cesati M, Trevisan L. On the efficiency of polynomial time approximation schemes. Information Processing Letters, 1997, 64: 165–171.
Khanna S, Motwani R. Towards a syntactic characterization of PTAS. In Proc. 28th Ann. ACM Symp. Theory of Computing (STOC’96), 1996, pp. 329–337.
Cai L, Fellows M, Juedes D, Rosamond F. Efficient polynomial-time approximation schemes for problems on planar structures: Upper and lower bounds. Manuscript, 2001.
Huang X. Parameterized complexity and polynomial-time approximation schemes [Dissertation]. Department of Computer Science, Texas A&M University, December, 2004.
This research is supported in part by the National Natural Science Foundation of China under Grants No.60373083 and No.60433020, and by the Changjiang Scholar Reward Project of Ministry of Education, P.R. China.
Jian-Er Chen got his B.S. degree in computer science in 1982 from Central South University, China, and his Ph.D. degree in computer science in 1987 from Courant Institute, New York University, USA, where he was awarded the Janet Fabri Award for the best Ph.D. dissertation. After graduation from NYU, he went to the Department of Mathematics at Columbia University, USA, where he received the Ph.D. degree in mathematics in 1990. Since then, he has been with the Department of Computer Science at Texas A&M University, USA, where he is a professor. Currently, he is a ChangJiang Scholar Professor at Central South University, China. His research interests include theoretical computer science, bioinformatics, computer networks, and computer graphics. He has published over 120 journal and conference papers in these areas, and received numerous awards, including the Research Initiation Award in 1991 from US National Science Foundation, TEES Select Young Faculty Award in 1993 and Distinguished Faculty Achievement Award in 1998 from Texas A&M University, Oversea Distinguished Young Scholars Award in 2000 from the National Natural Science Foundation of China, and Natural Science Award (first class) in 2003 from MOE, China.
About this article
Cite this article
Chen, J. Parameterized Computation and Complexity: A New Approach Dealing with NP-Hardness. J Comput Sci Technol 20, 18–37 (2005). https://doi.org/10.1007/s11390-005-0003-7
- parameterized computation
- approximation algorithm