Abstract
Exponential algorithms, whose time complexity is O(c n) for some constant c > 1, are inevitable when exactly solving NP-complete problems unless \(\mathbf{P} = \mathbf{NP}\). This chapter presents recently emerged combinatorial and algebraic techniques for designing exact exponential time algorithms. The discussed techniques can be used either to derive faster exact exponential algorithms or to significantly reduce the space requirements while without increasing the running time. For illustration, exact algorithms arising from the use of these techniques for some optimization and counting problems are given.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Recommended Reading
O. Amini, F.V. Fomin, S. Saurabh, Counting subgraphs via homomorphisms, in ICALP, Rhodes (1), 2009, pp. 71–82
L. Babai, W.M. Kantor, E.M. Luks, Computational complexity and the classification of finite simple groups, in FOCS, 1983, Tucson, pp. 162–171
E.T. Bax, Inclusion and exclusion algorithm for the Hamiltonian path problem. Inf. Process. Lett. 47(4), 203–207 (1993)
E.T. Bax, Algorithms to count paths and cycles. Inf. Process. Lett. 52(5), 249–252 (1994)
E.T. Bax, J. Franklin, A finite-difference sieve to count paths and cycles by length. Inf. Process. Lett. 60(4), 171–176 (1996)
R. Bellman, Bottleneck problems and dynamic programming. Proc. Natl. Acad. Sci. 39, 947–951 (1953)
R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957)
A. Björklund, Exact covers via determinants, in STACS, Nancy, 2010, pp. 95–106
A. Björklund, Determinant sums for undirected Hamiltonicity, in FOCS, Las Vegas, 2010, pp. 23–26
A. Björklund, Counting perfect matchings as fast as Ryser, in SODA, 2012, Kyoto, pp. 914–921
A. Björklund, T. Husfeldt, Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Fourier meets möbius: fast subset convolution, in STOC, San Diego, 2007, pp. 67–74
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Computing the tutte polynomial in vertex-exponential time, in FOCS, Philadelphia, 2008, pp. 677–686
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, The traveling salesman problem in bounded degree graphs, in ICALP, Reykjavik, 2008, pp. 198–209
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Counting paths and packings in halves, in ESA, Copenhagen, 2009, pp. 578–586
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Trimmed moebius inversion and graphs of bounded degree. Theory Comput. Syst. 47, 637–654 (2010)
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Covering and packing in linear space, in ICALP, Bordeaux, (1), 2010, pp. 727–737
A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Evaluation of permanents in rings and semirings. Inf. Process. Lett. 110(20), 867–870 (2010)
A. Bjorklund, T. Husfeldt, Inclusion-exclusion algorithms for counting set partitions, in FOCS, Berkeley, 2006, pp. 575–582
A. Bjorklund, T. Husfeldt, M. Koivisto, Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)
J.R. Bunch, J.E. Hopcroft, Triangular factorization and inversion by fast matrix multiplication. Math. Comput. 28, 231–236 (1974)
F.R.K. Chung, R.L. Graham, On the cover polynomial of a digraph. J. Comb. Theory B 65(2), 273–290 (1995)
F.R.K. Chung, P. Frankl, R.L. Graham, J.B. Shearer, Some intersection theorems for ordered sets and graphs. J. Comb. Theory A 43, 23–37 (1986)
D. Coppersmith, S. Winograd, Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)
S.E. Dreyfus, R.A. Wagner, The Steiner problem in graphs. Networks 1, 195–207 (1971/1972)
J. Edmonds, Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Stand. 71B(4), 241–245 (1967)
F.V. Fomin, D. Kratsch, Exact Exponential Algorithms (Springer, Berlin/Heidelberg, 2010)
B. Fuchs, W. Kern, D. Mölle, S. Richter, P. Rossmanith, X. Wang, Dynamic programming for minimum Steiner trees. Theory Comput. Syst. 41(3), 493–500 (2007)
M. Fürer, Faster integer multiplication, in STOC, San Diego, 2007, pp. 57–66
M. Held, R.M. Karp, A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10(1), 196–210 (1962)
Q.-S. Hua, D. Yu, Y. Wang, F.C.M. Lau, Exact algorithms for set multicover and multiset multicover problems, in ISAAC, Honolulu, 2009, pp. 34–44
Q.-S. Hua, Y. Wang, D. Yu, F.C.M. Lau, Set multi-covering via inclusion-exclusion. Theor. Comput. Sci. 410(38–40), 3882–3892 (2009)
Q.-S. Hua, Y. Wang, D. Yu, F.C.M. Lau, Dynamic programming based algorithms for set multicover and multiset multicover problems. Theor. Comput. Sci. 411(26–28), 2467–2474 (2010)
T. Husfeldt, Invitation to algorithmic uses of inclusion-exclusion, in ICALP, Zurich, (2), 2011, pp. 42–59
D.B. Johnson, Efficient algorithms for shortest paths in sparse networks. J. ACM 24, 1–13 (1977)
R.M. Karp, Dynamic programming meets the principle of inclusion and exclusion. Oper. Res. Lett. 1, 49–51 (1982)
R. Kennes, Computational aspects of the Moebius transform of a graph. IEEE Trans. Syst. Man Cybern. 22, 201–223 (1991)
D.E. Knuth, The art of computer programming, Vol. 3: Seminumerical algorithms, 3rd edn. (Addison-Wesley, Upper Saddle River, 1998)
S. Kohn, A. Gottlieb, M. Kohn, A generating function approach to the traveling salesman problem, in In ACM ’77: Proceedings of the 1977 Annual Conference (ACM, New York, 1977), pp. 294–300
M. Koivisto, An O ∗(2n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion, in FOCS, Berkeley, 2006, pp. 583–590
R. Lipton, Fast exponential algorithms, http://rjlipton.wordpress.com/2009/02/13/polynomial-vs-exponential-time/
D. Lokshtanov, J. Nederlof, Saving space by algebraization, in STOC, Cambridge, 2010, pp. 321–330
R. Motwani, P. Raghavan, Randomized Algorithms (Cambridge University Press, Cambridge/New York, 1995)
J. Nederlof, Fast polynomial-space algorithms using Möbius inversion: improving on Steiner tree and related problems, in ICALP, Rhodes, (1), 2009, pp. 713–725
J. Nederlof, J.M.M. van Rooij, Inclusion/exclusion branching for partial dominating set and set splitting, in IPEC, Chennai, 2010, pp. 204–215
S.i. Oum, Computing rank-width exactly. Inf. Process. Lett. 109(13), 745–748 (2009)
D. Paulusma, J.M.M. van Rooij, On partitioning a graph into two connected subgraphs, in ISAAC, Honolulu, 2009, pp. 1215–1224
G.C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368 (1964)
H.J. Ryser, Combinatorial Mathematics. Number 14 in Carus mathematical monographs (Mathematical Association of America, Buffalo, 1963)
U. Schöning, Algorithmics in exponential time, in STACS, Stuttgart, 2005, pp. 36–43
J.M.M. van Rooij, Polynomial space algorithms for counting dominating sets and the domatic number, in CIAC, Rome, 2010, pp. 73–84
J.M.M. van Rooij, H.L. Bodlaender, P. Rossmanith, Dynamic programming on tree decompositions using generalised fast subset convolution, in ESA, Copenhagen, 2009, pp. 566–577
J.M.M. van Rooij, J. Nederlof, T.C. van Dijk, Inclusion/exclusion meets measure and conquer: exact algorithms for counting dominating sets, in ESA, Copenhagen, 2009, pp. 554–565
V. Vassilevska, R. Williams, Finding, minimizing, and counting weighted subgraphs, in STOC, Bethesda, 2009, pp. 455–464
G.J. Woeginger, Exact algorithms for NP-hard problems: a survey, in Combinatorial Optimization, 2001, Springer-Verlag New York, pp. 185–208
G.J. Woeginger, Space and time complexity of exact algorithms: some open problems, in IWPEC, Bergen, 2004, pp. 281–290
G.J. Woeginger, Open problems around exact algorithms. Discret. Appl. Math. 156(3), 397–405 (2008)
F. Yates, The design and analysis of factorial experiments. Technical Communication No. 35, Commonwealth Bureau of Soil Science, Harpenden, UK, 1937
Acknowledgements
This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00302, the National Natural Science Foundation of China Grant 61103186, 61073174, 61033001, 61061130540, the Hi-Tech research and Development Program of China Grant 2006AA10Z216, and Hong Kong RGC-GRF grants 714009E and 714311.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this entry
Cite this entry
Yu, D., Wang, Y., Hua, QS., Lau, F.C. (2013). Faster and Space Efficient Exact Exponential Algorithms: Combinatorial and Algebraic Approaches. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_38
Download citation
DOI: https://doi.org/10.1007/978-1-4419-7997-1_38
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-7996-4
Online ISBN: 978-1-4419-7997-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering