On the Optimality of Pseudo-polynomial Algorithms for Integer Programming
In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m x n matrix A and an m-vector b=(b_1,..., b_m), there is a non-negative integer n-vector x such that Ax=b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH.
This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant.
Integer Programming
Strong Exponential Time Hypothesis
Branch-width of a matrix
Fine-grained Complexity
Theory of computation~Integer programming
31:1-31:13
Regular Paper
https://arxiv.org/abs/1607.05342
Fedor V.
Fomin
Fedor V. Fomin
Department of Informatics, University of Bergen, Norway
Fahad
Panolan
Fahad Panolan
Department of Informatics, University of Bergen, Norway
M. S.
Ramanujan
M. S. Ramanujan
University of Warwick, United Kingdom
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, HBNI, Chennai, India and University of Bergen, Norway
10.4230/LIPIcs.ESA.2018.31
William H. Cunningham and Jim Geelen. On integer programming and the branch-width of the constraint matrix. In Proceedings of the 12th International Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 4513 of Lecture Notes in Comput. Sci., pages 158-166. Springer, 2007.
Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as CNF-SAT. In Proceedings of the 27th IEEE Conference on Computational Complexity (CCC), pages 74-84. IEEE, 2012. URL: http://dx.doi.org/10.1109/CCC.2012.36.
http://dx.doi.org/10.1109/CCC.2012.36
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michał Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 150-159. IEEE, 2011.
Frederic Dorn. Dynamic programming and fast matrix multiplication. In Proceedings of the 14th Annual European Symposium on Algorithms (ESA), volume 4168 of Lecture Notes in Comput. Sci., pages 280-291. Springer, Berlin, 2006.
Friedrich Eisenbrand and Robert Weismantel. Proximity results and faster algorithms for integer programming using the steinitz lemma. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 808-816. SIAM, 2018.
G. B. Horn and Frank R. Kschischang. On the intractability of permuting a block code to minimize trellis complexity. IEEE Trans. Information Theory, 42(6):2042-2048, 1996. URL: http://dx.doi.org/10.1109/18.556701.
http://dx.doi.org/10.1109/18.556701
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Computer and System Sciences, 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
http://dx.doi.org/10.1006/jcss.2000.1727
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity. J. Computer and System Sciences, 63(4):512-530, 2001.
K. Jansen and L. Rohwedder. On Integer Programming and Convolution. ArXiv e-prints, 2018. URL: http://arxiv.org/abs/1803.04744.
http://arxiv.org/abs/1803.04744
Jisu Jeong, Eun Jung Kim, and Sang-il Oum. Constructive algorithm for path-width of matroids. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1695-1704. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch116.
http://dx.doi.org/10.1137/1.9781611974331.ch116
Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of operations research, 12(3):415-440, 1987.
Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538-548, 1983.
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs on bounded treewidth are probably optimal. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 777-789. SIAM, 2011.
Dániel Marx. Can you beat treewidth? Theory of Computing, 6(1):85-112, 2010. URL: http://arxiv.org/abs/toc:v006/a005.
http://arxiv.org/abs/toc:v006/a005
Christos H. Papadimitriou. On the complexity of integer programming. J. ACM, 28(4):765-768, 1981. URL: http://dx.doi.org/10.1145/322276.322287.
http://dx.doi.org/10.1145/322276.322287
Neil Robertson and Paul D. Seymour. Graph minors. X. Obstructions to tree-decomposition. J. Combinatorial Theory Ser. B, 52(2):153-190, 1991. URL: http://dx.doi.org/10.1016/0095-8956(91)90061-N.
http://dx.doi.org/10.1016/0095-8956(91)90061-N
Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In Proceedings of the 17th Annual European Symposium on Algorithms (ESA), volume 5757 of Lecture Notes in Comput. Sci., pages 566-577. Springer, 2009.
Fedor V. Fomin, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode