Near-Linear Time Algorithm for n-fold ILPs via Color Coding
We study an important case of ILPs max {c^Tx | Ax = b, l <= x <= u, x in Z^{n t}} with n * t variables and lower and upper bounds l, u in Z^{nt}. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s x t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and Delta, where Delta is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Omega(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs Delta)^{O(r^2s + s^2)} L^2 * nt log^{O(1)}(nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L.
Near-Linear Time Algorithm
n-fold ILP
Color Coding
Mathematics of computing~Integer programming
75:1-75:13
Track A: Algorithms, Complexity and Games
This work was supported by DFG project JA 612/20-1.
A full version can be found at https://arxiv.org/abs/1811.00950.
Klaus
Jansen
Klaus Jansen
Department of Computer Science, Kiel University, Kiel, Germany
Alexandra
Lassota
Alexandra Lassota
Department of Computer Science, Kiel University, Kiel, Germany
Lars
Rohwedder
Lars Rohwedder
Department of Computer Science, Kiel University, Kiel, Germany
10.4230/LIPIcs.ICALP.2019.75
Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM, 42(4):844-856, 1995. Previously appeared in STOC 1994.
Katerina Altmanová, Dušan Knop, and Martin Koutecký. Evaluating and Tuning n-fold Integer Programming. In 17th International Symposium on Experimental Algorithms, volume 103 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:14, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Alexandr Andoni and Krzysztof Onak. Approximating edit distance in near-linear time. SIAM Journal on Computing, 41(6):1635-1648, 2012. Previously appeared in STOC 2009.
William Cook, Jean Fonlupt, and Alexander Schrijver. An integer analogue of Carathéodory’s theorem. Journal of Combinatorial Theory, Series B, 40(1):63-70, 1986.
Jesús A. De Loera, Raymond Hemmecke, Shmuel Onn, and Robert Weismantel. N-fold integer programming. Discrete Optimization, 5(2):231-241, 2008.
Doratha E. Drake and Stefan Hougardy. A linear time approximation algorithm for weighted matchings in graphs. Journal of the ACM Transactions on Algorithms, 1(1):107-122, 2005.
Friedrich Eisenbrand, Christoph Hunkenschröder, and Kim-Manuel Klein. Faster Algorithms for Integer Programs with Block Structure. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 49:1-49:13, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Friedrich Eisenbrand and Robert Weismantel. Proximity results and faster algorithms for integer programming using the steinitz lemma. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 808-816. Society for Industrial and Applied Mathematics, 2018.
Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a Sparse Table with O(1) Worst Case Access Time. Journal of the ACM, 31(3):538-544, 1984. Previously appeared in FOCS 1982.
Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. N-fold integer programming in cubic time. Mathematical Programming, pages 1-17, 2013.
Raymond Hemmecke, Shmuel Onn, and Robert Weismantel. N-fold integer programming and nonlinear multi-transshipment. Optimization Letters, 5(1):13-25, 2011.
Klaus Jansen, Kim-Manuel Klein, Marten Maack, and Malin Rau. Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times. In ITCS, volume 124 of LIPIcs, pages 44:1-44:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019.
Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 217-226. SIAM, 2014.
Dušan Knop and Martin Kouteckỳ. Scheduling meets n-fold integer programming. Journal of Scheduling, pages 1-11, 2017.
Dušan Knop, Martin Koutecký, and Matthias Mnich. Combinatorial n-fold Integer Programming and Applications. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 54:1-54:14, 2017.
Martin Koutecký, Asaf Levin, and Shmuel Onn. A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 85:1-85:14, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and Near-Optimal Derandomization. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23-25 October 1995, pages 182-191, 1995.
Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and near-optimal derandomization. In Proceedings of IEEE 36th Annual Foundations of Computer Science, pages 182-191. IEEE, 1995.
Shmuel Onn and Pauline Sarrabezolles. Huge Unimodular n-Fold Programs. SIAM J. Discrete Math., 29(4):2277-2283, 2015.
Éva Tardos. A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs. Operations Research, 34(2):250-256, 1986. URL: http://dx.doi.org/10.1287/opre.34.2.250.
http://dx.doi.org/10.1287/opre.34.2.250
Klaus Jansen, Alexandra Lassota, and Lars Rohwedder
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode