Abstract
Integer programming in a variable dimension is a crucial research topic that has received a considerable attention in recent years. A series of fixed parameter tractable (FPT) algorithms have been developed for a variety of integer programming that has a special block structure, and such results were later applied successfully in many classical combinatorial optimization problems to derive FPT or approximation algorithms. From a theoretical point of view, it is important to understand the overall landscape, and distinguish the structures of integer programming that are tractable vs. intractable or unknown so far. From the application point of view, it is important to understand how the structure of such integer programming is related to the structure of concrete combinatorial optimization problems. The goal of this survey is to summarize recent progress in theory and application of integer programming that has a block structure and point to important open problems in this research direction.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Here we remark that by using proximity results or the standard bound from linear programming, we can always restrict that \(\log \|{\mathbf {u}}\|{ }_{\infty }+\log \|{\mathbf {l}}\|{ }_{\infty }\) is bounded by a polynomial.
References
Altmanová, K., Knop, D., Kouteckỳ, M.: Evaluating and tuning n-fold integer programming. arXiv preprint arXiv:1802.09007 (2018)
Aschenbrenner, M., Hemmecke, R.: Finiteness theorems in stochastic integer programming. Found. Comput. Math. 7(2), 183–227 (2007)
Chen, L., Marx, D.: Covering a tree with rooted subtrees–parameterized and approximation algorithms. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2801–2820. SIAM, Philadelphia (2018)
Chen, L., Marx, D., Ye, D., Zhang, G.: Parameterized and approximation results for scheduling with a low rank processing time matrix. In: Proceedings of the Thirty-Fourth Symposium on Theoretical Aspects of Computer Science, STACS, pp. 22:1–22:14 (2017)
Chen, L., Xu, L.. Shi, W.: On the graver basis of block-structured integer programming. arXiv preprint arXiv:1805.03741 (2018)
Cook, W., Fonlupt, J., Schrijver, A.: An integer analogue of Caratheodory’s theorem. J. Comb. Theory B 40(1), 63–70 (1986)
De Loera, J.A., Hemmecke, R., Onn, S., Rothblum, U.G., Weismantel, R.: Convex integer maximization via graver bases. J. Pure Appl. Algebra 213(8), 1569–1577 (2009)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer Science & Business Media, New York (2012)
Eisenbrand, F., Hunkenschröder, C., Klein, K.-M.: Faster algorithms for integer programs with block structure. arXiv preprint arXiv:1802.06289 (2018)
Fishburn, P.C.: Condorcet social choice functions. SIAM J. Appl. Math. 33(3), 469–489 (1977)
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969)
Graver, J.E.: On the foundations of linear and integer linear programming i. Math. Program. 9(1), 207–226 (1975)
Hemmecke, R., Onn, S., Weismantel, R.: A polynomial oracle-time algorithm for convex integer minimization. Math. Program. 126(1), 97–117 (2011)
Hemmecke, R., Onn, S., Romanchuk, L.: N-fold integer programming in cubic time. Math. Program. 137(1–2), 325–341 (2013)
Hemmecke, R., Köppe, M., Weismantel, R.: Graver basis and proximity techniques for block-structured separable convex integer minimization problems. Math. Program. 145(1–2), 1–18 (2014)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM (JACM) 34(1), 144–162 (1987)
Hoşten, S., Sullivant, S.: A finiteness theorem for Markov bases of hierarchical models. J. Comb. Theory A 114(2), 311–321 (2007)
Jansen, K., Klein, K.-M., Maack, M., Rau, M.: Empowering the configuration-IP-new PTAS results for scheduling with setups times. arXiv preprint arXiv:1801.06460 (2018)
Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer, Boston (1972)
Knop, D., Kouteckỳ, M.: Scheduling meets n-fold integer programming. J. Sched. 21(5), 493–503 (2018)
Knop, D., Kouteckỳ, M., Mnich, M.: Combinatorial n-fold integer programming and applications. arXiv preprint arXiv:1705.08657 (2017)
Kouteckỳ, M., Levin, A., Onn, S.: A parameterized strongly polynomial algorithm for block structured integer programs. arXiv preprint arXiv:1802.05859 (2018)
Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)
Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. Math. Program. 154(1–2), 533–562 (2015)
Onn, S.: Nonlinear Discrete Optimization. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich (2010)
Papadimitriou, C.H.: On the complexity of integer programming. J. ACM (JACM) 28(4), 765–768 (1981)
Sebö, A.: Hilbert bases, Caratheodory’s theorem and combinatorial optimization. In: Proceedings of the 1st Integer Programming and Combinatorial Optimization Conference, pp. 431–455. University of Waterloo Press, Ontario (1990)
Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34(2), 250–256 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Chen, L. (2019). On Block-Structured Integer Programming and Its Applications. In: Du, DZ., Pardalos, P., Zhang, Z. (eds) Nonlinear Combinatorial Optimization. Springer Optimization and Its Applications, vol 147. Springer, Cham. https://doi.org/10.1007/978-3-030-16194-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-16194-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16193-4
Online ISBN: 978-3-030-16194-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)