# A Fast Algorithm for Unbounded Monotone Integer Linear Systems with Two Variables per Inequality via Graph Decomposition

## Abstract

In this paper, we consider the feasibility problem of integer linear systems where each inequality has at most two variables. Although the problem is known to be weakly NP-complete by Lagarias, it has many applications and, importantly, a large subclass of it admits (pseudo-)polynomial algorithms. Indeed, the problem is shown pseudo-polynomially solvable if every variable has upper and lower bounds by Hochbaum, Megiddo, Naor, and Tamir. However, determining the complexity of the general case, pseudo-polynomially solvable or strongly NP-complete, is a longstanding open problem. In this paper, we reveal a new efficiently solvable subclass of the problem. Namely, for the *monotone* case, i.e., when two coefficients of the two variables in each inequality are opposite signs, we associate a directed graph to any instance, and present an algorithm that runs in \(O(n \cdot s \cdot 2^{O(\ell \log \ell )} + n + m)\) time, where *s* is the length of the input and \(\ell \) is the maximum number of the vertices in any strongly connected component of the graph. If \(\ell \) is a constant, the algorithm runs in polynomial time. From the result, it can be observed that the hardness of the feasibility problem lies on large strongly connected components of the graph.

## Keywords

Integer linear system Integer programming Two-variable-per-inequality system Monotone system## References

- 1.Bar-Yehuda, R., Rawitz, D.: Efficient algorithms for integer programs with two variables per constraint. Algorithmica
**29**(4), 595–609 (2001)MathSciNetCrossRefGoogle Scholar - 2.Bordeaux, L., Katsirelos, G., Narodytska, N., Vardi, M.Y.: The complexity of integer bound propagation. J. Artif. Intell. Res.
**40**, 657–676 (2011)MathSciNetCrossRefGoogle Scholar - 3.Chandrasekaran, R., Subramani, K.: A combinatorial algorithm for horn programs. Discrete Optim.
**10**(2), 85–101 (2013)MathSciNetCrossRefGoogle Scholar - 4.Fügenschuh, A.: A set partitioning reformulation of a school bus scheduling problem. J. Sched.
**14**(4), 307–318 (2011)MathSciNetCrossRefGoogle Scholar - 5.Hochbaum, D.S., Megiddo, N., Naor, J.S., Tamir, A.: Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Math. Program.
**62**(1–3), 69–83 (1993)MathSciNetCrossRefGoogle Scholar - 6.Hochbaum, D.S., Naor, J.S.: Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput.
**23**(6), 1179–1192 (1994)MathSciNetCrossRefGoogle Scholar - 7.Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res.
**12**(3), 415–440 (1987)MathSciNetCrossRefGoogle Scholar - 8.Kannan, R.: A polynomial algorithm for the two-variable integer programming problem. J. Assoc. Comput. Mach.
**27**(1), 118–122 (1980)MathSciNetCrossRefGoogle Scholar - 9.Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 3rd edn. Springer, Heidelberg (2005). https://doi.org/10.1007/3-540-29297-7. Japanese translation from EnglishCrossRefzbMATHGoogle Scholar
- 10.Lagarias, J.C.: The computational complexity of simultaneous diophantine approximation problems. SIAM J. Comput.
**14**(1), 196–209 (1985)MathSciNetCrossRefGoogle Scholar - 11.Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res.
**8**(4), 538–548 (1983)MathSciNetCrossRefGoogle Scholar - 12.Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
- 13.Shostak, R.: Deciding linear inequalities by computing loop residues. J. ACM
**28**(4), 769–779 (1981)MathSciNetCrossRefGoogle Scholar - 14.Upadrasta, R., Cohen, A.: A case for strongly polynomial time sub-polyhedral scheduling using two-variable-per-inequality polyhedra. In: IMPACT 2012–2nd Workshop on Polyhedral Compilation Techniques (associated with HiPEAC), Paris, France (2012)Google Scholar
- 15.Veinott, A.F.: Representation of general and polyhedral subsemilattices and sublattices of product spaces. Linear Algebra Appl.
**114–115**(1989), 681–704 (1989)MathSciNetCrossRefGoogle Scholar