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

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.