Trichotomy for integer linear systems based on their sign patterns

Abstract

In this paper, we consider solving the integer linear systems, i.e., given a matrix A ∈ R m×n , a vector b ∈ R m , and a positive integer d, to compute an integer vector x ∈ D n such that Ax ≥ b, where m and n denote positive integers, R denotes the set of reals, andThe problem is one of the most fundamental NP-hard problems in computer science.For the problem, we propose a complexity index η which is based only on the sign pattern of A. For a real γ, let ILS = (γ) denote the family of the problem instances I with η(I) = γ. We then show the following trichotomy: In this paper, we consider the problem of computing an integer vector x ∈ D n such that Ax ≥ b, which we denote by ILS. The ILS problem is one of the most fundamental and important problems in computer science, and have been studied extensively from both theoretical and practical points of view [18,26]. It is known that the ILS problem is strongly NP-hard, and can be solved in polynomial time, if m or n are bounded by some constant [22], or A is totally unimodular and b is integral [15]. When A is quadratic (also called TVPI, i.e., each row of A contains at most two nonzero elements) or Horn (i.e., each row of A contains at most one positive element), the ILS problem is known to be weakly NP-hard, but it can be solved in time polynomial in the input length and d, and hence in pseudo-polynomial time [20,14,29]. The best known bounds for quadratic and Horn systems require O(md) time [2] and O(n 2 md) time, respectively. For unit linear systems, i.e., A ∈ {0, −1, +1} m×n , it is known that the

Trichotomy for the Reconfiguration Problem of Integer Linear Systems

“…Namely, we are given an ILS I and two feasible solutions s and t of I, and then asked to transform s to t by changing a value of only one variable at a time, while maintaining a feasible solution of I throughout. We analyze the complexity of this problem using the complexity index described in the previous subsection and show the following three results: the reconfiguration problem of ILS is (i) always yes if the complexity index is less than one (ii) weakly coNP-complete and pseudo-polynomially solvable if the complexity index is exactly one As mentioned in the previous subsection, both Horn and TVPI ILSes are contained in ILS(1) [15]. Therefore, the reconfiguration problem of these ILSes are both in coNP and pseudo-polynomially solvable from this result.…”

“…Horn and TVPI ILSes arise in, e.g., program verification and scheduling, respectively, and many algorithms have been devised to solve the feasibility problems of these subclasses [1,7,10,19]. On the other hand, ILS(1) can be decomposed to Horn and TVPI ILSes in a certain way [15]; see also Section 3. It should be also noted that we can recognize which class a given ILS belongs to in linear time, without solving LP (1) [15].…”

“…On the other hand, ILS(1) can be decomposed to Horn and TVPI ILSes in a certain way [15]; see also Section 3. It should be also noted that we can recognize which class a given ILS belongs to in linear time, without solving LP (1) [15]. This is useful for practice, since if we recognized in linear time that the index of a given ILS is, say, less than one, then we could use the linear time algorithm to solve the feasibility problem.…”

Linear Satisfiability Preserving Assignments

“…For example, the problem can be solved in polynomial time, if n is bounded by some constant (Lenstra, Jr., 1983), or if G is totally unimodular (Hoffman & Kruskal, 1956). Moreover, it can be solved in pseudo-polynomial time if (1) m is bounded by some constant (Papadimitriou, 1981), (2) G corresponds to a two-variable-per-inequality (TVPI) system (i.e., each row of G contains at most two nonzero elements) (Hochbaum, Megiddo, Naor, & Tamir, 1993;Bar-Yehuda & Rawitz, 2001), (3) G is Horn (i.e., each row of G contains at most one positive element) (Glover, 1964;van Maaren & Dang, 2002), or (4) G is q-Horn (Kimura & Makino, 2016). It is also known that the problem is weakly NP-hard, even if m is bounded by some constant or the system is TVPI and Horn (also called monotone quadratic) (Lagarias, 1985).…”

“…A system Gx ≥ h is said to be TVPI if each row of G contains at most two nonzero elements, Horn if each row of G contains at most one positive element, and q-Horn if the optimal value of the following LP problem is at most one (Kimura & Makino, 2016).…”