Efficient Algorithms for Integer Programs with Two Variables per Constraint

Bar-Yehuda, Reuven; Rawitz, Dror

doi:10.1007/3-540-48481-7_11

Cited by 13 publications

(40 citation statements)

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“…We also show that η(I) < 1, = 1, and > 1 can be checked in linear time. This theorem implies the existing results [2,14,29] that quadratic (i.e., TVPI) systems and Horn systems can be solved in pseudo-polynomial time, since quadratic systems and Horn systems are included in ILS = (γ) with γ ≤ 1, which will be discussed later.…”

Section: The Results Obtained In This Papersupporting

confidence: 66%

“…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 problem is still strongly NP-hard, but it can be solved in O(nm) [21] and O(n log n + m) time [27] if A is in addition quadratic, and can be solved in O(n 2 m) time [9,28] if A is in addition Horn.…”

Section: Integer Linear Systemsmentioning

confidence: 99%

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Trichotomy for integer linear systems based on their sign patterns

Kimura

Makino

2016

Discrete Applied Mathematics

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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

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Section: The Results Obtained In This Papersupporting

confidence: 66%

Section: Integer Linear Systemsmentioning

confidence: 99%

Trichotomy for integer linear systems based on their sign patterns

Kimura

Makino

2016

Discrete Applied Mathematics

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“…We show that the equivalence between the paradigms continues to hold. We demonstrate the use of the extended frameworks on several algorithms: a 2.5-approximation algorithm for feedback vertex set in tournaments [16]; a 2-approximation algorithm for a noncovering problem called minimum 2-satisfiability [29,9]; and a 3-approximation algorithm for a bandwidth trading problem [15]. We show that the equivalence continues to hold in the maximization case.…”

Section: Our Resultsmentioning

confidence: 99%

“…Hochbaum et al [32] presented a 2-approximation algorithm for the two variables per constraint integer programming problem (2VIP) that generalizes min-2SAT. Later, Bar-Yehuda and Rawitz [9] presented a local ratio 2-approximation algorithm for 2VIP that is more efficient than the algorithm from [32]. On the special case of min-2SAT this algorithm is a variant of the Gusfield-Pitt algorithm.…”

Section: Minimum Weight 2-satisfiabilitymentioning

confidence: 99%

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On the Equivalence between the Primal-Dual Schema and the Local-Ratio Technique

Bar-Yehuda

Rawitz

2001

Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques

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Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primal-dual schema and the local ratio technique. Recently, primal-dual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primal-dual algorithm. This was done in the case of the 2-approximation algorithms for the feedback vertex set problem and in the case of the first primal-dual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447-478]. In this paper we answer this question by showing that the two paradigms are equivalent.

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Trichotomy for the Reconfiguration Problem of Integer Linear Systems

Kimura

Suzuki

2020

WALCOM: Algorithms and Computation

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In this paper, we consider the reconfiguration problem of integer linear systems. In this problem, we are given an integer linear system 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. Z(I) for I is the complexity index introduced by Kimura and Makino (Discrete Applied Mathematics 200:67-78, 2016), which is defined by the sign pattern of the input matrix. We analyze the complexity of the reconfiguration problem of integer linear systems based on the complexity index Z(I) of given I. We then show that the problem is (i) solvable in constant time if Z(I) is less than one, (ii) weakly coNP-complete and pseudo-polynomially solvable if Z(I) is exactly one, and (iii) PSPACE-complete if Z(I) is greater than one. Since the complexity indices of Horn and two-variable-par-inequality integer linear systems are at most one, our results imply that the reconfiguration of these systems are in coNP and pseudo-polynomially solvable. Moreover, this is the first result that reveals coNP-completeness for a reconfiguration problem, to the best of our knowledge.

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Efficient Algorithms for Integer Programs with Two Variables per Constraint

Cited by 13 publications

References 9 publications

Trichotomy for integer linear systems based on their sign patterns

Trichotomy for integer linear systems based on their sign patterns

On the Equivalence between the Primal-Dual Schema and the Local-Ratio Technique

Trichotomy for the Reconfiguration Problem of Integer Linear Systems

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