Farkas Bounds on Horn Constraint Systems

In this paper, we analyze the copy complexity of unsatisfiable Horn constraint systems, under the ADD refutation system. Recall that a linear constraint of the form ∑ i = 1 n a i · x i ≥ b , is said to be a horn constraint if all the a i ∈ { 0 , 1 , - 1 } and at most one of the a i s is positive. A...

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Published in:Theory of computing systems Vol. 68; no. 2; pp. 227 - 249
Main Authors: Subramani, K., Wojciechowski, Piotr, Velasquez, Alvaro
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2024
Springer Nature B.V
Subjects:
Approximation
Complexity
Computer Science
Empirical analysis
Integer programming
Program verification (computers)
Theory of Computation
Linear refutation
Approximation complexity
Horn constraints
Copy complexity
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary:In this paper, we analyze the copy complexity of unsatisfiable Horn constraint systems, under the ADD refutation system. Recall that a linear constraint of the form ∑ i = 1 n a i · x i ≥ b , is said to be a horn constraint if all the a i ∈ { 0 , 1 , - 1 } and at most one of the a i s is positive. A conjunction of such constraints is called a Horn constraint system (HCS). Horn constraints arise in a number of domains including, but not limited to, program verification, power systems, econometrics, and operations research. The ADD refutation system is both sound and complete . Additionally, it is the simplest and most natural refutation system for refuting the feasibility of a system of linear constraints. The copy complexity of an infeasible linear constraint system (not necessarily Horn) in a refutation system, is the minimum number of times each constraint needs to be replicated, in order to obtain a read-once refutation. We show that for an HCS with n variables and m constraints, the copy complexity is at most 2 n - 1 , in the ADD refutation system. Additionally, we analyze bounded-width HCSs from the perspective of copy complexity. Finally, we provide an empirical analysis of an integer programming formulation of the copy complexity problem in HCSs. (An extended abstract was published in FroCos 2021 [ 26 ].)
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-023-10156-6