In this paper, we study the problem of finding read-once refutations (ROR) of linear feasibility in a specialized class of constraint systems called UTVPI+ constraint systems (${\text{UCS}}^{+}$). The refutations in this paper are analyzed using the ADD inference rule. Recall that a Unit Two Variable Per Inequality (UTVPI) constraint is a constraint of the form $a_i\cdot x_i+a_j\cdot x_j\le b$, where $b\in \mathbb{Z}$, and $a_i,a_j\in \{0,1,-1\}$. A conjunction of such constraints is called a UTVPI constraint system (UCS). UCSs find applications in a number of domains such as abstract interpretation and scheduling. We examine a more general form of UCSs that allows for a limited number of non-UTVPI constraints to be added to a UCS. We refer to these more general UCSs as UTVPI+ constraint systems or ${\text{UCS}}^{+}\text{s}$. If a ${\text{UCS}}^{+}$ has only k non-UTVPI constraints, then we refer to it as a ${\text{UCS}}_k^{+}$. Our focus in this paper is on refutations, i.e., proofs of infeasibility in ${\text{UCS}}^{+}\text{s}$. In particular, we study read-once refutations of linear feasibility in ${\text{UCS}}^{+}\text{s}$. Although the problem of finding read-once refutations of UCSs is polynomial time solvable, the presence of non-UTVPI constraints makes the problem NP-hard. However, if the number of non-UTVPI constraints is fixed, then read-once refutations can be found in polynomial time. In fact, in this paper, we show that the ROR problem is fixed-parameter tractable (FPT) for ${\text{UCS}}_k^{+}\text{s}$, with respect to k, the number of non-UTVPI constraints in the system. We also provide a lower bound on the efficiency of a class of parameterized algorithms for this problem, based on the Strong Exponential Time Hypothesis.

在本文中，我们研究了在称为 UTVPI+ 约束系统的一类特殊约束系统中寻找线性可行性的一次性反驳（ROR）的问题。${\text{加州大学学院}}^{+}$）。本文的反驳使用ADD推理规则进行分析。回想一下，每不等式二单元变量 (UTVPI) 约束是以下形式的约束${\mathrm{一种}}_{\mathrm{一世}}\cdot X_{\mathrm{一世}}+{\mathrm{一种}}_j\cdot X_j\le 乙$， 在哪里 $乙\in \mathbb{Z}$， 和 ${\mathrm{一种}}_{\mathrm{一世}},{\mathrm{一种}}_j\in \{0,1,-1\}$. 此类约束的结合称为 UTVPI 约束系统 (UCS)。UCS 在许多领域都有应用，例如抽象解释和调度。我们研究了一种更通用的 UCS 形式，它允许将有限数量的非 UTVPI 约束添加到 UCS。我们将这些更通用的 UCS 称为 UTVPI+ 约束系统或${\text{加州大学学院}}^{+}\text{秒}$. 如果一个${\text{加州大学学院}}^{+}$只有k 个非 UTVPI 约束，那么我们将其称为${\text{加州大学学院}}_{克}^{+}$. 我们在本文中的重点是反驳，即不可行的证明${\text{加州大学学院}}^{+}\text{秒}$. 特别是，我们研究了线性可行性的一次性反驳${\text{加州大学学院}}^{+}\text{秒}$. 尽管找到 UCS 的一次性反驳的问题是多项式时间可解的，但非 UTVPI 约束的存在使问题变得NP-hard。但是，如果非 UTVPI 约束的数量是固定的，则可以在多项式时间内找到 read-once 反驳。事实上，在本文中，我们证明了 ROR 问题是固定参数易处理的 ( FPT )${\text{加州大学学院}}_{克}^{+}\text{秒}$，关于k，系统中非 UTVPI 约束的数量。基于强指数时间假设，我们还针对该问题提供了一类参数化算法的效率下限。