Unit Read-once Refutations for Systems of Difference Constraints

Abstract

In this paper, we investigate refutability in Difference Constraint Systems (DCS) under the Unit Read-Once Refutation (UROR) system. A difference constraint is a linear relationship of the form: $x_{i}-x_{j} \le b_{ij}$ and a DCS is a conjunction of such constraints. In the UROR refutation system, each constraint can be used by at most one inference. Additionally, each inference has to use at least one one-variable (absolute) constraint. Note that an unsatisfiable difference constraint system may not have a UROR. Thus, the UROR refutation system is incomplete for DCSs. The UROR refutation system is useful for proving that the infeasibility of a DCS is caused by the current variable domains. These domains are determined by the absolute constraints in the system. Thus, the UROR refutations of a DCS depend on these variable domains. This is in contrast to unrestricted refutations which do not need to depend on these domain constraints. Investigating weak (incomplete) refutation systems leads to a better understanding of the inference rules required for establishing contradictions in the given constraint system. Thus, this study is well-motivated. Likewise, difference constraint systems arise in a number of application domains such as program verification and scheduling. It follows that efficient refutation systems are of paramount interest. In this paper, we show that problem of checking if a DCS has a unit read-once refutation is NP-complete. Additionally, we provide parameterized and exact exponential algorithms for solving this problem. Finally, we show that the problem of finding the length of the shortest unit read-once refutation is NPO PB-complete.

Appendix A: Approximation Complexity Classes

We begin by defining the complexity class NPO [27].

Definition 6

The complexity class NPO is the set of optimization problems such that:

• 1. The set of instances can be recognized in polynomial time.

• 2. Solutions are polynomially sized and can be verified in polynomial time.

• 3. The objective function can be computed in polynomial time.

We next define the complexity class NPO PB [21].

Definition 7

NPO PB is the set of NPO problems for which the value of the objective function is polynomial in the size of the input.

Finally, we introduce the notion of PTAS reductions [27].

Definition 8

A PTAS reduction from problem A to problem B, is a trio of functions f, g, and $\alpha$ computable in polynomial time, such that:

• 1. f maps instances of problem A to instances of problem B.

• 2. g takes an instance x of problem A, an approximate solution to the corresponding problem f(x) in B, and an error parameter $\epsilon$ and produces an approximate solution to x.

• 3. $\alpha$ maps error parameters for solutions to instances of problem A to error parameters for solutions to problem B.

• 4. If the solution y to f(x) (an instance of problem B) is at most $(1+\alpha (\epsilon ))$ times worse than the optimal solution, then the corresponding solution $g(x,y,\epsilon )$ to x (an instance of problem A) is at most $(1+\epsilon )$ times worse than the optimal solution.

Definition 9

A problem P is NPO PB-hard under PTAS reductions, if every problem in NPO PB can be reduced to P by a PTAS reduction.

Unless otherwise stated, we assume that NPO PB-hardness is specified with respect to PTAS reductions.

The set of problems which are in the class NPO PB and are NPO PB-hard are called NPO PB-complete. Additionally, for every NPO PB-complete problem P there exists an $\epsilon >0$ such that P cannot be approximated to within a factor of $O(n^\epsilon )$ unless P $=$ NP [4]. Thus, if any NPO PB-complete problem can be approximated to within a polylogarithmic factor, then P $=$ NP.

An example of an NPO PB-complete problem is the Bounded Minimum 0-1 Programming problem. This problem is formulated as follows:

Given an integer program $\mathbf{A \cdot x \ge b}$, $\textbf{x} \in \{0,1\}^n$, find the minimum value of $\mathbf{1 \cdot x}$. This specific form of Minimum 0-1 Programming is known to be NPO PB-complete [21].