Analyzing Read-Once Cutting Plane Proofs in Horn Systems

Abstract

In this paper, we investigate variants of cutting plane proof systems for a class of integer programs called Horn constraint systems (HCS). Briefly, a system of linear inequalities $\mathbf{A \cdot x \ge b}$ is called a Horn constraint system, if each entry in $\mathbf{A}$ belongs to the set $\{0,1,-1\}$ and furthermore there is at most one positive entry per row. Our focus is on deriving refutations, i.e., proofs of unsatisfiability of such programs in variants of the cutting plane proof system. Horn systems generalize Horn formulas, i.e., CNF formulas with at most one positive literal per clause. A Horn system which results from rewriting a Horn clausal formula is called a Horn clausal constraint system (HClCS). The cutting plane calculus (CP) is a well-known calculus for deciding the unsatisfiability of propositional CNF formulas and integer programs. Usually, CP consists of the addition rule (ADD) and the division rule (DIV). We show that the cutting plane calculus with the addition rule only (CP-ADD) does not require constraints of the form $0 \le x_i \le 1$. We also investigate the existence of read-once refutations in Horn clausal constraint systems in the cutting plane proof system. We show that read-once refutations are incomplete. We show that the problem of finding a read-once refutation using only the ADD rule of an HCS is NP-hard even when the right-hand sides of the HCS belong to the set $\{0,1\}$. Additionally, we show that the problem of finding a read-once refutation using only the ADD rule of an HClCS is NP-hard. We then show that these problems remain hard when we can use both the ADD and DIV rules. We then show that the problem of finding a shortest read-once refutation of an HCS whose right-hand sides belong to the set $\{0,1\}$ is NPO BP-complete when the refutation can use only the ADD rule or the refutation can use both the ADD and DIV rules. Finally, we provide a parameterized exponential time algorithm for finding a read-once refutation of a system of Horn constraints using only the ADD rule.

Approximation Complexity

Definition A.1

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.

NPO PB is the subset of NPO in which the objective function is bounded by the size of the input problem [40].

Definition A.2

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

We next define the notion of a PTAS reduction [14].

Definition A.3

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

• 1. If I is an instance of problem P, then f(I) is an instance of problem Q.

• 2. If x is a solution to f(I), with approximation error bounded by $(1+\alpha (\epsilon ))$ for some $\epsilon > 0$. Then $g(I,x,\epsilon )$ is a solution to I with approximation error at most $(1+\epsilon )$.

We now define the notion of NPO PB-hardness under PTAS reductions. Note that any reduction between NPO PB problems has to maintain a relationship between the objective functions of the two problems, which necessitates the use of PTAS reduction.

Definition A.4

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

The set of problems which are in the class NPO PB and are NPO PB-hard are called NPO PB-complete. Observe that for every NPO PB-complete problem P there exists an $\epsilon >0$, such that P cannot be approximated to within a factor of $O(|I|^\epsilon )$, where I is an instance of P, unless P $=$ NP [44]. Thus, if any NPO PB-complete problem can be approximated to within a polylogarithmic factor of the input size, 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 $\mathbf{A \cdot x \ge b}$, $\mathbf{x} \in \{0,1\}^n$, find the minimum value of $\mathbf{1 \cdot x}$ such that $\mathbf{A \cdot x \ge b}$. This specific form of Minimum 0-1 Programming is known to be NPO PB-complete [40].