This paper is concerned with the computational complexities of three types of queries, namely, satisfiability, equivalence, and hull inclusion. The first two queries are analyzed over the domain of CNF formulas, while hull inclusion queries are analyzed over continuous and discrete sets defined by rational polyhedra. Although CNF formulas can be represented by polyhedra over discrete sets, we analyze them separately on account of their distinct structure. In particular, we consider the NAESAT and XSAT versions of satisfiability over HornCNF, 2CNF, and Horn2CNF formulas. These restricted families find applications in a number of practical domains. From the hull inclusion perspective, we are primarily concerned with the question of checking whether two succinct descriptions of a set of points are equivalent. In particular, we analyze the complexities of integer hull inclusion over 2SAT and Horn polyhedra. Hull inclusion problems are important from the perspective of deriving minimal descriptions of point sets. One of the surprising consequences of our work is the stark difference in complexities between equivalence problems in the clausal and polyhedral domains for the same polyhedral structure.

1. Introduction

The problem of testing the satisfiability of CNF formulas (or SAT) is ubiquitous in computer science and operations research. Applications of this problem abound from areas as diverse as econometrics and planning to graph theory and combinatorial optimization [1]. From the perspective of computational complexity, SAT was immortalized in [2] as the first natural NP-complete problem. Advances in SAT research have been along both theoretical and practical lines. On the theoretical side, there exist a number of algorithms running in time 𝑜(2𝑛), for instance, see [35]. On the practical front, greedy approaches based on random walks have been enormously successful [6, 7].

One of the approaches taken by SAT theoreticians is to identify structures of SAT families that are amenable to solution through polynomial time procedures. This approach has the benefit of defining yardsticks that categorize classes of SAT problems as polynomial time solvable or NP-complete [8]. Tractable families are distinguished by a specific clausal structure; these structures either limit the number of literals per clause [9, 10] or the number of times that a variable appears across all the clauses [11]. A completely orthogonal approach to SAT research is to study the complexities of SAT variants, that is, satisfiability problems with additional requirements on the type of solutions. For instance, in the NAESAT problem, we are required to find an assignment that satisfies all the clauses but falsifies at least one literal per clause. This paper is interested in such syntactic variants of the SAT problem, inasmuch as these variants arise naturally in the domains of graph theory and scheduling.

This paper also focuses on hull inclusion problems over continuous and discrete sets of points. Hull inclusion is checkable in polynomial time when the domain is continuous and NP-complete when the domain is discrete. However, there exist nontrivial cases of hull inclusion in discrete domains which are polynomial time solvable on account of the structure of the constraint matrix. These hull inclusion problems find applications in program verification [12, 13]. Although SAT problems can be cast as discrete domain polyhedral problems, we choose to treat them separately in order to exploit their structure. In this context, we will show that a particular problem is polynomial time solvable over Boolean CNF formulas, but provably hard over arbitrary polyhedra.

The principal contributions of this paper are as follows.

(a)Establishing the complexities of NAESAT and XSAT queries over restricted CNF families (see Section 2 for definitions of CNF restrictions).(b)Introducing the problems of NAE-equivalence and X-equivalence and establishing their complexities over restricted CNF families.(c)Developing a polynomial time algorithm for Linear hull inclusion. (d)Establishing the complexities of Integer hull inclusion over various polyhedral families.

The rest of this paper is organized as follows. Section 2 formally describes each of the problems considered in this paper. In Section 3, we discuss the motivation for our work as well as related approaches in the literature. The complexities of satisfiability queries over various clausal families are detailed in Section 4. Boolean equivalence queries and their variants are discussed in Section 5. An algorithm for Linear hull inclusion is discussed in Section 6. Section 7 is concerned with Integer hull inclusion over various polyhedral families. We conclude in Section 8 by summarizing our work in this paper and identifying avenues for future research.

2. Statement of Problems

We begin with definitions of satisfiability-related problems on clausal Boolean formulas.

Let 𝜙=𝐶1𝐶2𝐶𝑚 denote a Boolean formula in conjunctive normal form (CNF), where the 𝐶𝑖s are disjunctions on the literals {𝑥1,𝑥1,𝑥2,𝑥2,,𝑥𝑛,𝑥𝑛}. For the rest of the paper, we assume that our formulas are in CNF over the 𝑛 variables {𝑥1,𝑥2,𝑥𝑛}, unless otherwise stated.

Definition 2.1. The Boolean satisfiability  (SAT) problem is: Given 𝜙, is there a {true,false} assignment to the variables of 𝜙, such that at least one literal in each clause is set to true.

If such an assignment x exists for a given CNF formula 𝜙, then 𝜙 is said to be satisfiable and x is said to be a satisfying assignment. There are other variants of satisfiability of interest in the CNF case, where the number of true literals or the mix of true and false literals per clause is specified.

Definition 2.2. The Not-All-Equal satisfiability (NAESAT) problem is defined as follows: Given a Boolean formula 𝜙, does 𝜙 have a satisfying assignment, such that at least one literal in each clause is set to false. If such an assignment exists, then it is called a NAE-satisfying assignment and 𝜙 is said to be NAE-satisfiable.

Definition 2.3. The Exact satisfiability (XSAT) problem is defined as follows. Given a Boolean formula 𝜙, does 𝜙 have a satisfying assignment, such that exactly one literal in each clause is set to true. If such an assignment exists, then it is called an X-satisfying assignment and 𝜙 is said to be X-satisfiable.

If a Boolean formula 𝜙 is unsatisfiable, then it is neither NAE-satisfiable nor X-satisfiable; however, satisfiability does not imply NAE-satisfiability or X-satisfiability.

This paper will focus on Boolean formulas with restrictions on the structure of the formula. These restrictions are exploited in the design of polynomial time algorithms.

Definition 2.4. A Boolean formula 𝜙 is in 𝑘-CNF form, if every clause contains exactly 𝑘 literals.

Definition 2.5. A Boolean formula 𝜙 is Horn, if every clause contains at most one positive literal.

Definition 2.6. A Boolean formula 𝜙 is said to be Horn2CNF, if each clause contains at most two literals or at most one positive literal.

Given an assignment 𝑎1{true,false}𝑛 to the variables of a boolean formula 𝜙, let 𝑇(𝑎1) denote the set of variables that have been assigned true under 𝑎1.

Definition 2.7. A Boolean formula 𝜙 is said to be positively monotone, if for every pair of assignments 𝑎1,𝑎2{true,false}𝑛 to the variables of 𝜙, (𝑇(𝑎1)𝑇(𝑎2))(𝜙(𝑎1)𝜙(𝑎2)).

Definition 2.8. A Boolean formula 𝜙 is said to be negatively monotone, if for every pair of assignments 𝑎1,𝑎2{true,false}𝑛 to the variables of 𝜙, (𝑇(𝑎1)𝑇(𝑎2))(𝜙(𝑎2)𝜙(𝑎1)).

A Boolean formula 𝜙 is monotone if it is either positively monotone or negatively monotone. It is not hard to see that if 𝜙 is a CNF formula, then 𝜙 is monotone if all variables occur positively or all variables occur negatively.

It is well known that 3SAT, NAE3SAT, and X3SAT are NP-complete [11], whereas HornSAT and 2SAT are solvable in polynomial time [10]. Horn2CNF formulas have also been called Mixed Horn Formulas in the literature [14] and satisfiability checking in these formulas is also NP-complete [10]. Monotone SAT is trivial, since all variables can be set to true (or false). Other monotone problems, however, prove more interesting.

From the perspective of computational complexity, the following questions are open:

(i)What are the complexities of NAESAT over Horn and Mixed Horn formulas? These problems are called NAEHornSAT and NAEMixedHornSAT, respectively.(ii)What are complexities of XSAT over Horn and Mixed Horn formulas? These problems are called XHornSAT and XMixedHornSAT, respectively.

A problem closely related to satisfiability is the boolean equivalence problem.

Definition 2.9. Two Boolean formulas 𝜙1 and 𝜙2 are said to be equivalent (denoted 𝜙1𝜙2) if every assignment that satisfies 𝜙1 also satisfies 𝜙2 and vice versa.

The problem of checking whether two Boolean formulas are equivalent is denoted by BEQ.

Observe that the equivalence problem can be broken into two subproblems, namely, 𝜙1𝜙2 and 𝜙2𝜙1.

Consider the subproblem 𝜙1𝜙2. As before, we assume that 𝜙1=𝐶1𝐶2𝐶𝑚 and that 𝜙2=𝐶1𝐶2𝐶𝑚. Now, 𝜙1𝜙2𝜙1𝐶1𝐶2𝐶𝑚𝑚𝑖=1𝜙1𝐶𝑗.(2.1)

System (2.1) exploits the well-known propositional tautology, for arbitrary propositional formulas 𝐴, 𝐵, and 𝐶: ((𝐴𝐵)(𝐴𝐶))(𝐴(𝐵𝐶)).(2.2) Pick a particular clause 𝐶𝑖𝜙2. Applying the tautology (𝐴𝐵)𝐴𝐵false(2.3) for arbitrary propositional formulas 𝐴 and 𝐵, it follows that [𝜙1𝐶𝑖] if and only if [𝜙1𝐶𝑖] is unsatisfiable.

Observe that 𝐶𝑖 is a disjunction of literals, so that 𝐶𝑖 is a conjunction of unit literal clauses; hence 𝜙1𝐶𝑖 is in CNF form. Thus, in order to check whether 𝜙1𝜙2, we merely need to confirm that all the CNF formulas in the set {𝜙1𝐶𝑖,𝜙1𝐶2,,𝜙1𝐶𝑚} are unsatisfiable. In other words, the implication problem for CNF problems has been Turing reduced to the CNF unsatisfiability problem. Then, we have the following lemma.

Lemma 2.10. BEQ is Turing reducible to the problem of checking whether a CNF formula is unsatisfiable.

Proof. Given 𝜙1 and 𝜙2, first check whether 𝜙1𝜙2 and then whether 𝜙2𝜙1.

Observe that there exist polynomial time algorithms to decide (un)satisfiability in 2CNF and HornCNF formulas [10], and hence it follows that BEQ can be decided in polynomial time for 2CNF and HornCNF formulas.

Two problems that are closely related to the boolean equivalence problem are the NAE-equivalence problem and the X-equivalence problem.

Definition 2.11. Two Boolean formulas 𝜙1 and 𝜙2 are said to be NAE-equivalent (denoted 𝜙1𝑛𝜙2) if every assignment that NAE-satisfies 𝜙1 also NAE-satisfies 𝜙2 and vice versa.

The problem of checking whether two boolean formulas are NAE-equivalent is denoted by NAEEQ.

Definition 2.12. Two Boolean formulas 𝜙1 and 𝜙2 are said to be X-equivalent (denoted by 𝑥) if every assignment that X-satisfies 𝜙1 also X-satisfies 𝜙2 and vice versa.

The problem of checking whether two boolean formulas are X-equivalent is denoted by XEQ.

Problem 1. Are there classes of Boolean formulas for which NAE-equivalence and X-equivalence can be determined in polynomial time?

We now proceed to describe the linear and integer hull inclusion problems for particular types of polytopes. A polytope is a bounded polyhedron. A polyhedron is defined by a linear system Axb, where A is an 𝑚×𝑛 integer matrix, b is an integral 𝑚-vector, and x=[𝑥1,𝑥2,,𝑥𝑛]𝑇 is the variable vector. Polytopes and polyhedra are interesting because they capture a number of problems that arise in combinatorial optimization.

For the following definitions, assume that we are given two polytopes 𝑃1{A1bx1} and 𝑃2{A2bx2}.

Definition 2.13. The linear hull of a polytope 𝑃1{A1bx1}, denoted by 𝐿𝑃1, is defined as the convex hull of all the points contained in it.

Definition 2.14 (see [15]). The integer hull of a polytope 𝑃1{A1bx1}, denoted by 𝑆𝑃1, is defined as the convex hull of the lattice points contained in 𝑃1.

Note that for a given polytope A1bx, both the linear hull and the integer hull have exponentially many extreme points. However, just as we are interested in equivalence between formulas with respect to various satisfiability measures, we are interested in comparing polytopes via their convex hulls.

Definition 2.15. The linear hull inclusion (LHI) problem is defined as follows. Given polyhedra 𝑃1and 𝑃2,  is it the case that  𝐿𝑃1𝐿𝑃2?

Definition 2.16. The integer hull inclusion (IHI) problem is defined as follows. Given polyhedra 𝑃1and𝑃2,  is it the case that  𝑆𝑃1𝑆𝑃2?

We first observe that the IHI problem is hard in the general case.

Lemma 2.17. IHI is coNP-complete.

Proof. Let 𝜙 denote an arbitrary 3CNF formula. It is well known that this formula can be represented as an integer program over {0,1}, for instance, see [16]. Set 𝑃1 to be the polyhedral system representing the 3CNF formula and 𝑃2 to the empty polyhedron {𝑥11,𝑥10}. The integer hull of 𝑃1 is contained in the integer hull of 𝑃2 if and only if the 3CNF formula 𝜙 is unsatisfiable. Then the lemma follows.

The natural question then is the following.

Problem 2. Are there classes of polytopes for which IHI can be decided in polynomial time?

We now define some special classes of polytopes. The structure of these polytopes will be exploited to design polynomial time algorithms for one or both types of hull inclusion.

Definition 2.18. A polyhedral system 𝑃1{Axb} is said to be a 2SAT polytope if all entries of A belong to the set {0,1,1} and further, there are at most 2 nonzero entries per row of A.

Individual constraints of a 2SAT polytope have also been referred to as UTVPI constraints in the literature [13].

Definition 2.19. A polyhedral system of the form 𝑃1{Axb} is said to be a Horn polytope if all entries in A belong to {1,0,1}, and there exists at most one positive entry in each row.

Observe that 2SAT polytopes and Horn polytopes generalize 2CNF clauses and HornCNF clauses, respectively.

Definition 2.20. A matrix A is said to be totally unimodular (TUM) if every square submatrix of A has determinant 0, 1, or 1.

TUM matrices arise in network flow problems [17], scheduling problems [18], and a whole host of situations in which only strict difference constraints are permitted between program variables [19]. One of the more celebrated results about TUM matrices in the operations research literature is the following. Let P1bAx denote a polyhedral system, with A TUM and b integral. Then, the integer hull and linear hull of P1 are identical (see [15]).

Definition 2.21. A polyhedral system of the form 𝑃1{Axb}, where all entries in A belong to {1,0,1} and each row has exactly one +1 and one 1, is called a Difference polytope.

Note that the class of difference polytopes is a proper subset of the class of 2SAT polytopes and also a proper subset of the class of Horn polytopes. (See Figure 1.)

Difference polytopes are a special subset of the class of TUM polytopes. Conjunctions of difference constraints are used to capture requirements in a number of application domains such as symbolic model checking [20], verification of timed systems [21, 22], and timed automata [23, 24]. Difference constraint feasibility has also been studied as the Single Source Shortest Paths problem within the operations research and algorithms communities [25]. Additionally, separation relationships in a number of scheduling problems are captured through difference constraints [18, 26, 27]. In real-time software, temporal requirements are modeled through variants of difference constraints [28, 29].

The two variants of SAT that we are interested in are NAESAT and XSAT. Both these problems have a long and interesting history, inasmuch as they are closely linked to graph coloring problems [30]. Mixed Horn Formulas (MHFs) have been used in the formulation of level-planarity tests in planar graphs and crossing-minimization problems [31]. In [14], it has been argued that graph colorability can also be formulated as an MHF satisfiability problem.

Convex body inclusion has been well studied in the operations research literature [32, 33]. Convex body inclusion is related to the problem of estimating the volumes of convex bodies [34, 35]. In Section 6, we present an algorithm for polytope inclusion that depends on the fact that maximizing a linear function over a polytope can be accomplished in polynomial time. Integer hull inclusion is different from convex body inclusion since in general, the set of lattice points satisfying a system of linear inequalities do not form a convex set. 2SAT polytopes were introduced in [36], where it was shown that the existence of a lattice point could be determined in 𝑂(𝑛3) time, where 𝑛 is the dimension of the polytope. It is known that the problem of obtaining the integer maximum of a linear function over a 2SAT polytope is, in general, NP-hard. (This problem generalizes the vertex cover problem.) Horn polytopes generalize Horn clausal systems [37] and find wide application in linear complementarity and econometrics research [38].

A secondary motivation for the study of clausal equivalence problems is provided by classical computational complexity. One of the goals in complexity is to find the exact threshold at which problems become hard [10]. This paper shows that the NAE-equivalence and X-equivalence problems are solvable in polynomial time for 2CNF formulas but are provably hard for the other boolean families. Likewise, we show that the integer hull inclusion problem can be solved in polynomial time for 2SAT polytopes. A surprising consequence of our work is the apparent disparity between the clausal and integer programming versions of Horn equivalence. Clausal Horn equivalence can be determined in polynomial time, whereas integer hull inclusion over Horn polytopes is coNP-complete.

4. Boolean Satisfiability Queries

This section focuses on the computational complexity of the satisfiability queries detailed in Section 2. It has been shown that the NAE2SAT problem can be solved in polynomial time [39]. Indeed, the results in [40] establish that NAE2SAT is in the complexity class L (Deterministic Logarithmic Space). Likewise, NAE3SAT has been proven NP-complete in [8]. The arguments in [8] also establish that Monotone XSAT is NP-complete.

We are now ready to prove the following theorem.

Theorem 4.1. XHornSAT is NP-complete.

The proof is a reduction from Monotone XSAT to XHornSAT. Before we construct the reduction, we will prove a minor but useful lemma.

Lemma 4.2. Any X-satisfying assignment for the formula (𝑧1𝑧2)(𝑧1𝑧2𝑧3) sets 𝑧3 to false.

Proof. We note that any X-satisfying assignment for (𝑧1𝑧2) sets one of 𝑧1 and 𝑧2 to true and the other to false. Therefore, a consistent X-satisfying assignment to (𝑧1𝑧2𝑧3) must set 𝑧3 to false.

Note that the formula in Lemma 4.2 is Horn.

Proof of Theorem 4.1. Let 𝜑=𝐶1𝐶2𝐶𝑘 be a monotone CNF formula. Let 𝑧1, 𝑧2, and 𝑧3 be variables that do not occur in 𝜑, and assume without loss of generality that all variables in 𝜑 appear negated.
Let 𝑓(𝜑)=𝑧1𝑧2𝑧1𝑧2𝑧3𝑖𝑘¬𝐶𝑖𝑧3.(4.1)
Observe that if 𝐶𝑖=(𝑥1,𝑥2,𝑥3), then [(¬𝐶𝑖)𝑧3] has the form: [(𝑥1𝑥2𝑥3)𝑧3]. As discussed in Lemma 4.2, any X-satisfying assignment to 𝑓(𝜑) must set 𝑧3 to false. However, any X-satisfying assignment for a Horn clause having the form [(𝑥𝑖𝑥𝑗𝑥𝑘)false] must set exactly one literal of (𝑥𝑖𝑥𝑗𝑥𝑘) to true. It follows that 𝑓(𝜑) is X-satisfiable if and only if 𝜑 is. Since 𝑓(𝜑) can be computed in time linear in |𝜑|, this reduction is polynomial-time computable.
It follows that XHornSAT is NP-complete.

There is a similar reduction from monotone NAESAT to NAEHornSAT. The reduction itself is completely straightforward since a monotone clause 𝐶 is equivalent to (¬𝐶false). (Again, without loss of generality we assume that all variables in 𝐶 are negated.)

Corollary 4.3. Monotone NAESAT 𝑃𝑚 NAEHornSAT, where the 𝑃𝑚 stands for many-to-one polynomial time.

In order to show the complexity of these two satisfiability problems, we introduce a coloring problem.

Definition 4.4. The Set Splitting problem is: given a universe 𝑈={𝑥1,,𝑥𝑛} and 𝑆𝒫(𝑈) (subsets of 𝑈), is there a coloring of {𝑥1,,𝑥𝑛} by two colors, say red and blue, so that no set in 𝑆 is monochrome?

Sipser gives the NP completeness of Set Splitting as an exercise [41]. There is a straightforward mapping between the sets in 𝑆 and clauses in a monotone formula and between the colors and the values true and false.

Lemma 4.5. The Splitting Set problem is NP-complete, and Monotone NAESAT is equivalent to it.

Corollary 4.6. NAEHornSAT is NP-complete.

Note that any NAESAT assignment to a Horn clause must set at least one body variable to false. Otherwise, the NAE constraint would force the body to true and the head to false, and the assignment would not satisfy the clause.

Observe that NAEHornSAT and XHornSAT are both NP-complete as shown in Theorem 4.1 and Corollary 4.6, respectively. Furthermore, HornSAT is a subclass of MixedHornSAT. Finally, note that the reductions that established the NP-completeness of NAEHornSAT and XHornSAT have the properties that they map inputs either to NAE-satisfiable (resp, X-satisfiable) Horn formulas, or to NAE-unsatisfiable (resp, X-unsatisfiable) Horn formulas. Therefore, these very reductions establish the NP-completeness of NAEMixedHornSAT and XMixedHornSAT, respectively.

Corollary 4.7. NAEMixedHornSAT and XMixedHornSAT are NP-complete.

Table 1 summarizes our discussion on clausal satisfiability queries.

5. Boolean Equivalence Queries

The reasoning in Section 2 can be used to establish that BEQ is coNP-complete over 3CNF and Mixed Horn Formulas.

Theorem 5.1. BEQ is coNP-complete for 3CNF and Mixed Horn Formulas.

Proof. Let 𝜙1 denote an arbitrary 3CNF formula or Mixed Horn formula. We know that the problem of checking whether 𝜙1 is satisfiable is NP-complete; thus, if we set 𝜙2 to false, we can determine the satisfiability of 𝜙1 by checking whether 𝜙1𝜙2.

In similar fashion, it can be shown that the NAEEQ and XEQ problems are coNP-complete for 3CNF and Mixed Horn Formulas.

We now prove a general theorem relating the NAEEQ and BEQ problems. Let 𝜙 denote a formula in 𝑘-CNF form, and let 𝜙 denote the CNF formula obtained by negating every literal in every clause of 𝜙.

Lemma 5.2. 𝜙 is NAE-satisfiable if and only if 𝜙𝜙 is satisfiable.

Proof. Only if: Let 𝜙 be NAE-satisfiable and let x denote a NAE-satisfying assignment. Let 𝐶1 denote a clause of 𝜙. Since x is a NAE-satisfying assignment, it sets at least one literal to true and at least one literal to false in 𝐶1. Let 𝐶1 denote the clause in which each literal in 𝐶1 is complemented. Note that the literal that was set to false in 𝐶1 is set to true in 𝐶1 and vice versa; since 𝐶1 was chosen arbitrarily, the same argument applies to all clauses of 𝜙. In other words, 𝐱 NAE-satisfies 𝜙, and therefore, x NAE-satisfies 𝜙𝜙. Hence 𝜙𝜙 is satisfiable.
If: Let 𝜙𝜙 be satisfiable in the ordinary sense, and let x denote a satisfying assignment. Let us study the clauses 𝐶1 and 𝐶1 under the assignment x which is obtained by complementing every assignment in x. As per the definition of 𝜙, every literal that is set to true in 𝐶𝑖 is set to false in 𝐶1 and vice versa. It therefore follows that x also satisfies both 𝐶1 and 𝐶1. Inasmuch as 𝐶1 and 𝐶1 were chosen arbitrarily, the same argument holds for all clause pairs, that is, x satisfies 𝜙𝜙. Now, 𝜙𝜙 has a complementary pair of assignments and therefore is NAE-satisfiable. But this immediately implies the NAE-satisfiability of 𝜙.

Lemma 5.2 leads us directly to the following lemma.

Lemma 5.3. (𝜙𝑛𝜓) if and only if 𝜙(𝜙𝜓)(𝜓).

Proof. Suppose that (𝜙𝑛𝜓) but 𝜙(𝜙𝜓)(𝜓). Without loss of generality, assume that there exists an assignment x which satisfies 𝜙𝜙 but falsifies 𝜓𝜓. Since x does not satisfy 𝜓𝜓, it cannot NAE-satisfy 𝜓, as per Lemma 5.2. As per the hypothesis, it cannot NAE-satisfy 𝜙 either. But since 𝐱 satisfies 𝜙𝜙, it must NAE-satisfy 𝜙, as per Lemma 5.2, and thus we have a contradiction. It follows that (𝜙𝑛𝜓) implies that 𝜙(𝜙𝜓)(𝜓). The converse can be proved similarly.

Note that in case of 2CNF formulas, the NAEEQ and XEQ problems are identical.

The next theorem follows from the previous proofs.

Theorem 5.4. The NAEEQ and XEQ problems can be solved in polynomial time for 2CNF formulas.

Table 2 summarizes our discussion on clausal equivalence queries.

6. Linear Hull Inclusion

In this section, we focus on the problem of checking whether the linear hulls of two polyhedra defined by systems of linear inequalities are equivalent.

Consider two polyhedra represented by Axb,x0,(6.1) where

(1)A is an 𝑚×𝑛 rational matrix,(2)b is a rational 𝑚vector,(3)xn+,

and Cxd,x0,(6.2) where

(1)C, is an 𝑚×𝑛 rational matrix,(2)d is a rational 𝑚-vector,(3)xn+

The goal is to decide the following predicate: 0dxAxbCx?(6.3) Algorithm 1 represents our strategy to decide Query (6.3).

Function Polytope-Include  ( d A , b , C , )
(1) c { L e t t h e 𝑖 t h c o n s t r a i n t o f C x d , x 0 b e r e p r e s e n t e d a s 𝐢 x d 𝐢 } .
(2) for ( 𝑖 = 1   to   𝑚 ) do
(3)if ( m a x A 𝟎 𝐱 𝐛 , 𝐱 c 𝐢 x > d 𝐢 ) then
(4)return (false)
(5)end if
(6) end for
(7) return (true)

6.1. Analysis

Let (𝑚,𝑛) denote the polynomial running time of a linear programming algorithm on 𝑚 constraints and 𝑛 variables [42]. Since a total of 𝑚 calls are made, the running time of Algorithm 1 is 𝑂(𝑚), which is polynomial, since is a polynomial function of 𝑚 and 𝑛.

6.2. Correctness

Lemma 6.1. If Algorithm 1 returns true, then for all  𝐱𝐧+𝐝,𝐀𝐱𝐛𝐂𝐱.

Proof. Let us assume the contrary, that is, Algorithm 1 returns true, yet there exists a point x such that xAb, but xCd. We note that the notation xCd is used to indicate the fact that at least one of the 𝑚 constraints defining the polyhedron dCx is violated. Let cx𝑑 denote a violated constraint, that is, cx>𝑑. (See Figure 2.) Then  maxAxb,x0cx is greater than 𝑑, contradicting the hypothesis that true was returned by the algorithm.

Lemma 6.2. If Algorithm 1 returns false, there exists a point 𝐱𝐑𝐧+ such that 𝐱𝐀𝐛 and 𝐱𝐂𝐝.

Proof. See Figure 2. Let maxAxbcx exceed 𝑑 at point x. Then x is the required offending point, that is, x0Axb,x, but x0Cxd,x.

Corollary 6.3. Algorithm 1 decides if the polyhedron defined by 𝟎𝐀𝐱𝐛,𝐱 is contained in the polyhedron represented by 𝟎𝐂𝐱𝐝,𝐱

Note that the linear hull inclusion problem is decidable in polynomial time, irrespective of the polyhedral system involved, that is, the polyhedral system could be completely arbitrary (and not necessarily one of 2SAT, Horn or TUM).

7. Integer Hull Inclusion

Let us restate the Integer hull inclusion problem (IHI).

Given polyhedra  𝑃1{A1bx1}and  𝑃2{A2bx2}, is it the case that the integer hull of   𝑃1  is contained within the integer hull of   𝑃2, that is,  is  𝑆𝑃1𝑆𝑃2?

We point out some relevant information.

(1)Algorithm 1 shows that the hull inclusion problem (Linear or Integer) is Turing reducible to the problem of finding the maximum (linear or integer) of a linear function over a polyhedron.(2)It therefore follows that the IHI problem can be solved in polynomial time when A1,A2 are TUM. Note that when the vectors b1,b2 are integral, Algorithm 1 works as is, since the linear hull is identical to the integer hull. If the bi vectors are not integral, we make the observation that the integer hull of Aibxi is identical to the linear hull of Aibx0𝑥0𝑓8𝑓0i0𝑥0𝑓8𝑓𝑏 when the Ai are TUM, and use Algorithm 1 on the modified polyhedra. For the rest of this paper, we will assume that the vectors b1,b2 are integral.(3)Optimizing a linear function over an arbitrary 2SAT polytope is NP-hard since it subsumes the vertex cover problem [15, 43], and hence we cannot directly use the technique discussed in Section 6. Likewise, maximizing an arbitrary linear function over a Horn polytope is NP-hard, and thus the technique of Section 6 is inapplicable (see Section 7.2).
7.1.  2SAT Polytopes

Assume that A1 has dimension 𝑚×𝑛 and that A2 has dimensions 𝑚×𝑛. Note that 𝑃1 is constructed by taking the intersection of the 𝑚 half-spaces a11x𝑏11,a21x𝑏21a,m1x𝑏𝑚1. Likewise, 𝑃2 is constructed by taking the intersection of the 𝑚 half-spaces, a12x𝑏12,a22x𝑏22a,m2x𝑏𝑚2.

Observe that 𝑆𝑃1𝑆𝑃2 if and only if for all lattice points x,(x𝑃1x𝑃2). Now, for a lattice point x, x𝑃1x𝑃2x𝑃1x𝑚𝑗=1aj2x𝑏𝑗2𝑚𝑗=1x𝑃1aj2x𝑏𝑗2.(7.1)

Let us focus on proving (x𝑃1aj2x𝑏𝑗2), for a specific constraint of 𝑃2, that is, the 𝑗th half-space defining 𝑃2. As in the case of CNF equivalence, we observe that for lattice points x, (x𝑃1aj2x𝑏𝑗2), if and only if the set 𝑃1[aj2x𝑏𝑗2] is empty with respect to lattice points. Note that the constraint aj2x𝑏𝑗2 can be written as: 𝑙1aj2x>𝑏𝑗2. But we are only interested in lattice point solutions. Consequently, a lattice point solution will satisfy the constraint 𝑙1 if and only if it satisfies the constraint aj2x(𝑏𝑗2+1), which is a closed half-space. Thus, we can check that (x𝑃1aj2x𝑏𝑗2) for all lattice points x, by checking whether the polyhedron 𝑃1aj2x(𝑏𝑗2+1) is empty with respect to lattice points. The operation of negating a constraint is referred to as constraint complementation.

The discussion above implies the following theorem.

Theorem 7.1. Given polyhedra 𝑃1 and 𝑃2, the IHI problem turing reduces to the problem of checking whether a polyhedron does not have any lattice point.

Corollary 7.2. The IHI problem can be solved in polynomial time for 2SAT polytopes.

Proof. 2SAT polytopes are defined by constraints that are closed under constraint complementation, insofar as lattice points are concerned. For instance, the complement of the constraint 𝑥1𝑥24 is 𝑥1+𝑥2>4, which is equivalent to 𝑥1+𝑥23 if only integral values of 𝑥1 and 𝑥2 are permitted. In other words, the complement of a constraint in a 2SAT polytope is also a 2SAT constraint. The presence or absence of lattice points in a 2SAT algorithm can be checked using the algorithm in [36], which runs in 𝑂(𝑛3) time. The corollary follows.

7.2. Horn Polytopes

Observe that Horn constraints are not closed under constraint complementation. For instance, the negation of the Horn constraint 𝑥1𝑥2𝑥35 is 𝑥1𝑥2𝑥3<5, which is equivalent to 𝑥1𝑥2𝑥34 (since the 𝑥𝑖s are integral) and hence equivalent to 𝑥1+𝑥2+𝑥34. Note that the last constraint is not a Horn constraint but an anti-Horn constraint!

We will show that the IHI problem for Horn polytopes is coNP-complete. In order to establish this result, we study a related problem, namely, PK(Horn, 1).

Given a Horn system   bAx  and a single non-Horn constraint   cx𝑑,  does the polyhedral system  bAxcx𝑑, enclose a lattice point?

We will use a reduction from the following problem.

Definition 7.3. The Hitting Set (HS) problem is: Given a set 𝑆={𝑠1,𝑠2,,𝑠𝑛} called the ground set, a collection of 𝑚 subsets 𝑇𝑖𝑆, 𝑖=1,2,,𝑚, and a number 𝐾𝑛, does there exist a set 𝑆𝑆, such that |𝑆|𝐾 and 𝑇𝑖𝑆𝜙,𝑖=1,2,𝑚?

Theorem 7.4. HS is NP-complete.

The Hitting Set problem and a proof of its NP-completeness are provided in [11].

Corollary 7.5. PK(Horn, 1) is NP-complete.

Proof. PK(Horn, 1)    NP from the NP-completeness of Integer Programming [19]. We reduce the Hitting Set problem to PK(Horn, 1).
Given an instance of HS, with ground set 𝑆={𝑠1,𝑠2,,𝑠𝑛}, 𝑚 sets 𝑇𝑖𝑆 and the target 𝐾, we construct an instance of the following variation of CNF satisfiability called MCNF. Corresponding to the ground set 𝑆, we create the literal set 𝐿={𝑥1,𝑥2,,𝑥𝑛}. Corresponding to each subset 𝑇𝑖={𝑠𝑖1,𝑠𝑖2,,𝑠𝑖𝑝}, we create the clause 𝐶𝑖={𝑥𝑖1,𝑥𝑖2,,𝑥𝑖𝑝}, where each 𝑖𝑘 indexes the set {1,2,,𝑛}. The query in MCNF is as follows. Does there exist an assignment that satisfies the clause set 𝜙=𝐶1𝐶2𝐶𝑚 such that the number of variables set to true is at most 𝐾? Note that all literals are positive in the MCNF instance.
Assume that the instance of HS is a “yes” instance, that is, there exists a set 𝑆𝑆 such that |𝑆|𝑇𝑖𝜙, such that |𝑆|𝐾. Set the literals corresponding to the variables in 𝑆 to true and all other literals to false. By construction, each clause 𝐶𝑖 is satisfied. Since the number of literals set to true is 𝐾, the instance of MCNF is also a “yes” instance.
Now assume that the instance of MCNF is a “yes” instance, that is, there exists a {true,false} assignment to the literals of 𝜙, such that all clauses are satisfied and the number of literals set to true is at most 𝐾. Construct the set 𝑆 with those elements 𝑠𝑖𝑆, such that 𝑥𝑖=true. By construction, 𝑆 intersects each 𝑇𝑖 in at least one element (since each clause is satisfied), and the cardinality of 𝑆 is at most 𝐾. It follows that the instance of 𝐻𝑆 is also a “yes” instance.
We now transform the MCNF instance into an instance of PK(Horn, 1) as follows. Create 𝑛 variables 𝑦1,𝑦2,,𝑦𝑛, where each 𝑦𝑖{0,1}. Corresponding to the clause 𝐶𝑖=(𝑥𝑖1,𝑥𝑖2,𝑥𝑖3) (say), create the Horn constraint: (1𝑦𝑖1)+(1𝑦𝑖2)+(1𝑦𝑖3)1. Finally, add the non-Horn constraint: 𝑛𝑖=1𝑦𝑖𝑛𝐾. Thus, the MCNF instance is transformed into the following PK(Horn, 1) instance: bAy𝑛𝑖=1𝑦𝑖𝑛𝐾, where A has the Horn structure and b is an integral vector.
Assume that the MCNF instance is a “yes” instance. For each literal 𝑥𝑖 set to true, set 𝑦𝑖 to 0 and for each literal 𝑥𝑗 set to false, set 𝑦𝑗 to 1. Since each clause is satisfied, it must be the case that each Horn constraint is satisfied, as per construction of the Horn constraint. Further, since the number of variables set to true in the MCNF instance is at most 𝐾, the number of 𝑦𝑖s set to 0 is at most 𝐾 and hence the number of 𝑦𝑖s set to 1 is at least 𝑛𝐾, that is, we must have 𝑛𝑖=1𝑦𝑖𝑛𝐾. Thus, the PK(Horn, 1) instance is a “yes”-instance.
Now assume that the PK(Horn, 1) instance is a “yes” instance. Set 𝑥𝑖 to true, if 𝑦𝑖=0 and 𝑥𝑖 to false, if 𝑦𝑖=1. As per the construction, each clause must be satisfied, since each Horn constraint is satisfied. Further, the number of 𝑦𝑖s set to 1 is at least 𝑛𝐾, and hence the number of 𝑥𝑖s set to true is at most 𝐾. It follows that the MCNF instance is a “yes” instance.

We get a small bonus from this result which is useful for our next theorem.

Definition 7.6. Let HMax denote the following problem. Given a Horn polytope bAx and a number 𝐾, do there exist 𝑥1,𝑥2,,𝑥𝑛{0,1} such that xAb and 𝑛𝑖=1𝑥𝑖𝐾, where x=[𝑥1,𝑥2,,𝑥𝑛]𝑇?

Corollary 7.7. HMax is NP-complete.

Proof. This follows from Corollary 7.2.

We now use the NP-completeness of HMax to prove the following.

Theorem 7.8. The IHI problem for Horn polytopes, that is, IH-Horn, is coNP-complete.

Proof. From the proof that integer programming is in NP, we know that the IHI problem is in coNP [10].
Let A1bx1,𝑥𝑖{0,1},𝑖=1,2,,𝑛, and 𝐾 describe an instance of HMax. We create the following instance of the IHI-Horn problem:𝑃1A1bx1,𝑥𝑖{0,1},𝑖=1,2,,𝑛, and 𝑃2𝑛𝑖=1𝑥𝑖(1𝐾).
We now argue that the instance of HMax is true if and only if the instance of IHI-Horn is false, that is, x{0,1}𝑛A1xb𝑛𝑖=1𝑥𝑖𝐾𝑃1̸𝑃2.(7.2)
Note that 𝑃1 is a Horn system, as per the hypothesis and 𝑃2 is a Horn system by definition.
Assume that the IHI-Horn instance is true, that is, 𝑃1𝑃2.
We then have fol all xA1xb𝑛𝑖=1𝑥𝑖(1𝐾). It follows that for all xA1xb𝑛𝑖=1𝑥𝑖(𝐾1). Hence, the instance of HMax is false.
Now assume that the HMax instance is true, that is, 𝐱{0,1}𝑛A1bx𝑛𝑖=1𝑥𝑖𝐾. This implies that there exists x𝑃1, with 𝑛𝑖=1𝑥𝑖𝐾, that is, 𝑛𝑖=1𝑥𝑖𝐾. Thus, 𝑛𝑖=1𝑥𝑖<(𝐾+1), and hence 𝑛𝑖=1𝑥𝑖<(𝐾1). It follows that x𝑃1 is not contained in 𝑃2, and hence the instance of IHI-Horn is false.
The theorem follows.

Table 3 summarizes the discussion on Integer hull inclusion problems.

8. Conclusion

In this paper, we discussed clausal equivalence and hull inclusion (both linear and integer) from the perspectives of a number of specialized constraint clauses. We also detailed the complexities of some satisfiability variants. Finally, we showed that the Integer hull inclusion problem for Horn polytopes is coNP-complete. To the best of our knowledge, our results are the first of their kind.

The work in this paper is important from the orthogonal perspectives of providing efficient strategies for special cases of hard problems and exposing interesting avenues for future research. The following interesting open problems have arisen from this work.

(i)While the satisfiability of a Horn clause system can be checked by resolution in 𝑂(𝑛2) time, to date, the only known strategy for checking the feasibility of a general Horn polytope is linear programming [37]. Finding a simpler strategy for Horn polytope feasibility is of paramount importance since Horn polytopes find wide application in engineering domains.(ii)Although the IHI-Horn problem is coNP-complete, the complexity of the problem, when the number of nonzero variables defining each constraint is a fixed constant, is unknown. A polynomial time algorithm for this problem is of enormous practical significance.


The author is extremely grateful to Judy Goldsmith for numerous discussions and to Matthew Williamson for proofreading the paper. This research has been supported in part by the Air Force Office of Scientific Research under Grant FA9550-06-0050 and in part by the National Science Foundation through Award CCF-0827397.