In this section we describe another way to establish the infeasibility of constraint sys- tem. This method links the infeasibility of one constraint system to the feasibility of a second.
Typically, theorems of the alternative connect pairs of linear constraint systems and have the following form: Given two linear systems A and B, exactly one of them is feasible. System A is called the primal system and System B is called the dual system. It is not hard to see that theorems of the alternative provide certificates of infeasibility.
One famous theorem of the alternative is Farkas’ Lemma [Sch87]. Lemma 3.5.1. Let A denote an m × n matrix and let c denote an m-vector.
CHAPTER 3. REFUTATIONS 38 Then, either I : ∃x ∈ Rn A · x ≤ c (3.9) or (mutually exclusively) II : ∃y ∈ Rm+ yT· A = 0, yT· c < 0.
It is worth noting that there are several variants of Farkas’ lemma in the literature. A formal proof of the above lemma along with a geometric interpretation can be found in [Sch87]. Farkas’ lemma can be specialized to difference constraints through a constraint network representation, as described in [CLRS01]. Essentially, the variables become nodes and the constraints become directed edges in this setting. A consequence of Farkas’ Lemma is that the difference constraint system is feasible if and only if the corresponding constraint network does not have a negative cost cycle.
Farkas’ Lemma is not the only theorem of the alternative. There is Gordan’s Theorem [Man69]:
Theorem 3.5.1. Let A denote an m × n matrix. Then, either
I : ∃x ∈ Rn+ A · x = 0, x 6= 0
or (mutually exclusively)
II : ∃y ∈ Rm yT· A > 0
There is also Stiemke’s Theorem [Man69]: Theorem 3.5.2. Let A denote an m × n matrix.
CHAPTER 3. REFUTATIONS 39
Then, either
I : ∃x ∈ Rn A · x = 0, x > 0
or (mutually exclusively)
II : ∃y ∈ Rm yT· A ≥ 0, yT· A 6= 0
A graphical theorem of the alternative, on the other hand, relates infeasibility in a lin- ear system to the existence of particular paths in an appropriately constructed constraint network. Such theorems of the alternative are known to exist for selected classes of linear programs. For instance, it is well-known that a system of difference constraints is infeasible if and only if the corresponding constraint network contains a negative cost cycle [Sch87].
Example 31: Consider the DCS represented by System (3.10).
x1− x2 ≤ 3 x1− x3 ≤ 5 x2− x3 ≤ 0
x3− x1 ≤ −1 −x1 ≤ 1 x3 ≤ −7
(3.10)
CHAPTER 3. REFUTATIONS 40 x1 x2 x3 x0 3 5 0 −1 1 −7
Figure 3.4: Directed graph corresponding to DCS (3.10)
The graph in Figure 3.4 has the negative cycle x0
−7 → x3 5 → x1 1 → x0. This corresponds
to the constraints x3≤ −7, x1− x3≤ 5, and −x1≤ 1. Summing these constraints results in
the constraint 0 ≤ −1. This constraint is clearly unsatisfiable, thus the DCS is infeasible. Graphical theorems of the alternative for linear feasibility UTVPI constraints were de- scribed in [LM05, SW17b]. In this dissertation, we provide a graphical theorem of the alternative for integer feasibility in UTVPI constraints.
41
Chapter 4
Statement of Problems
In this chapter, we define the problems examined in this dissertation. These problems are concerned with satisfiability, refutability, and closure problems for the CSPs described in Chapter 2.
4.1
Satisfiability Problems
The first type of problems we look at are satisfiability problems. Recall that a constraint system S is satisfiable if there exists an assignment to the variables in S that satisfies all domains and all constraints in S.
However, it is not enough to simply say that a given constraint system is satisfiable. We also need to provide proof that it is satisfiable. This proof is known as a certificate of feasibility. In the case of a constraint system S, the certificate of feasibility is the assignment
x∗. Thus we can define the satisfiability problem as follows:
The Satisfiability problem: Given a constraint system S, find an assignment x∗ such
that x∗satisfies S.
We can restrict the satisfiability problem by looking for assignments that have certain properties or that satisfy the constraints in a certain way. In particular we are interested in
CHAPTER 4. STATEMENT OF PROBLEMS 42
assignments which NAE-satisfy the constraint system.
4.1.1
CSPs with side constraints
We now look at CSPs where there are restrictions on solutions that are not expressed within the set of constraints C. Such constraints are known as side constraints and CSPs with these additional constraints are known as CSPs with Side Constraints (CSPSCs).
In some cases side constraints can be expressed using the language of the original CSP. In such cases, the CSPSC is equivalent to a CSP where the side constraints are incorporated into the set of regular constraints.
Once such CSPSC is a constrained form of satisfiability of Boolean Constraint Systems known as Not-All-Equal satisfiability (NAE-satisfiability). In NAE-satisfiability, the side constraint is that no clause can have all of its literals assigned to the same value. This means that each clause φ has at least one literal set to true and at least one literal set to false.
This is equivalent to requiring that the negation of at least one literal in φ is true. This
requirement can be incorporated by adding a new clause φ0 consisting of the negations of
each of the literals in φ . Doing this for every clause in a CNF formula Φ generates a new formula that is satisfiable if and only if the original formula is NAE-satisfiable.
There are also forms of CSPSCs where the side constraints cannot be expressed using the language of the original CSP. Consider the case of a linear program L in which no two variables can be assigned the same value. This can be accomplished by adding the side
constraint xi6= xjfor each pair of variables in L. However theses side constraints cannot be
expressed using the language of linear programming. Thus, there is not necessarily an LP
L0equivalent to the CSPSC L.
In terms of CSPSCs, this dissertation focuses on the NAE-satisfiability problem which we now formally define.
Definition 4.1.1. Given a CSP S, an assignment x∗NAE-satisfies S if
CHAPTER 4. STATEMENT OF PROBLEMS 43
2. For every constraint Cj, there exist literals liand lkin Cjsuch that, under assignment
x∗, li6= lk.
Example 32: Consider the following Boolean formula (x1∨ x2) ∧ (¬x2∨ x3). The
assignment x1, x2, x3= true satisfies the formula. However, this assignment does not NAE-
satisfy the formula since both clauses have all of their literals assigned true.
The assignment x1= true, x2 = false, x3 = true does NAE-satisfy the formula since
each clause has one literal set to true and one literal set to false.
Just like for regular satisfiability, the assignment to the variables serves as a certificate of NAE-feasibility.Thus we can define the NAE-satisfiability problem as follows:
The NAE-satisfiability problem: Given a constraint system S, find an assignment x∗
such that x∗NAE-satisfies S.