IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

IEOR269 notes, Prof. Hochbaum, 2010 7
3.4 Disjunctive constraints
Disjunctive (or) constraints on continuous variables can be written as conjunctive (and) constraints
involving integer variables. Given u 2 > u 1 , the constraints:
{
x ≤ u 1 or
x ≥ u 2
can be rewritten as: {
x ≤ u 1 + M y and
x ≥ u 2 − (M + u 2 ) (1 − y)
where M is a ‘large number’ and y is defined by:
{
1 if x ≥ u 2
y =
0 otherwise
(6)
If y = 1, the first line in (6) is automatically satisfied and the second line requires that x ≥ u 2 .
If y = 0, the second line in (6) is automatically satisfied and the first line requires that x ≤ u 1 .
Reciprocally if x ≤ u 1 the first line is satisfied ∀y ∈ {0, 1}, and the second line is satisfied for y = 0.
If x ≥ u 2 , the second line is satisfied ∀y ∈ {0, 1} and the first line is satisfied for y = 1.
4 Some famous combinatorial problems
4.1 Max clique problem
Definition 4.1. Given a graph G(V, E) where V denotes the set of vertices and E denotes the set
of edges, a clique is a subset S of vertices such that ∀i, j ∈ S, (i, j) ∈ E.
Given a graph G(V, E), the max clique problem consists in finding the clique of maximal size.
4.2 SAT (satisfiability)
Definition 4.2. A disjunctive clause c i on a set of binary variables {x 1 , . . . , x n } is an expression
of the form c i = y 1 i ∨ . . . ∨ yk i
i
where y j i ∈ {x 1, . . . , x n , ¬x 1 . . . , ¬x n }.
Given a set of binary variables {x 1 , . . . , x n } and a set of clauses {c 1 , . . . , c p }, SAT consists of finding
an assignment of the variables {x 1 , . . . , x n } such that c i is true, ∀i ∈ {1, . . . , p}.
If the length of the clauses is bounded by an integer n, the problem is called n-SAT, and has
been proven to be NP-hard in general. Famous instance are 2-SAT and 3-SAT. The different level of
difficulty between these two problems illustrates general behavior of combinatorial problems when
switching from 2 to 3 dimensions (more later). 3-SAT is a NP-hard problem, but algorithms exist
to solve 2-SAT in polynomial time.
4.3 Vertex cover problem
Definition 4.3. Given a graph G(V, E), S is a vertex cover if ∀(i, j) ∈ E, i ∈ S or j ∈ S.
Given a graph G(V, E), the vertex cover problem consists of finding the vertex cover of G of
minimal size.