EMIS 8373: Integer Programming

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EMIS 8373 Integer
Programming

• NP-Complete Problems
• updated 21 April 2009

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The Class of NP-Complete Problems (NPC)

• A problem q is NP-complete if
• It is in NP and,
• Any other problem p in NP is polynomial reducible
  to q.
• Implications of q being NP-complete
• If q is NP-complete then q is at least as hard as
  every other problem in NP.
• If there is an efficient (polynomial) algorithm
  for q, then there is an efficient algorithm for
  every problem in NP (i.e., P NP).

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Boolean Algebra

• A Boolean variable can either be true or false.
• There are three operators in Boolean algebra
• x ? y true if x true and y true, and false
  otherwise.
• x ? y true if at least one of x and y is true.
• ?x true if x is false, and false if x is true.

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Boolean Formulae

• A Boolean formula is satisfiable if there exists
  at least one truth assignment of its variables
  such that it evaluates to true.
• Example f(x1,x2,x3) ?x3 ? (x1 ? ?x2 ? x3)
• f(true,true,false) true
• f(false,false,false) true
• f is satisfiable.
• (x1 ? ?x2 ? x3) is an example of a clause
  containing the literals x1 , ?x2 , and x3 .

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An Unsatisfiable Formula
Suppose x1 is true gt x3 must also be true to
satisfy clause 4 gt x2 must also be true to
satisfy clause 3 gt all three variables are true
gt clause 5 is false gt the formula evaluates to
false
Suppose x1 is false gt x2 must also be false to
satisfy clause 2 gt x3 must also be false to
satisfy clause 3 gt all three variables are
false gt clause 1 is false gt the formula
evaluates to false
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The Satisfiability (SAT) Problem

• Input A set of m clauses C1, C2, , Cm
  involving n Boolean variables x1, x2, , xn.
• Note that a clause contains only ? and ?
  operators.
• Question Is C1? C2 ? ? Cm satisfiable?
• Cooks Theorem (1971) SAT is NP-Complete.
• Every problem in NP is polynomial reducible to
  SAT.
• If there is a polynomial-time algorithm for SAT,
  then there is a polynomial-time algorithm for
  every problem in NP.

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Binary Integer Programming (BIP)

• Instance
• A set of n binary variables x1, x2, , xn, and
• m constraints
  ai1x1 ai2x2 ainxn ? (, or
  ?) bi
• Question Is the following problem feasible?

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BIP is NP-Complete

• Show that BIP is in NP
• The size of a BIP instance is O(mn).
• The complexity of checking a proposed solution to
  an instance of BIP is O(mn).
• Each constraint requires n multiplications, n-1
  additions, and one comparison to verify.
• Show that some other NP-complete problem can be
  formulated as BIP with a polynomial reduction.

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BIP is NP-Complete Reduction from SAT

• Let x1, x2, , xn, be the binary variables in the
  BIP instance.
• For each clause Ci in the SAT instance let ci be
  a constraint in the BIP instance
• Let Bi be the set of literals in Ci with a ?
  operator and let Ai be the other literals in Ci
• Add the following constraint to the BIP instance
• The reduction is clearly polynomial since the BIP
  has the same size as the SAT instance.

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Example SAT to BIP Reduction
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Proof yes SAT ? yes BIP
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Proof yes BIP ? yes SAT
Setting literal xi true in the SAT formula if
variable xi 1 in the BIP solution and xi
false otherwise ensures that each clause
evaluates to true since at least one Ai literal
will be true or at least one Bi literal will be
false.
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Complexity of BIP

• SAT is NP-complete (Cooks theorem)
• BIP is in NP
• BIP is at least as hard as SAT (reduction)
• BIP is NP-complete

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The Clique Problem

• Input An undirected graph G (V, E), and an
  integer k.
• Question Is there a k-clique in G?
• A k-clique is a subset U ? V of k nodes such that
  for every pair of nodes in i ? U and j ? U, the
  edge (i, j) ? E.

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Show that CLIQUE is NP-Complete

• Show that CLIQUE is in NP
• We can verify whether or not a given set of k
  nodes is a clique in O(k2) time.
• Show that some other NP-complete problem is
  polynomial reducible to CLIQUE.
• Give an efficient formulation of some other
  NP-complete problem as CLIQUE
• We will use a reduction from SAT

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Reduction from SAT to CLIQUE

• Let f C1? C2 ? ? Cm be the SAT instance where
  Ci (?i1 or ?i2 or or ?i,s(i)) and s(i) is the
  number of literals in clause i.
• Say that a pair of literals x and ?x are
  complementary.
• Let G (V, E) where
• V ?ij 1 ? i ? m and 1 ? j ? s(i)
• (?ij, ?hk) is an edge in E if i ? h, and ?ij and
  ?hk are not complementary
• Let k m

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Example Reduction of SAT to CLIQUE
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Example Reduction SAT to CLIQUE
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Example Reduction SAT to CLIQUE
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Correctness of the Reduction

• The graph will have at most mn nodes and fewer
  than (mn)(mn-1)/2 edges. Therefore the time to
  construct the graph is polynomial in the number
  of variables and clauses of the SAT instance.

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Correctness of the Reduction Feasible SAT ?
Feasible Clique

• Consider a truth assignment that makes the
  Boolean formula true.
• At least one literal in each clause must be true.
• Select one such literal from each clause.
• This gives us k literals.
• Since no pair of these k literals can be
  complementary, the corresponding set of k nodes
  in G forms a clique.
• Thus, a yes instance of SAT yields a yes
  instance of Clique.

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Correctness of the Reduction Feasible Clique
? Feasible SAT

• Suppose that G contains a k-clique.
• Consider the k literals that correspond to the
  nodes in the k-clique.
• Each literal must come from a different clause.
• No pair of the k literals can be complementary
  (otherwise there would be no edge between the
  corresponding nodes).
• Setting each of the k literals to true makes the
  formula evaluate to true.
• Thus, the given SAT instance must be satisfiable.

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STABLE SET

• Input A graph G (V, E), and an integer k.
• Question Is there a set of k vertices U ? V such
  that there are no edges between any pair of
  vertices in U?

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STABLE SET is NP-Complete

• Given a set of k nodes in V, we can verify that
  there are no edges between them in O(k2) time.
  Thus, STABLE SET is in NP.
• Reduction Let G (V, E), k be an instance of
  CLIQUE and let H (V, F) be the complement of G
  where (i, j) ? F if (i, j) ? E, and (i, j) ? F if
  (i, j) ? E.
• If v1, v2, , vk is a clique in G, then it is a
  stable set in H.
• If v1, v2, , vk is a stable set in H, then it
  is a clique in G.

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Reducing CLIQUE to STABLE SET
G
3
2
H
3
2
1
1
5
4
5
4
3-Clique 2, 3, 4
Stable Set 2, 3, 4
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Tree of Reductions (Karp 72)
SAT
CLIQUE
3-SAT
BIP
STABLE SET
TSP
3D-Matching
3-Exact Cover
0-1 Knapsack
Integer Knapsack
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Cooks theorem
SAT
CLIQUE
BIP
STABLE SET
TSP
0-1 Knapsack
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Recognition vs. Optimization TSP Example
Is there a TSP tour of length k or less?
3
3
2
4
5
1
k 35
yes
7
3
1
7
no
5
k 5
2
yes
k (35 5)/2 20
5
4
1
yes
k ? (205)/2 ?13
no
k (135)/2 9
no
k (139)/2 11
yes
k (1311)/2 12
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Optimization vs. Recognition

• An optimization problem is said to be NP-hard if
  its recognition problem is NP-complete.
• Many important optimization problems are NP-hard.
• NP-hard problems are considered theoretically
  intractable (difficult to solve).
• A problem can be NP-hard in general, but have
  special cases that are polynomial.
• Example MCNFP (easy) is a special case of ILP
  (hard).