NP-Completeness (36.4-5/34.4-5)

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NP-Completeness (36.4-5/34.4-5)

• P yes and no in pt
• NP yes in pt
• NPH ? NPC

NP-hard
NPC
P
NP
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Satisfiability

• Boolean formulas ?x, (x ? y) (x ? y) (x?y) (x?y)
• Satisfiability Problem (SP)
• given a Boolean formula
• is there any 0-1 input (0-1 assignment to
  variables) s.t. formula is true (1)?
• Cooks Theorem SP is NP-complete.
• SP is an NP-problem (why?)
• SP is NP-hard (w/o proof)

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3-CNF

• Conjunctive normal form (CNF)
• (l11 ? l12 ?... ?l1s1) ?... ? (ln1 ? ln2 ?...
  ?lnsn)
• each literal l is either variable or negation
• 3-CNF each si3
• 3-CNF Satisfiability is NPC
• (?x1 ?x2) ? (x1 ? ? x4 )
• Corresponding tree
• y3?(y1 ? (?x1 ? x2))
• ?(y2 ? ( x1 ? ? x4 ))
• Truth assignment for each clause using tables.

y3?
y2?
y1?
? x1
x2
x1
? x4
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Independent Set

• Independent set in a graph G
• pairwise nonadjacent vertices
• Max Independent Set is NPC
• Is there independent set of size k?
• Construct a graph G
• literal -gt vertex
• two vertices are adjacent iff
• they are in the same clause
• they are negations of each other
• 3-CNF with k clauses is satisfiable iff G has
  independent k-set
• assign 1s to literals-vertices of independent
  set
• Example f (xzy) (xza) (axy)

x, z, y independent
F is satisfiable f 1 if x z y 1
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MAX Clique

• Max Clique (MC)
• Find the maximum number of pairwise adjacent
  vertices
• MC is in NP
• for the answer yes there is certificate of
  polynomial length clique which can be checked
  in polynomial time
• MC is in NPC
• Polynomial time reduction from MIS
• For any graph G any independent set in G 1-1
  corresponds to clique in the complement graph G

red clique
noedge ? edge edge ? noedge
G
Complement G
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Minimum Vertex Cover

• Vertex Cover
• the set of vertices which has at least one
  endpoint in each edge
• Minimum Vertex Cover (MVC)
• the set of vertices which has at least one
  endpoint in each edge
• MIN Vertex Cover is NPC
• If C is vertex cover, then V - C is an
  independent set

red independent set
blue vertex cover
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Set Cover

• Given a set X and a family F of subsets of X, F
  ? 2X, s.t. X covered by F
• Find subfamily G of F such that G covers X and
  G is minimize
• Set Cover is NPC
• reduction from Vertex Covert
• Graph representation

edge between set A? F and element x ? X means x
? A
d
a
b
c
red elements of ground set X
blue subsets in family F
A
B
A a,b,c, B c,d
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Intermediate Classes
Dense Set Cover is NP but not in P neither in NPC
Dense Set Cover Each element of X belongs to at
least half of all sets in F
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Runtime Complexity Classes

• Runtime order
• constant
• almost constant
• logarithmic
• sublinear
• linear
• pseudolinear
• quadratic
• polynomial
• subexponential
• exponential
• superexponential

• Example
• adding an element in a queue/stack
• inverse Ackerman function O(logloglog n)
• n times
• extracting minimum from binary heap
• n1/2
• traversing binary search tree, list
• O(n log n) sorting n numbers, closest pair, MST,
  Dijkstra shortest paths
• adding two n?n matrices
• e n (1/2)
• e n, , n!
• Ackerman function