MCS 312: NP Completeness and Approximation algorithms

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MCS 312 NP Completeness and Approximation
algorithms

• Instructor
• Neelima Gupta
• ngupta_at_cs.du.ac.in

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Table of Contents

• Hamiltonian Cycle

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Hamiltonian Cycle Problem

• A Hamiltonian cycle in a graph is a cycle
• that visits each vertex exactly once
• Problem Statement
• Given A directed graph G (V,E)
• To Find If the graph contains a Hamiltonian
  cycle

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Hamiltonian Cycle Problem

• Hamiltonian Cycle Problem is
• NP-Complete
• Hamiltonian Cycle Problem is in NP.
• Given a directed graph G(V,E), and a
• certificate containing an ordered list of
  vertices on
• a Hamiltonian Cycle. It can be verified
• in polynomial time that the list contains each
  vertex
• exactly once and that each consecutive pair in
  the
• ordering is joined by an edge.

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Hamiltonian Cycle Problem

• Hamiltonian Cycle Problem is NP Hard
• 3-SAT p Hamiltonian Cycle
• We begin with an arbitrary instance of 3-
• SAT having variables x1,.,xn and
• clauses C1,.,Ck
• We model one by one, the 2n different ways in
  which
• variables can assume assignments, and the
• constraints imposed by clauses.

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Hamiltonian Cycle Problem

• To correspond to the 2n truth assignments,
• we describe a graph containing 2n different
• Hamiltonian cycles.
• The graph is constructed as follows
• Construct n paths P1,.,Pn.
• Each Pi consists of nodes vi1,.., vib,
• where b 2k (k being the number of clauses)

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Hamiltonian Cycle Problem

• Draw edges from vij to vi,j1

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Hamiltonian Cycle Problem

• Draw edges from vi,j1 to vi,j

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Hamiltonian Cycle Problem

• For each i 1,2,..,n-1, define edges from vi1
  to vi1,1 and to vi1,b.
• Also, define edges from vib to vi1,1 and vi1,b

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• Add two extra nodes s and t. Define edges from s
  to v11 and v1b, from
• vn1and vnb to t, and from t to s

s
t
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Hamiltonian Cycle Problem

• Observations
• Any Hamiltonian cycle must use the edge (t,s)
• Each Pi can either be traversed left to right or
  right to left. This gives rise to 2n Hamiltonian
  cycles.
• We have therefore modeled the n independent
  choices of how to set each variable if Pi is
  traversed left to right, xi1, else xi0

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Hamiltonian Cycle Problem

• We now add nodes to model the clauses
• Consider the clause C (x1? x2 ? x3)
• What is the interpretation of the clause?
• The path P1 should be traversed right to left, or
  P2 should be traversed left to right, or P3 right
  to left.

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• We add a node that does this

C
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Hamiltonian Cycle Problem

• In general
• We define a node cj for each clause Cj.
• In each path Pi, positions 2j-1 and 2j are
  reserved for variables that participate in clause
  Cj
• If Cj contains xi, add edges (vi,2j-1 , cj)and
  (cj , vi,2j)
• If Cj contains xi, add edges (vi,2j , cj)and
  (cj, vi,2j-1)
• Gadget constructed !

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Hamiltonian Cycle Problem

• Consider an instance of 3-SAT having
• 4 variables x1,x2 x3,x4
• 3 clauses C1 (x1 v x2 v x3)
• C2 (x2 v x3 v x4)
• C3 (x1 v x2 v x4)

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Hamiltonian Cycle Problem

• We reduce the given instance as follows
• n 4 k 3 b 23 6
• Construct 4 paths P1, P2, P3, P4
• P1 consists of nodes v1,1, v1,2 ,.., v1,6
• P2 consists of nodes v2,1, v2,2 ,.., v2,6
• P3 consists of nodes v3,1, v3,2 ,.., v3,6
• P4 consists of nodes v4,1, v4,2 ,.., v4,6

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C1
P1
1
2
6
5
4
3
P2
1
2
6
5
4
3
P3
1
2
6
4
3
5
P4
1
2
6
5
4
3
C3
C2
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Hamiltonian Cycle Problem

• Claim 3-SAT instance is satisfiable if and only
  if
• G has a Hamiltonian cycle
• Proof Part I
• Given A satisfying assignment for the 3-SAT
  instance
• If xi 1, traverse Pi left to right, else right
  to left.
• Since each clause Cj is satisfied by the
  assignment,
• there has to be at least one path Pi that moves
  in the
• right direction to be able to cover node cj. This
  Pi can be
• spliced into the tour there via edges incident on
  vi,2j-1 and
• vi,2j

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Hamiltonian Cycle Problem

• Let us try to verify this with our example
• Given A satisfying assignment for 3-SAT, say x1
  1 x2 0 x3 1 x4 0
• Let us check out a corresponding Hamiltonian
  cycle

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C1
s
P1
1
2
6
5
4
3
P2
1
2
6
5
4
3
P3
1
2
6
4
3
5
P4
1
2
6
5
4
3
t
C3
C2
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Hamiltonian Cycle Problem

• Part II
• Given A Hamiltonian cycle in G.
• Observe that if the cycle enters a node cj on
• an edge from vi,2j-1 it must depart on an edge
• to vi,2j.
• Why?

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Hamiltonian Cycle Problem

• Because otherwise, the tour will not be able
• to cover this node while still maintaining the
• Hamiltonian property
• Similarly, if the path enters from vi,2j, it has
  to
• depart immediately to vi,2j-1.

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Hamiltonian Cycle Problem

• However, in some situations it may so happen that
  the path
• enters cj from the first (or last) node of Pi
  and departs at the
• first (or last) node of Pi1.
• In either case the following holds true
• The nodes immediately before and after any cj
• in the cycle are joined by an edge in G, say e.
• Let us consider the following Hamiltonian cycle
• given on our graph

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C1
s
P1
1
2
6
5
4
3
P2
1
2
6
5
4
3
P3
1
2
6
4
3
5
P4
1
2
6
5
4
3
t
C3
C2
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Hamiltonian Cycle Problem

• Obtain a Hamiltonian cycle on the
• subgraph G c1,ck by removing
• cj and adding e as shown below

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C1
s
P1
1
2
6
5
4
3
P2
1
2
6
5
4
3
P3
1
2
6
4
3
5
P4
1
2
6
5
4
3
t
C3
C2
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Hamiltonian Cycle Problem

• We now use this new cycle on the subgraph
• to obtain the truth assignments for the
• 3-SAT instance.
• If it traverses Pi left to right, set xi 1, else
  
• set xi 0.
• We therefore get the following assignments
• x1 1 x2 0 x3 0
  x4 1

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Hamiltonian Cycle Problem

• Can we claim that the assignment thus
• determined would satisfy all clauses.
• YES !
• Since the larger cycle visited each clause
• node cj, at least one Pi was traversed in the
• right direction relative to the node cj

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Acknowledgements

• BIG Thanks to Sonika Arora