3/26/08  NPcomplete partitioning pro...
Reduction from 3SAT to HAM. CYCLE
Theorem. HAM CYCLE is NPComplete
Proof.
(1) It’s in NP.
Show me the cycle, and I can verify it in O(n) time.
(2) The reduction runs in poly(n) time.
Let b = length of each path = 3m + 3
O(b) work to construct each path.
O(1) work to construct each clause gadget.
O(1) work to create s, t and their edges.
Total work is O(bn + m + 1)
n = # vars
m = # clauses
= O(nm)
(3) If 3SAT instance
ϕ
is satisfiable,
∃
a Ham. cycle
Proof. By construction
(4) If
∃
a Ham. cycle in G, then
ϕ
has a satisfying assignment.
(A) Define a notion of “intended cycles”
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(B) Show every Ham. cycle is intended.
(C) From an intended Ham cycle you get a sat. assignment.
Def. A cycle in G is
intended if
(a)
∀
1
≤
i < n the cycle visits
every vtx in P
i
before
visiting first vertex of P
i+1
(b)
∀
c
i
the cycle arrives at c
i
from some path P
j
and leaves
c
i
by returning to P
j
.
Prop. Every Hamiltonian cycle in G is an intended cycle.
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 Spring '08
 KLEINBERG
 Algorithms, #, Glossary of graph theory, Travelling salesman problem, Hamiltonian path problem, Ham. cycle

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