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08 - NP-complete partitioning problems- 3-dimensional matching

08 - NP-complete partitioning problems- 3-dimensional matching

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3/26/08 - NP-complete partitioning pro... Reduction from 3SAT to HAM. CYCLE Theorem. HAM CYCLE is NP-Complete 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” 28-1 28
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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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