WINSEM2015-16_CP0739_25-Jan-2016_RM02_Lecture-5.pdf - 1 TSP...

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TSP and SubSet Sum
Don.S
[email protected]

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Traveling-salesman problem is
NPC
• TSP={<G, c , k >:
G=(V,E) is a complete graph,
c is a function from V  V  Z,
k  Z, and G has a traveling salesman
tour with cost at most k .}
• Theorem 34.14 : (page 1012)
– TSP is NP-complete.
Material from Internet
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TSP
• TSP belongs to NP:
– Given a certificate of a sequence of vertices in the tour, the
verifying algorithm checks whether each vertex appears once,
sums up the cost and checks whether at most k . in poly time.
• TSP is NP-hard (show HAM-CYCLE 
p
TSP)
– Given an instance G=(V,E) of HAM-CYCLE, construct a TSP
instance <G',c,0) as follows (in poly time):
• G'=(V,E'), where E'={< i , j >: i , j  V and i  j } and
• Cost function c is defined as c ( i , j )=0 if ( i , j )  E, 1, otherwise.
– If G has a hamiltonian cycle h , then h is also a tour in G' with
cost at most 0.
– If G' has a tour h ' of cost at most 0, then each edge in h ' is 0, so
each edge belong to E, so h ' is also a hilmitonian cycle in G.

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Subset Sum is NPC
• SUBSET-SUM={<S, t