chapter 10 complexity of approximation (1) l-reduction
DESCRIPTION
Chapter 10 Complexity of Approximation (1) L-Reduction. Ding-Zhu Du. Traveling Salesman. Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once. Definition. Theorem. Proof: Given a graph G=(V,E), define a distance table on V as follows:. - PowerPoint PPT PresentationTRANSCRIPT
Traveling Salesman
• Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.
. distance
total tour witha find ,and cities those
between tabledistance a with cities Given
:TSP-Approx-
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r
Definition
Proof:
Given a graph G=(V,E), define a distance table on V as follows:
EvurV
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hard.- is TSP-Approx- ,1any For NPrr
Theorem
solvable. time-polynomial being HC implies solvable
time-polynomial being TSP-Approx- r
Contradiction Argument
• Suppose r-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow:
r-approximation solution < r |V|
if and only if
G has a Hamiltonian cycle
Special Case
• Traveling around a minimum spanning tree is a 2-approximation.
solvable. time-polynomial
is TSP-Approx-2 ,inequality triangular
thesatisfies tabledistance n the Whe Theorem
• Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5-approximation
solvable. time-polynomial
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thesatisfies tabledistance n the Whe Theorem
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hint
DS-2-MinVC pm
• Min-2-DS is MAX SNP-complete in the case that all given pools have size at most 2.
4 Prove that
5. Is TSP with triangular inequality MAX SNP-complete?