Transcript The Traveling Salesman Problem

The Traveling Salesman Problem
Approximation
Rachit Shah
March 3 Class Notes
CSE Department
UTA Arlington
CSE 6311 Adv Comp Models and Algorithms
Prof: Dr Gautam Das
Traveling Salesman Problem
• Traveling Salesman Problem (TSP): Given a
complete graph with nonnegative edge costs,
Find a minimum cost cycle visiting every vertex
exactly once.
• Application (Example): Given a number of cities
and the costs of traveling from any city to any
other city, what is the cheapest round-trip route
that visits each city exactly once and then
returns to the starting city
Triangle Inequality
a
c
a+b≥c
b
• For example, cost of an edge as the euclidian distance
between two points.
• Metric Traveling Salesman Problem (metric TSP): Given
a complete graph with edge costs satisfying triangle
inequalities, Find a minimum cost cycle visiting every
vertex exactly once.
Metric TSP approximation
• The cost function satisfies the triangle inequality
for all vertices u, v, w in V
– c(u,w) <= c(u,v) + c(v,w)
• Eulerian graphs and Eulerian circuits:
– Eulerian circuit: cycle that uses each edge exactly
once
– Eulerian graph: graph with a Eulerian circuit
– An undirected graph is Eulerian iff each vertex has
even degree
Approximate-TSP-tour (G)
1. Find MST T of G
2. Double every edge of the MST to get a
Eulerian graph G’
Idea: Get a tour from Minimum spanning
tree without increasing its cost too much
(at most twice).
3. Find an Eulerian circuit E on G’
4. Output the vertices of G in order of
appearance in E
First let’s compare the optimal solutions of MST
and TSP for any problem instance G=(V,E)
Optimal TSP sol-n
A tree obtained from the tour
Optimal MST sol-n
(*)
Cost (Opt. TSP sol-n) ≥ Cost (of this tree) ≥ Cost (Opt. MST sol-n)
What is the cost of the tour
compared to the cost of MST?
5
6
3
1
4
2
cost 
 c(e)
dashed arcs e
Triangle inequality

2   c(e)
arcs e
 2  cost(MST)
TSP - e  MST

2  cost(optim al TSP)
Analysis
• This is a 2 approximation algorithm
• MST < Euler Cycle = 2 * MST <= 2.0 TSP
• Can we improve this?
– The previous algorithm was a factor 2 algorithm.
• Recall Step 2:
Double every edge of the MST to get a Eulerian graph G’
– If we can work avoid this, we could possibly have a
better solution !
Perfect Matching
• Matching: a matching is a subset M
of E such that for all vertices v in V,
at most one edge is incident on v.
• Perfect Matching: is a matching M
such that for all vertices v in V
exactly one edge of M is incident on
v.
Minimum Weight Matching
• Perfect matching with minimum sum of
weights
• Input: Complete weighted graph s, |v| is
even
• Output: A pair of values (vi, vj) such that
the sum of weights is smallest.
1.5 approximation algorithm
(Known as Christofides Heuristics)
1. Find a MST T of G
2. Find a minimum weight (perfect) matching
M on the set of odd degree vertices in T.
3. Add M to T to get the Eulerian graph G’
4. Find an Eulerian circuit E on G’ by skipping
vertices already seen (shortcutting)
5. Output the vertices of G in order of appearance
in E
Step 2:
Even nodes with odd degree??
• How can I know that I always have even number of nodes,
which have odd degree, for me to do the MATCHING?
• Let Sum(d) = 2m, where m= number of edges. Therefore
Sum(d) is even.
• Let SumEven(d) to be the sum of degrees of the vertices
which have even degree, SumEven(d) is also even.
• Therefore SumOdd(d)=Sum(d)-SumEven(d) = 2k,
k=1,2,…, which means that the sum of degrees of the
vertices which have odd degree each is also an even
number. Thus there are even numbers of vertices which
have odd number of degree.
Analysis
TSP = match1 + match2 (we can divide TSP in two
mutually exclusive matchings)
MWM <= min{ match1, match2}
 Match1 + match 2 = TSP
 MWM + MWM <= TSP (MWM <= match1 and MWM <=
match2)
 MWM <= .5 TSP
MWM <= .5 TSP
MST < Euler Cycle
= MST + MWM <= TSP + .5 TSP
= 1.5 TSP