The TSP : NP-Completeness Approximation and Hardness of Approximation

1
The TSP NP-Completeness Approximation and
Hardness of Approximation

All exact science is dominated by the idea of
approximation. -- Bertrand Russell (1872 - 1970)
Based upon slides of Dana Moshkovitz, Kevin Wayne
and others some old slides
TSP Traveling Salesman Problem
2
A related problem HC

• HAM-CYCLE given an undirected graph G (V, E),
  does there exist a simple cycle ? that contains
  every node in V.

3
Hamiltonian Cycle

• HAM-CYCLE given an undirected graph G (V, E),
  does there exist a simple cycle ? that contains
  every node in V.

1
1'
2
2'
3
3'
4
4'
5
NO bipartite graph with odd number of nodes.
4
Directed Hamiltonian Cycle

• DIR-HAM-CYCLE given a digraph G (V, E), does
  there exists a simple directed cycle ? that
  contains every node in V?
• Claim. DIR-HAM-CYCLE ? P HAM-CYCLE.
• Pf. Given a directed graph G (V, E), construct
  an undirected graph G' with 3n nodes. Each node
  splits up into an output node, a regular node and
  an input node.

5
Example
ain
a
aout
a
cin
cout
c
c
b
bin
G
b
bout
6
Example
ain
a
aout
a
cin
cout
c
c
b
bin
G
b
bout
7
Directed Hamiltonian Cycle

• Claim. G has a Hamiltonian cycle iff G' does.
• Pf. ?
• Suppose G has a directed Hamiltonian cycle ?.
• Then G' has an undirected Hamiltonian cycle (same
  order).
• Pf. ?
• Suppose G' has an undirected Hamiltonian cycle
  ?'.
• ?' must visit nodes in G' using one of following
  two orders
• , B, G, R, B, G, R, B, G, R, B,
• , B, R, G, B, R, G, B, R, G, B,
• Blue nodes in ?' make up directed Hamiltonian
  cycle ? in G, or reverse of one. ?

8
3-SAT Reduces to Directed Hamiltonian Cycle

• Claim. 3-SAT ? P DIR-HAM-CYCLE.
• Pf. Given an instance ? of 3-SAT, we construct
  an instance of DIR-HAM-CYCLE that has a
  Hamiltonian cycle iff ? is satisfiable.
• Construction. First, create graph that has 2n
  Hamiltonian cycles which correspond in a natural
  way to 2n possible truth assignments.

9
3-SAT Reduces to Directed Hamiltonian Cycle

• Construction. Given 3-SAT instance ? with n
  variables xi and k clauses.
• Construct G to have 2n Hamiltonian cycles.
• Intuition traverse path i from left to right ?
  set variable xi 1.

xi 1
s
x1
x2
x3
t
3k 3
10
3-SAT Reduces to Directed Hamiltonian Cycle

• Construction. Given 3-SAT instance ? with n
  variables xi and k clauses.
• For each clause add a node and 6 edges.

clause node
clause node
s
x1
x2
x3
t
11
3-SAT Reduces to Directed Hamiltonian Cycle

• Claim. ? is satisfiable iff G has a Hamiltonian
  cycle.
• Pf. ?
• Suppose 3-SAT instance has satisfying assignment
  x.
• Then, define Hamiltonian cycle in G as follows
• if xi 1, traverse row i from left to right
• if xi 0, traverse row i from right to left
• for each clause Cj , there will be at least one
  row i in which we are going in "correct"
  direction to splice node Cj into tour

12
3-SAT Reduces to Directed Hamiltonian Cycle

• Claim. ? is satisfiable iff G has a Hamiltonian
  cycle.
• Pf. ?
• Suppose G has a Hamiltonian cycle ?.
• If ? enters clause node Cj , it must depart on
  mate edge.
• thus, nodes immediately before and after Cj are
  connected by an edge e in G
• removing Cj from cycle, and replacing it with
  edge e yields Hamiltonian cycle on G - Cj
• Continuing in this way, we are left with
  Hamiltonian cycle ?' inG - C1 , C2 , . . . ,
  Ck .
• Set xi 1 iff ?' traverses row i left to right.
• Since ? visits each clause node Cj , at least one
  of the paths is traversed in "correct" direction,
  and each clause is satisfied. ?

13
3-SAT ? P Directed HC ? P HC

• Objectives
• To explore the Traveling Salesman Problem.
• Overview
• TSP Examples and Defn.
• Is TSP NP-complete?
• Approximation algorithm for special cases
• Hardness of Approximation in general.

14
Traveling Salesman Problem A Tour around USA

• TSP. Given a set of n cities and a pairwise
  distance function d(u, v), is there a tour of
  length ? D?

All 13,509 cities in US with a population of at
least 500Reference http//www.tsp.gatech.edu
15
Traveling Salesperson Problem

• TSP. Given a set of n cities and a pairwise
  distance function d(u, v), is there a tour of
  length ? D?

Optimal TSP tour Reference http//www.tsp.gatech
.edu
16
Traveling Salesperson Problem

• TSP. Given a set of n cities and a pairwise
  distance function d(u, v), is there a tour of
  length ? D?

11,849 holes to drill in a programmed logic
array Reference http//www.tsp.gatech.edu
17
Traveling Salesperson Problem

• TSP. Given a set of n cities and a pairwise
  distance function d(u, v), is there a tour of
  length ? D?

Optimal TSP tour Reference http//www.tsp.gatech
.edu
18
TSP

• Given a weighted graph G(V,E)
• V Vertices Cities
• E Edges Distances between cities
• Find the shortest tour that visits all cities

19
TSP
3
2
1
10
4
3
5

• Instance A complete weighted undirected graph
  G(V,E)
• (all weights are non-negative).
• Problem To find a Hamiltonian cycle of minimal
  cost.

20
Naïve Solution

• Try all possible tours and pick the minimum
• Dynamic Programming

Definitely we need something better
21
Approximation Algorithms

• A good algorithm is one whose running time is
  polynomial in the size of the input.
• Any hope of doing something in polynomial time
  for NP-Complete problems?

22
c-approximation algorithm

• The algorithm runs in polynomial time
• The algorithm always produces a solution which is
  within a factor of c of the value of the optimal
  solution

c
(1/c) A(x) Opt(x) A(x)
For all inputs x. OPT(x) here denotes the optimal
value of the minimization problem
23
c-approximation algorithm

• The algorithm runs in polynomial time
• The algorithm always produces a solution which is
  within a factor of c of the value of the optimal
  solution

c
c A(x) Opt(x) A(x)
For all inputs x. OPT(x) here denotes the optimal
value of the maximization problem
24
So why do we study Approximation Algorithms

• As algorithms to solve problems which need a
  solution
• As a mathematically rigorous way of studying
  heuristics
• Because they are fun! ?
• Because it tells us how hard problems are

25
Vertex Cover

• Any guess on how to design approximation
  algorithms for vertex cover?

26
Vertex Cover Greedy
27
Vertex Cover Greedy
28
Vertex Cover Greedy
29
Vertex Cover Greedy
30
Vertex Cover Greedy
31
Vertex Cover Greedy
Greedy VC Approx 8 Opt 6 Factor 4/3 HW 2
Problem Example can be extended to O(log n)
approximation
32
A Simpler Approximation Algorithm

• Choose an edge e in G
• Add both endpoints to the Approximate VC
• Remove e from G and all incident edges and repeat.

Cover generated is at most twice the optimal
cover!

• Nothing better than 2-factor known.
• If P ltgt NP, there is no poly-time algorithm that
  achieves
• an approximation factor better than 1.1666
  Has97.

33
What Next?

• Well show an approximation algorithm for TSP,
• with approximation factor 2
• for cost functions that satisfy a certain
  property.

34
Polynomial Algorithm for TSP?
What about the greedy strategy At any point,
choose the closest vertex not explored yet?
35
The Greedy Strategy Fails
10
?
?
5
12
2
3
?
0
1
36
The Greedy Strategy Fails
10
?
?
5
12
2
3
?
0
1
37
Another ExampleGreedy strategy fails
Dont be greedy Always!
0
1
-1
3
7
-5
-11
Even monkeys can do better than this !!!
38
TSP is NP-hard

• The corresponding decision problem
• Instance a complete weighted undirected graph
  G(V,E) and a number k.
• Problem to find a Hamiltonian path whose cost is
  at most k.

39
TSP is NP-hard

• Theorem HAM-CYCLE ?p TSP.
• Proof By the straightforward efficient reduction
  illustrated below

1
cn
1
1
cn
1
HAM-CYCLE
TSP
n k V
40
TSP

• Is a minimization problem.
• We want a 2-approximation algorithm
• But only for the case when the
• cost function
• satisfies the triangle inequality.

41
The Triangle Inequality

• Cost Function Let c(x,y) be the cost of going
  from city x to city y.
• Triangle Inequality In most situations, going
  from x to y directly is no more expensive than
  going from x to y via an intermediate place z.

42
The Triangle Inequality

• Definition Well say the cost function c
  satisfies the triangle inequality, if
• ?x,y,z?V c(x,z)c(z,y)?c(x,y)

43
Approximation Algorithm

• 1. Grow a Minimum Spanning Tree (MST) for G.
• 2. Return the cycle resulting from a preorder
  walk on that tree.

44
Demonstration and Analysis
The cost of a minimal Hamiltonian cycle ? the
cost of a MST
?
45
Demonstration and Analysis
The cost of a preorder walk is twice the cost of
the tree
46
Demonstration and Analysis
Due to the triangle inequality, the Hamiltonian
cycle is not worse.
47
The Bottom Line
optimal HAM cycle
our HAM cycle
preorder walk
?
? ½
½
MST
48
What About the General Case?

• Well show TSP cannot be approximated within any
  constant factor ??1
• By showing the corresponding gap version is
  NP-hard.

Inapproximability
49
gap-TSP?

• Instance a complete weighted undirected graph
  G(V,E).
• Problem to distinguish between the following two
  cases
• There exists a Hamiltonian cycle, whose cost
  is at most V.
• The cost of every Hamiltonian cycle is more
  than ?V.

YES
NO
50
Instances
min cost
51
What Should an Algorithm for gap-TSP Return?
min cost
DONT-CARE...
52
gap-TSP Approximation

• Observation Efficient approximation of factor ?
  for TSP implies an efficient algorithm for
  gap-TSP?.

53
gap-TSP is NP-hard

• Theorem For any constant ??1,
• HAM-CYCLE ?p gap-TSP?.
• Proof Idea Edges from G cost 1. Other edges cost
  much more.

54
The Reduction Illustrated
1
?V1
1
1
?V1
1
HAM-CYCLE
gap-TSP
Verify (a) correctness (b) efficiency
55
Approximating TSP is NP-hard

• gap-TSP? is NP-hard

Approximating TSP within factor ? is NP-hard
56
Summary
?

• Weve studied the Traveling Salesman Problem
  (TSP).
• Weve seen it is NP-hard.
• Nevertheless, when the cost function satisfies
  the triangle inequality, there exists an
  approximation algorithm with ratio-bound 2.

57
Summary
?

• For the general case weve proven there is
  probably no efficient approximation algorithm for
  TSP.
• Moreover, weve demonstrated a generic method for
  showing approximation problems are NP-hard.