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Transcript of 1 The TSP : NP-Completeness Approximation and Hardness of Approximation All exact science is...
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The TSP : NP-Completeness Approximation and Hardness of Approximation
All exact science is dominated by the idea of approximation.-- Bertrand Russell (1872 - 1970)
*
*TSP = Traveling Salesman Problem
Based upon slides of Dana Moshkovitz,Kevin Wayne and others + some old slides
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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.
YES: vertices and faces of a dodecahedron.
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Hamiltonian Cycle HAM-CYCLE: given an undirected graph G = (V, E), does there
exist a simple cycle that contains every node in V.
1
3
5
1'
3'
2
4
2'
4'
NO: bipartite graph with odd number of nodes.
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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.
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Example
c
a
b
G
ain
a
aout
bin
b
bout
cin coutc
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Example
c
a
b
G
ain
a
aout
bin
b
bout
cin coutc
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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. ▪
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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.
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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.
s
t
3k + 3
x1
x2
x3
xi = 1
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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.
s
t
clause nodeclause node
x1
x2
x3
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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 x*i = 1, traverse row i from left to right
• if x*i = 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
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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 x*i = 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. ▪
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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.
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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
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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 tourReference: http://www.tsp.gatech.edu
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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 arrayReference: http://www.tsp.gatech.edu
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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 tourReference: http://www.tsp.gatech.edu
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TSP
Given a weighted graph G=(V,E)
V = Vertices = Cities
E = Edges = Distances between cities
Find the shortest tour that visits all cities
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TSP
Instance: A complete weighted undirected graph G=(V,E)
(all weights are non-negative).
Problem: To find a Hamiltonian cycle of minimal cost.
3
432
5
1 10
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Naïve Solution
Try all possible tours and pick the minimum
Dynamic Programming
))!1(( nO
)2( nO
Definitely we need something better
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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?
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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
)(
)(
xOPT
xA c
For all inputs x.OPT(x) here denotes the optimal value of the minimization problem
(1/c) A(x) ≤ Opt(x) ≤ A(x)
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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
( )
( )
OPT x
A x c
For all inputs x.OPT(x) here denotes the optimal value of the maximization problem
c A(x) ≥ Opt(x) ≥ A(x)
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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
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Vertex Cover
Any guess on how to design approximation algorithms for vertex cover?
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Vertex Cover: Greedy
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Vertex Cover: Greedy
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Vertex Cover: Greedy
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Vertex Cover: Greedy
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Vertex Cover: Greedy
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Vertex Cover: Greedy
Greedy VC Approx = 8Opt = 6Factor 4/3HW 2 Problem : Example can be extended to O(log n) approximation
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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 <> NP, there is no poly-time algorithm that achieves an approximation factor better than 1.1666 [Has97].
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What Next?
We’ll show an approximation algorithm for TSP,
with approximation factor 2
for cost functions that satisfy a certain property.
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Polynomial Algorithm for TSP?
What about the greedy strategy:
At any point, choose the closest vertex not explored
yet?
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The Greedy Strategy Fails
5
0
3
1
12
10
2
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The Greedy Strategy Fails
5
0
3
1
12
10
2
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Another ExampleGreedy strategy fails
0 1-1 3 7-5-11
Even monkeys can do better than this !!!
Don’t be greedy Always!
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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.
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TSP is NP-hard
Theorem: HAM-CYCLE p TSP.
Proof: By the straightforward efficient reduction illustrated below:
HAM-CYCLE TSP
1 cn1
1
1
n = k = |V|
cn
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TSP
Is a minimization problem. We want a 2-approximation algorithm But only for the case when the
cost function
satisfies the triangle inequality.
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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.
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The Triangle Inequality
Definition: We’ll say the cost function c satisfies the triangle inequality, if
x,y,zV : c(x,z)+c(z,y)c(x,y)
x
yz
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Approximation Algorithm
1. Grow a Minimum Spanning Tree (MST) for G.
2. Return the cycle resulting from a preorder walk on that tree.
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Demonstration and Analysis
The cost of a minimal
Hamiltonian cycle the cost of a
MST
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Demonstration and Analysis
The cost of a preorder walk is twice the cost of
the tree
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Demonstration and Analysis
Due to the triangle inequality, the
Hamiltonian cycle is not worse.
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The Bottom Line
optimal HAM cycle
MSTpreorder
walk
our HAM cycle
= ½· ½·
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What About the General Case?
We’ll show TSP cannot be approximated within any constant factor 1
By showing the corresponding gap version is NP-hard.
Inapproximability
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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|.
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Instances
min cost
|V| |V|
1
1
1
0+1
0
0
1
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What Should an Algorithm for gap-TSP Return?
|V| |V|
YES! NO!
min cost
gap
DON’T-CARE...
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gap-TSP & Approximation
Observation: Efficient approximation of factor for TSP implies an efficient algorithm for gap-TSP[].
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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.
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The Reduction Illustrated
HAM-CYCLE gap-TSP
1 |V|+11
1
1
|V|+1
Verify (a) correctness (b)
efficiency
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Approximating TSP is NP-hard
gap-TSP[] is NP-hard
Approximating TSP within factor is NP-hard
![Page 56: 1 The TSP : NP-Completeness Approximation and Hardness of Approximation All exact science is dominated by the idea of approximation. -- Bertrand Russell.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649e705503460f94b6e556/html5/thumbnails/56.jpg)
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Summary
We’ve studied the Traveling Salesman Problem (TSP).
We’ve seen it is NP-hard. Nevertheless, when the cost function
satisfies the triangle inequality, there exists an approximation algorithm with ratio-bound 2.
![Page 57: 1 The TSP : NP-Completeness Approximation and Hardness of Approximation All exact science is dominated by the idea of approximation. -- Bertrand Russell.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649e705503460f94b6e556/html5/thumbnails/57.jpg)
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Summary
For the general case we’ve proven there is probably no efficient approximation algorithm for TSP.
Moreover, we’ve demonstrated a generic method for showing approximation problems are NP-hard.