Transcript Folie 1 - ETH - Cadmo

Weighted Matching-Algorithms,
Hamiltonian Cycles and TSP
Graphs & Algorithms
Lecture 6
Weighted bipartite matching
• Given:
– Kn, n (complete bipartite graph on 2n vertices)
– n £ n weight matrix with entries wi, j ¸ 0
• Want: perfect matching M maximizing the total
weight
• Weighted cover (u, v): choice of vertex labels
u = u1,…,un and v = v1,…,vn , u, v 2 Rn
such that, for all 1 · i, j · n, we have
ui + vj ¸ wi, j .
• Cost c(u, v) := i ui + vi .
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Duality: max. weighted matching
and min. weighted vertex cover
• For any matching M and weighted cover (u, v) of a
weighted bipartite graph, we have
w(M) · c(u, v) .
• Also, w(M) = c(u, v) if and only if M consists of
edges {i, j} such that ui + vj = wi, j. In this case, M
and (u, v) are optimal.
• Equality subgraph Gu, v of a fixed cover (u, v):
– spanning subgraph of Kn, n ,
– {i, j} 2 E(Gu, v) , ui + vj = wi, j .
• Idea: a perfect matching of Gu, v corresponds to a
maximum weighted matching of Kn, n.
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Hungarian Algorithm
Kuhn (1955), Munkres (1957)
W eight edB ipar t it eM at ching(Kn ,n = (A [ B, A £ B), w[n , n ])
1 for each i 2 [n ]
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do u[i ] Ã maxj w[i , j ]
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v[i ] Ã 0
4 while Gu ,v has no perfect mat ching
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do X Ã minimum
vert ex cover of Gu ,v
©
ª
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" Ã min u[i ] + v[j ] - w[i , j ] : fi , j g 2 (A n X) £ (B n X)
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for each i 2 A n X
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do u[i ] Ã u[i ] - "
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for each i 2 B \ X
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do v[i ] Ã v[i ] + "
11 r et ur n perfect mat ching of Gu ,v
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Correctness of the Hungarian Method
Theorem
The Hungarian Algorithm finds a maximum weight
matching and a minimum cost cover.
Proof
• The statement is true if the algorithm terminates.
• Loop invariant: consider (u, v) before and (u', v') after
the while loop
– (u, v) is cover of G ) (u', v') is a cover of G
– c(u, v) ¸ c(u', v') +  ¸ w(M)
• For rational weights,  is bounded from below by an
absolute constant.
• In the presence of irrational weights, a more careful
selection of the minimum vertex cover is necessary.
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Hamiltonian Cycles
• A graph on n vertices is Hamiltonian if it contains a
simple cycle of length n.
• The Hamiltonian-cycle problem is NP-complete
(reduction to the vertex-cover problem).
• The naïve algorithm has running time (n!) = (2n).
• What are sufficient conditions for Hamiltonian
graphs?
Theorem (Dirac 1952)
Every graph with n ¸ 3 vertices and minimum
degree at least n/2 has a Hamiltonian cycle.
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Proof of Dirac’s Theorem (Pósa)
• Suppose G is a graph with (G)  n/2 that contains no
Hamiltonian cycle.
• Insert as many edges into G as possible
 Embedding of G into a saturated graph G’ that
contains a Hamilton path
x1
x2
x3
xi-1
xi
xn
• Neighbourhood (x1) yields n/2 forbidden neighbours for
xn in {x1, …, xn – 2}.
• Since xn cannot connect to itself, there is not enough
space for all of its n/2 neighbours.
Weaker degree conditions
• Let G be a graph on n vertices with degrees
d 1 · … · dn.
• (d1, …, dn) is called the degree sequence
• An integer sequence (a1, …, an) is Hamiltonian if
every graph on n vertices with a pointwise greater
degree sequence (d1, …, dn) is Hamiltonian.
Theorem (Chvátal 1972)
An integer sequence (a1, …, an) such that
0 · a1 · … · an < n and n ¸ 3 is Hamiltonian iff,
for all i < n/2, we have: ai · i ) an – i ¸ n – i.
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Traveling Salesman Problem (TSP)
• Given n cities and costs c(i, j) ¸ 0 for going from city
i to city j (and vice versa).
• Find a Hamiltonian cycle H*of minimum cost
c(H*) = e 2 E(H*) c(e) .
• Existence of a Hamiltonian cycle is a special case.
• Brute force: n! = (nn + 1/2 e-n) time complexity
• Better solutions:
– polynomial time approximation algorithm for TSP with
triangle inequality approximation ratio 2
– optimal solution with running time O(n22n)
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Approximation algorithms
• Consider a minimization problem. An algorithm ALG
achieves approximation ratio (n) ¸ 1 if, for every
problem instance P of size n, we have
ALG(P) / OPT(P) · (n) ,
where OPT(P) is the optimal value of P.
• An approximation scheme takes one additional
parameter  and achieves approximation ratio
(1 + ) on every problem instance.
• A polynomial time approximation (PTAS) scheme runs in
polynomial time for every fixed  ¸ 0.
• The running time of a fully polynomial time
approximation scheme (FPTAS) is also polynomial in -1.
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2-approximation algorithm for TSP
• Running time
– Prim's algorithm with Fibonacci heaps: O(E + V logV)
– Kruskal's algorithm: O(E logV)
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An optimal TSP algorithm
• For each S µ {2,…,n} and k 2 S, define
P(S, k) ´ "minimum cost of a Hamiltonian path
on S starting in 1 and ending in k"
• Let V = {1,…,n} and positive costs c(i, j) be given.
TSP = min{P(Vn{1}, k) + c(k, 1) : k 2 Vn{1}}
• Recursive computation of P(S, k) :
P({k}, k) = c(1, k)
P(S, k) = min{P(Sn{k}, j) + c(j, k) : j 2 Sn{k}}
• Compute P(S, k) buttom-up (dynamic programming)
• Number of distinct P(S, k) values is (n – 1)2n – 2 .
• Each time at most n operations are necessary.
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Example
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More positive and negative results on
TSP
• Slight modifications yield a 1.5-approximation
algorithm for TSP with -inequality. (Exercise)
• Arora (1996) gave a PTAS for Euclidean TSP with
running time nO(1/) .
• For general TSP, there exists no polynomial time
approximation algorithm with any constant
approximation ratio  ¸ 1, unless P = NP. (Exercise)
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