Weighted Matching-Algorithms, Hamiltonian Cycles and TSP

Graphs & AlgorithmsLecture 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 labelsu = u1,…,un and v = v1,…,vn , u, v 2 Rn

such that, for all 1 · i, j · n, we haveui + vj ¸ wi, j .

• Cost c(u, v) := i ui + vi . 2

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. 3

Hungarian AlgorithmKuhn (1955), Munkres (1957)

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Weight edBipart it eMat ching(Kn ,n = (A [ B,A £ B),w[n,n])1 for each i 2 [n]2 do u[i]Ã maxj w[i, j]3 v[i] Ã 04 while Gu ,v has no perfect matching5 do X Ã minimum vertex cover of Gu ,v

6 " Ã min©u[i]+v[j]- w[i, j] : fi, jg2 (A nX) £ (B nX)

ª

7 for each i 2 A nX8 do u[i] Ã u[i]- "9 for each i 2 B \ X

10 do v[i] Ã v[i]+"11 return perfect matching of Gu ,v

Correctness of the Hungarian Method

TheoremThe 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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• 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

• 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.

Proof of Dirac’s Theorem (Pósa)

x1 x2 xnxi-1 xix3

Weaker degree conditions

• Let G be a graph on n vertices with degrees d1 · … · 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 costc(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, defineP(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. 12

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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