1. Weighted Matching-Algorithms, Hamiltonian Cycles and TSP
Graphs & Algorithms
Lecture 6
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2. 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 .

3. 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.

4. Hungarian Algorithm Kuhn (1955), Munkres (1957)

5. 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.

6. 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.

7. 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
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. x1 x2 xn
xi-1 xi x3
Theorem of Chvatal?

8. 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.

9. 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)

10. 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.

11. 2-approximation algorithm for TSP
Running time
Prim's algorithm with Fibonacci heaps: O(E + V logV)
Kruskal's algorithm: O(E logV)

12. 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.

13. Example
Put on an extra slide

14. 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)