Vertex cover problemS V such that for every {u,v} E uS or vS (or both)

Vertex cover problemS V such that for every {u,v} E uS or vS (or both)

Vertex cover problemS V such that for every {u,v} E uS or vS (or both)

OPTIMIZATION VERSION: INPUT: graph G OUTPUT: vertex cover S of minimum-size

DECISION VERSION: INSTANCE: graph G, integer k QUESTION: does G have vertex cover of size k ?

Vertex cover problemDECISION VERSION: INSTANCE: graph G, integer k QUESTION: does G have vertex cover of size k ?

complement of a graph G G

vertex cover S in G V-S is _________ in G ?

Vertex cover problemDECISION VERSION: INSTANCE: graph G, integer k QUESTION: does G have vertex cover of size k ?

complement of a graph G G

vertex cover S in G V-S is clique in G ?

Vertex cover problemDECISION VERSION: INSTANCE: graph G, integer k QUESTION: does G have vertex cover of size k ?

complement of a graph G G

vertex cover S in G V-S is clique in G ?

Clique Vertex Cover Vertex Cover is NP-complete

Vertex cover problemOPTIMIZATION VERSION: INPUT: graph G OUTPUT: vertex cover S of minimum-size

Vertex cover problemOPTIMIZATION VERSION: INPUT: graph G OUTPUT: vertex cover S of minimum-size

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

Algorithm 2: find a maximal matching M in G, for each {u,v}M put both u,v in S

Algorithm 2: find a maximal matching M in G, for each {u,v}M put both u,v in S

k edges|S| = 2k

Algorithm 2: find a maximal matching M in G, for each {u,v}M put both u,v in S

k edges|S| = 2k OPT k

Algorithm 2: find a maximal matching M in G, for each {u,v}M put both u,v in S

k edges|S| = 2k

OPT k|S| 2 OPT

2-approximation algorithm

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

n

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

n/2

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

n/2 n/3

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

n/k k=2

n=

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

n/k k=2

n (n/k – 1) (n ln n) – 2n =(n ln n)

k=2

n

OPT = n

Algorithm 1 has approximation ratio (ln n)

Vertex cover problemOPTIMIZATION VERSION: INPUT: graph G OUTPUT: vertex cover S of minimum-size

Algorithm 1: pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat

Algorithm 2: find a maximal matching M in G, for each {u,v}M put both u,v in S 2-approximation

ln n)-approximation

Hamiltonian cycle problemHamiltonian cycle in (undirected) graph G=(V,E) C=u1,u2,...,un, such that every vertex vV occurs in C exactly once ui,ui+1 E for i=1,...,n-1 u1,un E

Hamiltonian cycle problemHamiltonian cycle in (undirected) graph G=(V,E) C=u1,u2,...,un, such that every vertex vV occurs in C exactly once ui,ui+1 E for i=1,...,n-1 u1,un E

Hamiltonian cycle problemHamiltonian cycle in (undirected) graph G=(V,E) C=u1,u2,...,un, such that every vertex vV occurs in C exactly once ui,ui+1 E for i=1,...,n-1 u1,un E

NP-complete problem

Travelling salesman (TSP)INSTANCE: complete graph with edge weights G=(V,E,w)

SOLUTION: hamiltonian cycle C in G

OBJECTIVE: sum of the weights of the cycle C

Travelling salesman (TSP)INSTANCE: complete graph with edge weights G=(V,E,w)

SOLUTION: hamiltonian cycle C in G

OBJECTIVE: sum of the weights of the cycle C

Travelling salesman (TSP)INSTANCE: complete graph with edge weights G=(V,E,w)

SOLUTION: hamiltonian cycle C in G

OBJECTIVE: sum of the weights of the cycle C

Is there an approximation algorithm ?

Metric TSPINSTANCE: complete graph with edge weights G=(V,E,w)

SOLUTION: cycle C in G, repeated vertices,edges allowed

OBJECTIVE: sum of the weights of the cycle C

Is there an approximation algorithm ?

Metric TSPd(u,v) = cheapest way of getting from u to v

d(u,v) = d(v,u)d(u,v) d(u,w)+ d(w,u)

Metric TSPcompute the d(u,v)compute MST T weight(T) OPT

Metric TSPcompute the d(u,v)compute MST T weight(T) OPT

2-approximation algorithm

Euler tourwhen can a graph be drawn withoutlifting a pen, and without drawing thesame edge twice?

Euler tourwhen can a graph be drawn withoutlifting a pen, and without drawing thesame edge twice?

if we want to end where we started?

Metric TSPcompute the d(u,v)compute MST Tfind a min-weight perfect matching on odd-degree vertices of T

weight(T) OPTweight(M) OPT/2

1.5-approximation algorithm

Optimization problemsINSTANCE

FEASIBLE SOLUTIONS

c: SOLUTIONS R+

OPT= min c(T)T FEASIBLE SOLUTIONS

APPROXIMATION ALGORITHM INSTANCE T

c(T) OPT

PTASPolynomial-time approximation schemepolynomial-time (1+)-approximation algorithm for any constant >0

FPTASFully polynomial-time approximation scheme

(1+)-approximation algorithm running in time poly(INPUT,1/)

APPROXIMATION ALGORITHM INSTANCE T

c(T) OPT