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## Vertex cover problem

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**Vertex cover problem**S V such that for every {u,v} E uS or vS (or both)**Vertex cover problem**S V such that for every {u,v} E uS or vS (or both)**Vertex cover problem**S 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 problem**DECISION 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 problem**DECISION 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 problem**DECISION 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 problem**OPTIMIZATION VERSION: INPUT: graph G OUTPUT: vertex cover S of minimum-size**Vertex cover problem**OPTIMIZATION 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 OPT k |S| = 2k**Algorithm 2:**find a maximal matching M in G, for each {u,v}M put both u,v in S 2-approximation algorithm k edges OPT k |S| = 2k |S| 2 OPT**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 n/k = k=2**Algorithm 1:**pick a vertex v with the largest degree, put v in S, remove v and adjacent edges from G, repeat n n n/k (n/k – 1) (n ln n) – 2n =(n ln n) k=2 k=2 OPT = n Algorithm 1 has approximation ratio (ln n)**Vertex cover problem**OPTIMIZATION 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 (ln n)-approximation 2-approximation**Hamiltonian cycle problem**Hamiltonian 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 problem**Hamiltonian 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 problem**Hamiltonian 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 TSP**INSTANCE: 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 TSP**d(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 TSP**compute the d(u,v) compute MST T weight(T) OPT**Metric TSP**compute the d(u,v) compute MST T weight(T) OPT 2-approximation algorithm**Euler tour**when can a graph be drawn without lifting a pen, and without drawing the same edge twice?**Euler tour**when can a graph be drawn without lifting a pen, and without drawing the same edge twice? if we want to end where we started?**Metric TSP**weight(T) OPT weight(M) OPT/2 compute the d(u,v) compute MST T find a min-weight perfect matching on odd-degree vertices of T 1.5-approximation algorithm**Optimization problems**INSTANCE FEASIBLE SOLUTIONS c: SOLUTIONS R+ OPT= min c(T) T FEASIBLE SOLUTIONS -APPROXIMATION ALGORITHM INSTANCE T c(T) OPT**-APPROXIMATION ALGORITHM**INSTANCE T c(T) OPT PTAS Polynomial-time approximation scheme polynomial-time (1+)-approximation algorithm for any constant >0 FPTAS Fully polynomial-time approximation scheme (1+)-approximation algorithm running in time poly(INPUT,1/)