The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Lecture 21 - vertex cover Thus finding the largest clique is NP-complete: If VC is a vertex cover in G, then V-VC is a clique in G'. If C is a clique in G, V-C is a vertex cover in G'. Listen To Part 25-4 36.5-1 Prove that subgraph isomorphism is NP-complete. 1. Guessing a subgraph of G and proving it is isomorphism to h takes time, so it is in NP. 2. Clique and subgraph isomorphism. We must transform all instances of clique into some instances of subgraph isomorphism. Clique is a special case of subgraph isomorphism! Thus the following reduction suffices. Let G=G' and , the complete subgraph on k nodes. Listen To Part 25-5 Instance: A set of integers S and a target integer t. Integer Partition (Subset Sum) Problem: Is there a subset of S which adds up exactly to t? Example: and T=3754 Answer: 1+16+64+256+1040+1093+1284 = T Observe that integer partition is a number problem, as opposed to the graph and logic problems we have file:///E|/LEC/LECTUR16/NODE21.HTM (5 of 7) [19/1/2003 1:35:28]
Lecture 21 - vertex cover seen to date. Theorem: Integer Partition is NP-complete. Proof: First, we note that integer partition is in NP. Guess a subset of the input number and simply add them up. To prove completeness, we show that vertex cover integer partition. We use a data structure called an incidence matrix to represent the graph G. Listen To Part 25-6 How many 1's are there in each column? Exactly two. How many 1's in a particular row? Depends on the vertex degree. The reduction from vertex cover will create n+m numbers from G. The numbers from the vertices will be a base-4 realization of rows from the incidence matrix, plus a high order digit: ie. becomes . The numbers from the edges will be . The target integer will be file:///E|/LEC/LECTUR16/NODE21.HTM (6 of 7) [19/1/2003 1:35:28]
- Page 973 and 974: Lecture 14 - data structures for gr
- Page 975 and 976: Lecture 15 - DFS and BFS Next: Lect
- Page 977 and 978: Lecture 15 - DFS and BFS In a DFS o
- Page 979 and 980: Lecture 15 - DFS and BFS It could u
- Page 981 and 982: Lecture 15 - DFS and BFS Algorithm
- Page 983 and 984: Lecture 16 - applications of DFS an
- Page 985 and 986: Lecture 16 - applications of DFS an
- Page 987 and 988: Lecture 16 - applications of DFS an
- Page 989 and 990: Lecture 17 - minimum spanning trees
- Page 991 and 992: Lecture 17 - minimum spanning trees
- Page 993 and 994: Lecture 17 - minimum spanning trees
- Page 995 and 996: Lecture 17 - minimum spanning trees
- Page 997 and 998: Lecture 17 - minimum spanning trees
- Page 999 and 1000: Lecture 18 - shortest path algorthm
- Page 1001 and 1002: Lecture 18 - shortest path algorthm
- Page 1003 and 1004: Lecture 18 - shortest path algorthm
- Page 1005 and 1006: Lecture 18 - shortest path algorthm
- Page 1007 and 1008: Lecture 18 - shortest path algorthm
- Page 1009 and 1010: Lecture 19 - satisfiability Next: L
- Page 1011 and 1012: Lecture 19 - satisfiability Why? Th
- Page 1013 and 1014: Lecture 19 - satisfiability between
- Page 1015 and 1016: Lecture 19 - satisfiability Note th
- Page 1017 and 1018: Lecture 20 - integer programming Ex
- Page 1019 and 1020: Lecture 20 - integer programming Ne
- Page 1021 and 1022: Lecture 21 - vertex cover Proof: VC
- Page 1023: Lecture 21 - vertex cover Question:
- Page 1027 and 1028: Lecture 22 - techniques for proving
- Page 1029 and 1030: Lecture 22 - techniques for proving
- Page 1031 and 1032: Lecture 22 - techniques for proving
- Page 1033 and 1034: Lecture 22 - techniques for proving
- Page 1035 and 1036: Lecture 23 - approximation algorith
- Page 1037 and 1038: Lecture 23 - approximation algorith
- Page 1039 and 1040: Lecture 23 - approximation algorith
- Page 1041 and 1042: Lecture 23 - approximation algorith
- Page 1043 and 1044: Lecture 23 - approximation algorith
- Page 1045 and 1046: About this document ... Up: No Titl
- Page 1047 and 1048: 1.1.6 Kd-Trees ● Point Location
- Page 1049 and 1050: 1.1.3 Suffix Trees and Arrays Send
- Page 1051 and 1052: 1.5.1 Clique ● Vertex Cover Go to
- Page 1053 and 1054: 1.6.2 Convex Hull Related Problems
- Page 1055 and 1056: Visual Links Index file:///E|/WEBSI
- Page 1057 and 1058: Visual Links Index file:///E|/WEBSI
- Page 1059 and 1060: Visual Links Index file:///E|/WEBSI
- Page 1061 and 1062: Visual Links Index file:///E|/WEBSI
- Page 1063 and 1064: Visual Links Index file:///E|/WEBSI
- Page 1065 and 1066: Visual Links Index file:///E|/WEBSI
- Page 1067 and 1068: Visual Links Index file:///E|/WEBSI
- Page 1069 and 1070: Visual Links Index file:///E|/WEBSI
- Page 1071 and 1072: Visual Links Index file:///E|/WEBSI
- Page 1073 and 1074: Visual Links Index file:///E|/WEBSI
Lecture 21 - vertex cover<br />
Thus finding the largest clique is NP-complete:<br />
If VC is a vertex cover in G, then V-VC is a clique in G'. If C is a clique in G, V-C is a vertex cover in G'.<br />
Listen To Part 25-4<br />
36.5-1 Prove that subgraph isomorphism is NP-complete.<br />
1. Guessing a subgraph of G and proving it is isomorphism to h takes time, so it is in NP.<br />
2. Clique and subgraph isomorphism. We must transform all instances of clique into some instances<br />
of subgraph isomorphism. Clique is a special case of subgraph isomorphism!<br />
Thus the following reduction suffices. Let G=G' and , the complete subgraph on k nodes.<br />
Listen To Part 25-5<br />
Instance: A set of integers S and a target integer t.<br />
Integer Partition (Subset Sum)<br />
Problem: Is there a subset of S which adds up exactly to t?<br />
Example: and T=3754<br />
Answer: 1+16+64+256+1040+1093+1284 = T<br />
Observe that integer partition is a number problem, as opposed to the graph and logic problems we have<br />
file:///E|/LEC/LECTUR16/NODE21.HTM (5 of 7) [19/1/2003 1:35:28]
Change language