Fundamental CS Skill 3 Oral Skill 12 NPCompleteness

Fundamental CS Skill 3 Oral Skill 12

NP-Completeness n NP-complete problem Balanced Complete Bipartite Subgraph n n INSTANCE: Bipartite graph G = (V, E), positive integer K |V| QUESTION: Are there two disjoint subsets V 1, V 2 V such that |V 1| = |V 2| = K and such that, for each u V 1 and each v V 2 , {u, v} E? A E B F C G D H

NP-Completeness n Finding Large K-itemsets n n INSTANCE: Given a database of transaction records QUESTION: Is there an f-frequent itemset of size J Egg 1 0 0 Rice 1 1 1 Oil 1 1 1 Juice 0 0 1 1 0

NP-Completeness n n We can transform the graph problem into itemset problem. n For each vertex in V 1, create a transaction n For each vertex in V 2, create an item n For each edge (u, v), create a purchase of item v in transaction u n f K n J K Is there a K-frequent itemset of size K? A E B F C G D H A B C D E 1 1 1 0 F 1 1 G 1 1 1 0 H 0 0 1 0

NP-Completeness n n It is easy to verify that solving the problem Finding Large K-itemsets is equal to solving problem Balanced Complete Bipartite Subgraph Finding Large K-itemsets is NP-hard.

Methods to prove that a problem P is NP-hard n n n Step 1: Find an existing NP-complete problem (e. g. , complete bipartite graph) Step 2: Transform this NP-complete problem to P (in polynomial-time) Step 3: Show that solving the “transformed” problem is equal to solving “original” NP-complete problem

Exercise 1 n Maximum Route Problem n n INSTANCE: We are given n airports. Let V be the set of airports. We are also given m plane routes from one airport to another airport. We are also given an integer J and an integer K QUESTION: Can we find a subset V’ V such that |V’| <= K and the total number of plane routes within V’ is at least J. Definition: A plane route within V’ is defined to be a plan route from an airport in V’ to another airport in V’ Prove that the above problem is NP-hard.

Exercise 1 n HINTS n CLIQUE n n INSTANCE: Given a graph G = (V, E) and a positive integer K |V| QUESTION: Is there a subset V' V such that |V'| K and every two vertices in V' are joined by an edge in E ?

Exercise 2 n Clustering Problem n n n INSTANCE: We are given a table T with m attributes. There are n tuples in table T. We are also given an integer J and an integer K QUESTION: Can we find a set of clusters such that (1) all tuples in all clusters form the table T (2) each cluster has at least J tuples (3) the total cost of clustering is at most K Prove that the above problem is NP-hard.

Exercise 2 n Cost of a cluster t 1 t 2 t 3 n A 1 5 6 7 A 2 2 A 3 4 A 4 1 1 1 The cost of a cluster is equal to the total number of “different” attributes multiplied by the total number of tuples. In this example, it is equal to 2 x 3 =6. Total Cost of Clustering is equal to the sum of costs of all clusters

Exercise 2 n HINTS n EDGE PARTITION INTO TRIANGLES n n Given a graph G = (V, E) with |E| = 3 m for some integer m, Can the edges of G be partitioned into m edgedisjoint triangles?

How can the paper be accepted? n First, you need to understand why you need to conduct research. Only for graduation? You want to let others know that you have some contributions to the field. n Then, after knowing the above answer, you know the reason why the paper can be accepted.

How can the paper be accepted? n What factor can improve the chance of the paper acceptance? Improving the quality of the paper including more technical materials presenting the paper clearly including more real applications including comprehensive experiments giving the motivations clearly stating your contributions clearly

How can the paper be accepted? n Is it true that if you try your best to improve the quality of the paper, then your paper MUST be accepted? NO. Why?

How can the paper be accepted? n There a lot of factors which can affect the paper acceptance. Quality of the paper If the quality of the paper is higher, then the chance of the paper acceptance is higher. Competition on the same topic Good/Bad? If there are many papers on the same topic, it is more difficult that the paper is accepted. Bad: If the quality of the paper is low, then the chance of acceptance is low Good: If the quality of the paper is high, then the chance of acceptance is high. Besides, the number of citations will be higher in the future.

How can the paper be accepted? n There a lot of factors which can affect the paper acceptance. Luck from reviewers Usually, there are 3~5 reviewers. If one of the reviewers is very picky, then it is more difficult that the paper is accepted. Limited number of papers accepted in some venues Within a conference, there are only a limited number of papers to be accepted. Even if the paper has a high quality, it is possible that the paper will be rejected at the end.

How can the paper be accepted? n What can we do after we know these real factors? After submitting a paper, you need to think out that it is difficult that the paper is accepted. Of course, we hope that the paper is accepted because we have already tried our best to improve the quality of the paper. Even if the paper is rejected, then it is OK. Don’t be sad. We can receive a review report from reviewers. We can improve our paper again according to reasonable points raised by reviewers. After we submit a paper, we can still work on “another” paper so that the “expected” number of papers accepted can be increased (even though there is a luck factor inside. )

Exercise 3 n Please listen to the tape and follow the intonation of the speaker n n audio 15. mp 3 For example, the physically thinner, ectomorphic body types tended to be more sensitive, selfconscious, introspective, and generally quiet people. At the opposite end, the physically heavier, endomorhpic body types were sociable, easygoing, relaxed and pleasure-loving.

Reminder n Please do exercise 1, exercise 2 and exercise 3