CSCI 2670Introduction to Theory of

Computing

November 23, 2004

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Agenda

• Today– Section 7.3

• Next week– Section 7.4

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Announcement

• Office hours today 2:30 – 3:30• No homework this week

– No more graded homework– I will assign problems next week, but

you won’t have to turn them in

• Quiz next Wednesday (last quiz!)

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Last week

• Big-O and small-o notation• Computational complexity depends

on– Type of machine– Encoding

• The class P– Algorithms that can be solved in O(nk)

time on a single-tape deterministic TM using reasonable encoding

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Solving vs. verifying

• What if we can’t solve the problem in O(nk) time?

• Given a problem and a potential solution, can we verify the solution is correct?

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Example

• The bin-packing problem– Given a set of n items with weights w1,

w2, …, wn, and k bins that can hold a maximum weight of 1, can we place these items the bins?

• There is no known O(nk) solution to this problem

• What if we have a potential solution– b1, b2, …, bn 1 bi k

• Can we verify it in O(nk) time?

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Verifier

M = “On input <w1, …, wn, b1, …, bn, k>

1. Initialize s1, s2, …, sk to 0

2. For i = 1, …, n3. if bi {1, 2, …, k} reject

4. sb_i = sb_i + wi

5. if sb_i > 1 reject

6. Next i7. Accept

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The class NP

Definition: A verifier for a language A is an algorithm V, whereA={w|V accepts <w,c> for some

string c}The string c is called a certificate of membership in A.

Definition: NP is the class of languages that have polynomial-time verifiers.

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Why NP?

• The N in NP stands for non-deterministic

• Any language in NP can be non-deterministically solved in polynomial time using the verifier– Guess the certificate– Verify

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Other problems in NP

• The vertex cover problem– Given a graph G = <V,E> and a

number k in N, does there exist a subset V’ of V such that• |V’| = k• For every (u,v)E, either uV’ or vV’

• There is no known polynomial solution to this problem

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Vertex cover

• Can we verify vertex cover in polynomial time?– Yes

• What should the certificate be?– The subset V’

• How do we verify?– Check |V’| = k– Test that each (u,v)E has uV’ or

vV’• Takes O(|E|×k) time

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Group problem 1

• Find a verifier for the subset-sum problem– Given a finite set S N and a target t

N, does there exist a subset S’ S such that the sum of all elements in S’ is equal to t?

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Group problem 2

• Find a verifier for the traveling-salesman problem– Given a weighted graph G (i.e., a

graph where each edge has a associated weight) and a distance d, does there exist a cycle through the graph that visits each vertex exactly once (except for the start/end vertex) and has a total distance d?

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Group problem 3

• Find a verifier for the set partition problem– Given a set S of number, does there

exist a subset A of S such that the sum of all elements in A is equal to the sum of all elements in S-A?

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Group problem 4

• Find a verifier for the independent set problem– Given a graph G = <V,E> and a

number k, does there exist a subset V’ of V such that |V’|=k and no edge in E connects 2 elements of V’?

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Group problem 5

• Find a verifier for the graph 3-colorability problem– Given a graph G = <V,E> is there a

way of assigning the colors red, blue and green to the vertices of G such that no edge connects two nodes of the same color?

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Group problem 6

• Find a verifier for the cyclic ordering problem– Given a finite set A and a collection C

of three-tuples of distinct elements of A (i.e., for each (x,y,z)C, the elements x, y, and z are all in A and they are all different), is there a ono-to-one mapping f:A→{1, 2, …, |A|} such that for each (a,b,c) in C, f(a) < f(b) < f(c)?