NP-Completeness Greg Plaxton Theory in Programming Practice, Spring - - PowerPoint PPT Presentation

NP-Completeness

Greg Plaxton Theory in Programming Practice, Spring 2004 Department of Computer Science University of Texas at Austin

Decision Problems

• A decision problem is a computational problem where the answer to

any given instance of the problem is either “yes” or “no” – Instances of the problem with a “yes” answer are called positive instances – Instances of the problem with a “no” answer are called negative instances

• Here are some examples:

– Given a text string t and a pattern string p, does p occur in t? – Given an instance of an LP, and a real value B, is there a feasible solution to the LP for which the value of the objective function is at least B? – Maximum Independent Set: Given an undirected graph G and a nonnegative integer k, does G contain a set U of at least k vertices such that no two vertices in U are connected by an edge?

Theory in Programming Practice, Plaxton, Spring 2004

Polynomial-Time Algorithms

• A polynomial-time algorithm is an algorithm with worst-case complexity

that is upper bounded by some polynomial in the input size

• Virtually all of the algorithms studied in this course run in polynomial

time – Can you think of an exception?

Theory in Programming Practice, Plaxton, Spring 2004

The Class P

• The acronym “P” stands for “polynomial time”
• A decision problem is in the class P if there is a polynomial-time

algorithm to determine the answer (“yes” or “no”) for any given instance

Theory in Programming Practice, Plaxton, Spring 2004

The Class NP

• The acronym “NP” stands for nondeterministic polynomial time
• There are several equivalent ways to formally define the class NP
• One approach involves the notion of nondeterministic computation,

thereby justifying the choice of the acronym NP

• We will instead define the class NP in terms of polynomial-time

verifiability

Theory in Programming Practice, Plaxton, Spring 2004

The Class NP

• A decision problem X is in NP if for every positive instance I of X,

there is a short and polynomial-time verifiable certificate proving that I is a positive instance – The certificate is just a string (particular to the instance I) – By “short” we mean that the size of the certificate is bounded by some polynomial in the size of the instance I – By “polynomial-time verifiable” we mean that there is a polynomial- time algorithm that can check whether any given string is indeed a certificate proving that I is a positive instance

• We will discuss the notion of “polynomial-time verifiability” in greater

detail momentarily

• But first let’s look at a few examples of problems in NP

Theory in Programming Practice, Plaxton, Spring 2004

Examples of Problems in NP

• All decision problems in P also belong to NP

– We can use the empty string as the short certificate – The polynomial-time verifier can simply run the polynomial-time decision procedure for determining whether the given instance is a positive instance

• Satisfiability: Given a propositional formula f (such as the input to

the Davis-Putnam procedure discussed earlier in the course), is f satisfiable?

• Clique: Given an undirected graph G, does G contain a clique (i.e., a

completely interconnected set of vertices) of size at least k?

• Hamiltonian Cycle: Given an undirected graph G, does G contain a

simple cycle visiting all the vertices?

Theory in Programming Practice, Plaxton, Spring 2004

Polynomial-Time Verification

• Let X be a decision problem
• We say that X is polynomial-time verifiable if there exists a polynomial-

time algorithm A with the following properties – The input to A consists of an instance I of X and a string s such that the length of s is bounded by some polynomial in the size of I – The output of A is either “yes” or “no” – If I is a negative instance of X, then the output of A is “no”, regardless of the value of s – If I is a positive instance of X, then there is at least one choice of s for which A outputs “yes”

• We think of A as the “verifier” and s as a purported “short proof”

that the given instance I is a positive instance of X

Theory in Programming Practice, Plaxton, Spring 2004

The Complexity of Problems in NP

• Since problems in P also belong to NP, some problems in NP are “easy”

in the sense that they are known to be solvable by a polynomial-time algorithm

• It is an open problem whether all problems in NP can be solved in

polynomial time, i.e., whether NP is equal to P

• It is widely conjectured that some problems in NP require exponential

time, so that NP is not equal to P

• Still, the theory of NP-completeness, to be discussed next, sheds

considerable light on the structure of the class NP

Theory in Programming Practice, Plaxton, Spring 2004

NP-Completeness

• A decision problem X in NP is defined to be NP-complete if existence
• f a polynomial-time algorithm for X implies that P equals NP, i.e.,

that every problem in NP admits a polynomial-time algorithm

• Qualitatively, such an NP-complete problem is as hard to solve (up to

polynomial factors) as any problem in NP

• A priori, there is no particular reason to believe that NP-complete

problems exist – An alternative hypothesis might be that there is no such “hardest” problem in NP – Instead, we might imagine that there are two or more problems in NP of incomparable hardness – In fact, NP-complete problems exist, as discussed on the next slide

Theory in Programming Practice, Plaxton, Spring 2004

The Cook-Levin Theorem

• The satisfiability problem is NP-complete
• The proof of this theorem is not outrageously long but is beyond the

scope of the present course

Theory in Programming Practice, Plaxton, Spring 2004

Other NP-Complete Problems

• Given that Cook-Levin have done the “heavy lifting” of establishing

that satisfiability is NP-complete, we can now use the following simpler approach to establish the NP-completeness of a given problem X (that actually is NP-complete) – We still need to argue that X is in NP; Cook-Levin doesn’t help us here but in practice this step tends to be easy – We then prove that a polynomial-time algorithm for X could be used as a subroutine to obtain a polynomial-time algorithm for satisfiability (or some other known NP-complete problem)

• The literature contains literally thousands of examples of problems that

have been proven to be NP-complete in the above manner

• The maximum independent set, maximum clique, and Hamiltonian

cycle problems mentioned earlier in this lecture are all examples of NP-complete problems

Theory in Programming Practice, Plaxton, Spring 2004

NP-Hardness

• Thus far we have restricted our discussion to decision problems
• A problem X is said to be NP-hard if a polynomial-time algorithm for

X implies that P equals NP – An NP-hard problem need not be a decision problem – An NP-hard decision problem need not be in NP

• There is a significant body of literature devoted to the design

and analysis of provably-good approximation algorithms for NP-hard problems – Interestingly, the approximation versions of NP-complete problems can differ dramatically in terms of approximability

Theory in Programming Practice, Plaxton, Spring 2004

Other Complexity Classes

• Complexity theorists have defined a wide variety of complexity classes
• Some of these classes properly contain NP, i.e., they include problems

that are harder than NP-complete problems

• Some of these classes refine the class P, i.e., they define problems that

are in some sense easier than the hardest problems in P (the P-complete problems)

• Still, the P versus NP question is considered the central problem in

complexity theory – It is felt to best represent the boundary between “tractable” and “intractable” problems

Theory in Programming Practice, Plaxton, Spring 2004