P vs. NP

Theorem 8.15.

If a decision problem P is NP-complete and P ∈ P, then P = NP.

Proof: See definition of NP-completeness and Lemma 8.6.

There are two possible szenarios for the shape of the complexity world:

NP

PNP-c

scenario A

P = NP = NP-c

scenario B

I It is widely believed that P 6= NP, i. e., scenario A holds.

I Deciding whether P = NP or P 6= NP is one of the seven milleniumprize problems established by the Clay Mathematics Institute in 2000.

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Complexity Class coNP and coNP-Complete Problems

Definition 8.16.

i The complement of decision problem P = (X ,Y ) is P̄ := (X ,X \ Y ).ii coNP := {P̄ | P ∈ NP}iii A problem P ∈ coNP is coNP-complete if all problems in coNP

polynomially transform to P.

Theorem 8.17.

i P is NP-complete if and only if P̄ is coNP-complete.ii Unless NP = coNP, no coNP-complete problem is in NP.

iii Unless P = NP, there are problems in NP that are neither in P norNP-complete.

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Complexity Landscape (Widely Believed)

coNPNP

PNP-c coNP-c

Remark.I Almost all problems that are known to be in NP ∩ coNP are also

known to be in P.I One of few exceptions used to be the problem Primes (given a

positive integer, decide whether it is prime), before it was shown tobe in P by Agrawal, Kayal, and Saxena in 2002.

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The Asymmetry of Yes and NoExample:For a given TSP instance, does there exist a tour of cost ≤ 1573084?

Pictures taken from www.tsp.gatech.edu

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Partition Problem

Partition Problem

Given: positive integers a1, . . . , an ∈ Z>0.

Task: decide whether there exists S ⊆ {1, . . . , n} with∑i∈S

ai =∑

i∈{1,...,n}\S

ai .

Theorem 8.18.

i The Partition problem can be solved in pseudopolynomial time.

ii The Partition Problem is NP-complete.

Proof of (i): . . .

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www.tsp.gatech.edu

Strongly NP-Complete Problems

Let P = (X ,Y ) be a decision problem, p : Z→ Z a polynomial function.I Let Xp ⊂ X denote the subset of instances x where the absolute value

of every (integer) number in the input x is at most p(size(x)).

I Let Pp := (Xp,Y ∩ Xp).

Definition 8.19.

A problem P ∈ NP is strongly NP-complete if Pp is NP-complete forsome polynomial function p : Z→ Z.

Theorem 8.20.

Unless P = NP, there is no pseudopolynomial algorithm for any stronglyNP-complete problem P.

Proof: Such an algorithm would imply that the NP-complete problem Ppis in P and thus P = NP.

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Strongly NP-Complete Problems (cont.)Examples:

I The decision problem corresponding to the Traveling SalespersonProblem (TSP) is strongly NP-complete.

I The Partition problem is NP-complete but not strongly NP-complete(unless P = NP, by Theorem 8.20).

I The 3-Partition problem below is strongly NP-complete.

3-Partition Problem

Given: positive integers a1, . . . , a3n,B ∈ Z>0 with∑3n

i=1 ai = nB.

Task: decide whether the index set {1, . . . , 3n} can be partitioned into ndisjoint subsets S1, . . . ,Sn such that

∑i∈Sj ai = B for j = 1, . . . , n.

Remark.I The 3-Partition problem remains strongly NP-complete if one adds

the requirement that B/4 < ai < B/2 for all i = 1, . . . , 3n.I Notice that each Sj must contain exactly three indices in this case.

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NP-Hardness

Definition 8.21.

Let P be an optimization or decision problem.i P is NP-hard if all problems in NP polynomially reduce to it.ii P is strongly NP-hard if Pp is NP-hard for some polynomial

function p.

Remarks.I A sufficient criterion for an optimization problem to be (strongly)

NP-hard is the (strong) NP-completeness or (strong) NP-hardness ofthe corresponding decision problem.

I It is open whether each NP-hard decision problem P ∈ NP isNP-complete (difference between Turing and Karp reduction!).

I An example of an NP-hard decision problem that does not appear tobe in NP (and thus might not be NP-complete) is the following:

Given an instance of SAT, decide whether the majority of all truthassignments satisfy all clauses.

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NP-Hardness (cont.)

Theorem 8.22.

Unless P = NP, there is no polynomial time algorithm for any NP-harddecision or optimization problem.

Proof: Clear.

Remark: It is thus widely believed that problems like the TSP and all otherNP-hard optimization problems cannot be solved in polynomial time.

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Complexity of Linear Programming

I As discussed in Chapter 4, so far no variant of the simplex methodhas been shown to have a polynomial running time.

I Therefore, the complexity of Linear Programming remainedunresolved for a long time.

I Only in 1979, the Soviet mathematician Leonid Khachiyan provedthat the so-called ellipsoid method earlier developed for nonlinearoptimization can be modified in order to solve LPs in polynomial time.

I In November 1979, the New York Times featured Khachiyan and hisalgorithm in a front-page story.

I We will give a sketch of the ellipsoid method and its analysis.

I More details can, e. g., be found in the book of Bertsimas & Tsitsiklis(Chapter 8) or in the book Geometric Algorithms and CombinatorialOptimization by Grötschel, Lovász & Schrijver (Springer, 1988).

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New York Times, Nov. 27, 1979 238

IntroductionLinear Programming BasicsThe Geometry of Linear ProgrammingThe Simplex MethodDuality TheoryOptimal Trees and PathsMaximum Flow ProblemsMinimum-Cost Flow ProblemsNP-Completeness