3.10 Otros contratos

Problems

3.1h _{Consider the following decision problem.}

PARTITION

Input: a finite set of positive integersA.

Question: is there a partition ofA=B_∪˙ Csuch that

b∈B

b=

c∈C

c?

Show that PARTITION≤mSUBSET SUM. (SUBSET SUM is defined on page 43.)

3.2h If A⊆∗₀ then Ac_{= {x∈}∗

0 |x∈ A}. Show that A≤m B implies

Ac_≤ m Bc.

3.3h Show that ifP=NPthen there is a polynomial time algorithm which, when given a SAT formula f, will output ‘unsatisfiable’ if f is unsat- isfiable or a satisfying truth assignment if one exists.

3.4h _{Show thatk-COL isNP-complete fork≥4.}

3.5h _{Given a graphG=(V,E) and an integerk≥1 avertex coverof orderk}

3.7 NPrevisited – non-deterministic Turing machines 65

at least one vertex fromW. Show that the problem VERTEX COVER defined below isNP-complete.

VERTEX COVER

Input: a graphGand an integerk.

Question: doesGhave a vertex cover of orderk?

3.6h _{Show that the following subproblem of 3-COL is stillNP-complete.}

3-COL MAX DEGREE 4

Input: a graphGin which every vertex has degree at most 4. Question: isG3-colourable?

3.7a _{Does the following decision problem belong toPorNP?}

GOLDBACH

Input: an even integern ≥2.

Question: do there exist prime numberspandqsuch thatn= p+q? 3.8 The following decision problems are not known to belong toNP. In

each case explain why it is difficult to produce a suitable certificate. (a) UNSAT

Input: a Boolean CNF formula f. Question: is f unsatisfiable? (b) MAX CLIQUE

Input: a graphGand an integerk.

Question: iskthe maximum order of a clique inG? 3.9h Prove that MAX CLIQUE belongs toPSPACE. 3.10b _{Consider the following problem.}

TRAVELLING SALESMAN

Input: a list of citiesc1, . . . ,cnand ann×nsymmetric matrix of pos- itive integers giving the distances between each pair of cities.

Output: a shortest tour of the cities, where a tour is an ordering of the cities and the length of a tour is the sum of the distances between consecutive cities (including the distance from the last back to the first). Assuming that HAMILTON CYCLE (defined on page 43) is NP- complete show that TRAVELLING SALESMAN isNP-hard.

3.11h _{The chromatic number of a graphGis defined by} χ(G)=min{k|Gisk-colourable}. Show that computingχ(G) isNP-hard.

66 3 Non-deterministic computation

3.13h _{Two languages are said to be Turing equivalent if they are Turing}

reducible to each other. Prove that any twoNP-complete languages are Turing equivalent.

3.14h Prove that ifA∈co-NPandBisNP-complete thenA≤T B. 3.15a _{LetNPCdenote the class of NP-complete languages and let NPTC}

denote the set of languages inNPwhich are complete under Turing reductions. Prove thatNPC⊆NPTC. Is the containment strict?

Further notes

The notions of both polynomial and Turing reducibility were familiar tools in recursive function theory, as was the notion of nondeterminism. The class of languagesNPwas introduced in 1971 by S. Cook who proved that SAT was NP-complete under Turing reducibility. Karp (1972) then used SAT to show that 21 other natural problems wereNP-complete under polynomial reductions. These included VERTEX COVER, CLIQUE, HAMILTON CYCLE andk-COL (k≥3).

Independently Levin (1973) developed a similar theory using tilings rather than satisfiability, with the result that Theorem 3.10 is sometimes referred to as the Cook–Levin theorem.

It should also be noted that several authors/texts use Turing reducibility rather than polynomial reducibility in their definition ofNP-completeness. It is also interesting to note that G¨odel may have been the first to consider the complexity of anNP-complete problem as, according to Hartmanis (1989), he asked von Neumann in a (1956) letter how many Turing machine steps are needed to verify that a Boolean formula is true.

The book by Garey and Johnson (1979) contains a vast array ofNP-complete problems from a wide range of disciplines.

The proof that PRIMES is in Pby Agrawal, Kayal and Saxena (2002) aroused widespread interest in both cryptographic and complexity communi- ties. Whether it will lead to a fast (practical) deterministic algorithm for testing primality is a question of ongoing research interest.

Both PRIMES and GRAPH ISOMORPHISM were discussed in Cook’s original 1971 paper and it is intriguing to consider whether there will one day be a proof that the latter is also inP.

It is now more than twenty years since Luks (1982) showed that testing graph isomorphism for graphs of degree at mostdis polynomial for any fixedd. (The algorithm of Luks is polynomial in the number of vertices but exponential ind.)

4