DOCSLIB.ORG

1 RP and BPP

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 RP and BPP princeton university cos 522: computational complexity Lecture 6: Randomized Computation Lecturer: Sanjeev Arora Scribe:Manoj A randomized Turing Machine has a transition diagram similar to a nondeterministic TM: multiple transitions are possible out of each state. Whenever the machine can validly more than one outgoing transition, we assume it chooses randomly among them. For now we assume that there is either one outgoing transition (a deterministic step) or two, in which case the machine chooses each with probability 1/2. (We see later that this simplifying assumption does not effect any of the theory we will develop.) Thus the machine is assumed to have a fair random coin. The new quantity of interest is the probability that the machine accepts an input. The probability is taken over the coin flips of the machine. 1 RP and BPP Definition 1 RP is the class of all languages L, such that there is a polynomial time randomized TM M such that 1 x ∈ L ⇔ Pr[M accepts x] ≥ (1) 2 x ∈ L ⇔ Pr[M accepts x]=0 (2) We can also define a class where we allow two-sided errors. Definition 2 BPP is the class of all languages L, such that there is a polynomial time randomized TM M such that 2 x ∈ L ⇔ Pr[M accepts x] ≥ (3) 3 1 x ∈ L ⇔ Pr[M accepts x] ≤ (4) 3 As in the case of NP, we can given an alternative definiton for these classes. Instead of the TM flipping coins by itself, we think of a string of coin flips provided to the TM as an additional input. Definition 3 BPP is the class of all languages L, such that there is a polynomial time deterministic TM M and c>0 such that 2 x ∈ L ⇔ Pr |x|c [M accepts x] ≥ (5) r∈{0,1} 3 1 x ∈ L ⇔ Pr |x|c [M accepts x] ≤ (6) r∈{0,1} 3 1 A. Q. October 5, 2005 a 22 We make a few observations about randomized classed. First, RP NP, since a 0 We make a few observations about randomized classed. First, RP ⊆ NP, since a to collide is Thus, the number of There are How many for but once these are chosen, exactly one choice ⊆ 2 randomrandom string string that that causes causes the the machine machine to to accept accept is is a a certificate certificate that that the the input input is is in in the the language.language. We We do do not not know know if if BPPBPP isis in in NPNP.. Note Note that that BPPBPP isis closed closed under under complemen- complemen- tation.tation.We make We We do do a not not few know know observations of of any any complete complete about randomized problems problems for for classed. these these classes. classes. First, RP One One di difficultyffiNPculty, since seems seems a ⊆ randomtoto be be that that string the the problem problem that causes of of determining determining the machine whether whether to accept or or not not is a a a certificate given given randomized randomized that the polynomial polynomial input is in time time the = language.TMTM is is an an RP RPWe machine machinedo not know is is undecidable. undecidable. if BPP is in NP. Note that BPP is closed under complemen- tation. We do not know of any complete problems for these classes. One difficulty seems to1.11.1 be that Probability Probability the problem Amplification Amplification of determining whether or not a given randomized polynomial time TM is an RP machine is undecidable. 22 11 Proof (completed) InIn the the definition definition of of BPPBPP aboveabove the the actual actual values values in in the the place place of of 3 andand 3 areare not not crucial, crucial, 3 n 3 n asas we we observe observe below. below. These These two two numbers numbers can can be be replaced replaced by by 1 1 − 11//22n andand 1 1//22n withoutwithout ⎝ ⎜ ⎜ ⎛ − a 1.1 Probability Amplification changingchanging the the class. class. By repeatedly running a machine M accepting a BPP language2 L with1 probabilities In theBy definition repeatedly of runningBPP above a machine the actualM accepting values in a theBPP placelanguage of 3 andL with3 are probabilities not crucial, as above, and taking the majority vote of the different decisions it makes,n wen can get a asas we above, observe and below. taking Thesethe majority two numbers vote of can the be different replaced decisions by 1 1 it/2 makes,and 1 we/2 canwithout get a − ⎝ ⎜ ⎜ ⎛ much better probability of a correct answer. Using a polynomial (in input size) number 0 changingmuch better the class.probability of a correct answer. Using a polynomial (in input size) number ofof repeated repeatedBy repeatedly trials, trials, werunning we get get an an a exponentially exponentially machine M accepting small small probability probability a BPP oflanguage of deciding decidingL the thewith input input probabilities wrongly. wrongly. asThisThis above, follows follows and from from taking bounding bounding the majority the the tail tail vote of of a a binomial binomialof the diff distribution distributionerent decisions using using it Cherno Chernoff makes,ff webounds. bounds. can get a much better probability of a correct answer. Using a polynomial (in input size) number − causes of2 repeated Theorems trials, we about get an exponentially BPP small probability of deciding the input wrongly. This2 follows Theorems from bounding about the BPP tail of a binomial distribution using Chernoff bounds. NowNow we we show show that that all all BPPBPP languageslanguages have have polynomial polynomial sized sized circuits. circuits. 2TheoremTheorem Theorems 1 1 about BPP ∑ i BPPBPP ⊆ PP//poly Now we⊆ showpoly that all BPP languages have polynomial sized circuits. Proof:Proof: For any language LL ∈ BPPBPP consider a TM MM (taking a random string as additional = For any language ∈ consider a TM (taking a random string as additional r Theorem 1 input) which decides any input xx with an exponentially low probability of error; such an M BPPinput) whichP/poly decides any input with an exponentially low probability of error; such an M Copyright © 2001-5 by Erik D. De existsexists⊆ by by the the probability probability amplification amplification arguments arguments mentioned mentioned earlier. earlier. In In particular particular let let the the (n+1) 1 probability M(x, r) being wrong (taken over r)be 1/2 (n+1), where n = x . Say that an Proof:probabilityForM any(x, language r) beingL wrongBPP (takenconsider over ar)be TM M≤ 1(taking/2 a, random where n string= |x|. as Say additional that an ∈ ≤ | | input)rr isis badbad whichforfor xx decidesifif MM((x,x, any r r)) is is input an an incorrect incorrectx with an answer. answer. exponentially Let Let tt bebe low the the probability total total number number of error; of of choices choices such anfor for Mrr.. m (n+1) existsForFor each each byx thex,, at at probability most most t/t/22(n amplification+1) valuesvalues of of argumentsrr areare bad. bad. mentioned Adding Adding over over earlier. all all xx,, In we we particular conclude conclude that thatlet the at at n (n+1) probabilitymostmost 2 2n × t/t/M22((x,n+1) r)valuesvalues being wrong of of rr areare (taken bad bad for for over some somer)bexx.. In In1/ other other2(n+1) words, words,, where at atn least least= x .t/t/ Say22 choices choices that an of of a × ≤ | | rr isnotnotbad bad badfor for forx if every everyM(x,xx.. r) In In is particular particular an incorrect there there answer. is is some some Let one onet be value value the of of totalrr forfor number which which ofMM choices((x,x, r r)) decides decides for r. (n+1) ForxxHecorrectly.correctly.nc eache if wxe ,ch at We Weoos moste can cana ran hard-wiret/ hard-wiredo2m elemvaluesen this thist r ofr rt ofhintointoesre tare/ a2 a c polynomial polynomialh bad.oices, Addingthen th size sizee p overro circuit. circuit.bab allilityx t Givenh, Givena wet the concludere an an is a input inputny x thatfoxxr, ,w the theh atich n (n+1) mostcircuitcircuit 2 simulates simulatest/2 MMvaluesonon ( (x,x, of r r).).r Hence Henceare badLL for∈ PP some/poly./poly.x.✷2 In other words, ath least t/2 choices of r is bad is ×< 1. This provides a nonconstructive∈ proof that there is some one value of r for which M(x,r) decides i r notThe bad next for theorem every x. relates In particularBPPBPP to there the is polynomial some one hierarchy.value of r for which M(x, r) decides m The next theorem relates to the polynomial hierarchy. x correctly. We can hard-wire this r into a polynomial size circuit. Given an input x, the TheoremTheorem 2 2 choices for each of ✷ ( circuit simulates M on (x, r). Hence L P/poly. p p a BPPBPP ⊆ ΣΣp ∩ΠΠp ∈ The⊆ next22 ∩ theorem22 relates BPP to the polynomial hierarchy. x pp Proof:Proof: It is enough to prove that BPPBPP ⊆ ΣΣ2 because BPPBPP is closed under complemen- It is enough to prove that 2 because is closed’s cause under complemen- x Theorem 2 ⊆ tation. p p r BPPtation. Σ Π Suppose2 LL ∈2 BPPBPP, and MM is a randomized TM for LL, that uses mm random bits such Suppose⊆ ∩ ∈ , and is a randomized TM for , that uses random bits such that x L Pr [M(x, r) accepts ] 1 2 −nnpand x L Pr [M(x, r) accepts ] 2 −nn. Proof:that x ∈ItL is⇒ enoughPrrr[M( tox, prover) accepts that ]BPP≥ 1 − 2−Σ2 andbecausex ∈ LBPP⇒ Prisrr[ closedM(x, r under) accepts complemen- ] ≤ 2− .
Load more

Recommended publications
  • Complexity Theory Lecture 9 Co-NP Co-NP-Complete
    Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP.
    [Show full text]
  • On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
    On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly.
    [Show full text]
  • EXPSPACE-Hardness of Behavioural Equivalences of Succinct One
    EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets Petr Janˇcar1 Petr Osiˇcka1 Zdenˇek Sawa2 1Dept of Comp. Sci., Faculty of Science, Palack´yUniv. Olomouc, Czech Rep. [email protected], [email protected] 2Dept of Comp. Sci., FEI, Techn. Univ. Ostrava, Czech Rep. [email protected] Abstract We note that the remarkable EXPSPACE-hardness result in [G¨oller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity ques- tion for simulation preorder of succinct one counter nets (i.e., one counter automata with no zero tests where counter increments and decrements are integers written in binary). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjust- ing a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard “defender-choice” technique. 1 Introduction arXiv:1801.01073v1 [cs.LO] 3 Jan 2018 We concentrate on our contribution, without giving a broader overview of the area here. A remarkable result by G¨oller, Haase, Ouaknine, Worrell [2] shows that model checking a fixed CTL formula on succinct one-counter automata (where counter increments and decre- ments are integers written in binary) is EXPSPACE-hard. Their proof is interesting and nontrivial, and uses two involved results from complexity theory.
    [Show full text]
  • Dspace 6.X Documentation
    DSpace 6.x Documentation DSpace 6.x Documentation Author: The DSpace Developer Team Date: 27 June 2018 URL: https://wiki.duraspace.org/display/DSDOC6x Page 1 of 924 DSpace 6.x Documentation Table of Contents 1 Introduction ___________________________________________________________________________ 7 1.1 Release Notes ____________________________________________________________________ 8 1.1.1 6.3 Release Notes ___________________________________________________________ 8 1.1.2 6.2 Release Notes __________________________________________________________ 11 1.1.3 6.1 Release Notes _________________________________________________________ 12 1.1.4 6.0 Release Notes __________________________________________________________ 14 1.2 Functional Overview _______________________________________________________________ 22 1.2.1 Online access to your digital assets ____________________________________________ 23 1.2.2 Metadata Management ______________________________________________________ 25 1.2.3 Licensing _________________________________________________________________ 27 1.2.4 Persistent URLs and Identifiers _______________________________________________ 28 1.2.5 Getting content into DSpace __________________________________________________ 30 1.2.6 Getting content out of DSpace ________________________________________________ 33 1.2.7 User Management __________________________________________________________ 35 1.2.8 Access Control ____________________________________________________________ 36 1.2.9 Usage Metrics _____________________________________________________________
    [Show full text]
  • The Complexity Zoo
    The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/.
    [Show full text]
  • Glossary of Complexity Classes
    App endix A Glossary of Complexity Classes Summary This glossary includes selfcontained denitions of most complexity classes mentioned in the b o ok Needless to say the glossary oers a very minimal discussion of these classes and the reader is re ferred to the main text for further discussion The items are organized by topics rather than by alphab etic order Sp ecically the glossary is partitioned into two parts dealing separately with complexity classes that are dened in terms of algorithms and their resources ie time and space complexity of Turing machines and complexity classes de ned in terms of nonuniform circuits and referring to their size and depth The algorithmic classes include timecomplexity based classes such as P NP coNP BPP RP coRP PH E EXP and NEXP and the space complexity classes L NL RL and P S P AC E The non k uniform classes include the circuit classes P p oly as well as NC and k AC Denitions and basic results regarding many other complexity classes are available at the constantly evolving Complexity Zoo A Preliminaries Complexity classes are sets of computational problems where each class contains problems that can b e solved with sp ecic computational resources To dene a complexity class one sp ecies a mo del of computation a complexity measure like time or space which is always measured as a function of the input length and a b ound on the complexity of problems in the class We follow the tradition of fo cusing on decision problems but refer to these problems using the terminology of promise problems
    [Show full text]
  • The Weakness of CTC Qubits and the Power of Approximate Counting
    The weakness of CTC qubits and the power of approximate counting Ryan O'Donnell∗ A. C. Cem Sayy April 7, 2015 Abstract We present results in structural complexity theory concerned with the following interre- lated topics: computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting. The first result is a new characterization of the lesser known complexity class BPPpath in terms of more familiar concepts. Precisely, BPPpath is the class of problems that can be efficiently solved with a nonadaptive oracle for the Approximate Counting problem. Similarly, PP equals the class of problems that can be solved efficiently with nonadaptive queries for the related Approximate Difference problem. Another result is concerned with the compu- tational power conferred by CTCs; or equivalently, the computational complexity of finding stationary distributions for quantum channels. Using the above-mentioned characterization of PP, we show that any poly(n)-time quantum computation using a CTC of O(log n) qubits may as well just use a CTC of 1 classical bit. This result essentially amounts to showing that one can find a stationary distribution for a poly(n)-dimensional quantum channel in PP. ∗Department of Computer Science, Carnegie Mellon University. Work performed while the author was at the Bo˘gazi¸ciUniversity Computer Engineering Department, supported by Marie Curie International Incoming Fellowship project number 626373. yBo˘gazi¸ciUniversity Computer Engineering Department. 1 Introduction It is well known that studying \non-realistic" augmentations of computational models can shed a great deal of light on the power of more standard models. The study of nondeterminism and the study of relativization (i.e., oracle computation) are famous examples of this phenomenon.
    [Show full text]
  • A Short History of Computational Complexity
    The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns.
    [Show full text]
  • On Unapproximable Versions of NP-Complete Problems By
    SIAM J. COMPUT. () 1996 Society for Industrial and Applied Mathematics Vol. 25, No. 6, pp. 1293-1304, December 1996 O09 ON UNAPPROXIMABLE VERSIONS OF NP-COMPLETE PROBLEMS* DAVID ZUCKERMANt Abstract. We prove that all of Karp's 21 original NP-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate log (k) of the magnitude of these problems to within any constant factor, where log (k) denotes the logarithm iterated k times, unless NP is recognized by slightly superpolynomial randomized machines. We use the same technique to improve the constant such that MAX CLIQUE is hard to approximate to within a factor of n. Finally, we show that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of -1, 0, 1 matrices. Key words. NP-complete, unapproximable, randomized reduction, clique, counting problems, permanent, 2SAT AMS subject classifications. 68Q15, 68Q25, 68Q99 1. Introduction. 1.1. Previous work. The theory of NP-completeness was developed in order to explain why certain computational problems appeared intractable [10, 14, 12]. Yet certain optimization problems, such as MAX KNAPSACK, while being NP-complete to compute exactly, can be approximated very accurately. It is therefore vital to ascertain how difficult various optimization problems are to approximate. One problem that eluded attempts at accurate approximation is MAX CLIQUE.
    [Show full text]
  • Quantum Computational Complexity Theory Is to Un- Derstand the Implications of Quantum Physics to Computational Complexity Theory
    Quantum Computational Complexity John Watrous Institute for Quantum Computing and School of Computer Science University of Waterloo, Waterloo, Ontario, Canada. Article outline I. Definition of the subject and its importance II. Introduction III. The quantum circuit model IV. Polynomial-time quantum computations V. Quantum proofs VI. Quantum interactive proof systems VII. Other selected notions in quantum complexity VIII. Future directions IX. References Glossary Quantum circuit. A quantum circuit is an acyclic network of quantum gates connected by wires: the gates represent quantum operations and the wires represent the qubits on which these operations are performed. The quantum circuit model is the most commonly studied model of quantum computation. Quantum complexity class. A quantum complexity class is a collection of computational problems that are solvable by a cho- sen quantum computational model that obeys certain resource constraints. For example, BQP is the quantum complexity class of all decision problems that can be solved in polynomial time by a arXiv:0804.3401v1 [quant-ph] 21 Apr 2008 quantum computer. Quantum proof. A quantum proof is a quantum state that plays the role of a witness or certificate to a quan- tum computer that runs a verification procedure. The quantum complexity class QMA is defined by this notion: it includes all decision problems whose yes-instances are efficiently verifiable by means of quantum proofs. Quantum interactive proof system. A quantum interactive proof system is an interaction between a verifier and one or more provers, involving the processing and exchange of quantum information, whereby the provers attempt to convince the verifier of the answer to some computational problem.
    [Show full text]
  • Complexity Theory
    Mapping Reductions Part II -and- Complexity Theory Announcements ● Casual CS Dinner for Women Studying Computer Science is tomorrow night: Thursday, March 7 at 6PM in Gates 219! ● RSVP through the email link sent out earlier this week. Announcements ● Problem Set 7 due tomorrow at 12:50PM with a late day. ● This is a hard deadline – no submissions will be accepted after 12:50PM so that we can release solutions early. Recap from Last Time Mapping Reductions ● A function f : Σ1* → Σ2* is called a mapping reduction from A to B iff ● For any w ∈ Σ1*, w ∈ A iff f(w) ∈ B. ● f is a computable function. ● Intuitively, a mapping reduction from A to B says that a computer can transform any instance of A into an instance of B such that the answer to B is the answer to A. Why Mapping Reducibility Matters ● Theorem: If B ∈ R and A ≤M B, then A ∈ R. ● Theorem: If B ∈ RE and A ≤M B, then A ∈ RE. ● Theorem: If B ∈ co-RE and A ≤M B, then A ∈ co-RE. ● Intuitively: A ≤M B means “A is not harder than B.” Why Mapping Reducibility Matters ● Theorem: If A ∉ R and A ≤M B, then B ∉ R. ● Theorem: If A ∉ RE and A ≤M B, then B ∉ RE. ● Theorem: If A ∉ co-RE and A ≤M B, then B ∉ co-RE. ● Intuitively: A ≤M B means “B is at at least as hard as A.” Why Mapping Reducibility Matters IfIf thisthis oneone isis “easy”“easy” ((RR,, RERE,, co-co-RERE)…)… A ≤M B …… thenthen thisthis oneone isis “easy”“easy” ((RR,, RERE,, co-co-RERE)) too.too.
    [Show full text]
  • Some NP-Complete Problems
    Appendix A Some NP-Complete Problems To ask the hard question is simple. But what does it mean? What are we going to do? W.H. Auden In this appendix we present a brief list of NP-complete problems; we restrict ourselves to problems which either were mentioned before or are closely re- lated to subjects treated in the book. A much more extensive list can be found in Garey and Johnson [GarJo79]. Chinese postman (cf. Sect. 14.5) Let G =(V,A,E) be a mixed graph, where A is the set of directed edges and E the set of undirected edges of G. Moreover, let w be a nonnegative length function on A ∪ E,andc be a positive number. Does there exist a cycle of length at most c in G which contains each edge at least once and which uses the edges in A according to their given orientation? This problem was shown to be NP-complete by Papadimitriou [Pap76], even when G is a planar graph with maximal degree 3 and w(e) = 1 for all edges e. However, it is polynomial for graphs and digraphs; that is, if either A = ∅ or E = ∅. See Theorem 14.5.4 and Exercise 14.5.6. Chromatic index (cf. Sect. 9.3) Let G be a graph. Is it possible to color the edges of G with k colors, that is, does χ(G) ≤ k hold? Holyer [Hol81] proved that this problem is NP-complete for each k ≥ 3; this holds even for the special case where k =3 and G is 3-regular.
    [Show full text]