DOCSLIB.ORG

1 Counting Complexity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Counting Complexity This is page Printer Opaque this Counting Complexity Lance Fortnow ABSTRACT Traditionally computational complexity theorists have lo oked at NP problems like threecoloring to determine whether some solution ex ists In counting complexity we ask how many solutions exist This pap er surveys the vast area of counting complexity concentrating on a few of the most imp ortant results in the area Intro duction In Valiant Vala dened the complexity class P as a function class computing the numb er of accepting paths of a nondeterministic Turing machine Valiant used this class to capture the complexity of the p ermanent function Counting complexity has since played an imp ortant role in computational complexity theory and theoretical computer science The techniques used in counting complexity have signicant applications in circuit complexity and the series of recent results on interactive pro of systems In the rst Complexity Theory Retrospective Schoning Sch gave us a taste of the p ower of counting Schonings survey lo oked at counting as a technique in complexity theory We instead will lo ok at counting as a goal and study the techniques mostly algebraic that help us understand the computational complexity of counting In this short survey we cannot hop e to cover adequately the wealth of research in counting complexity We fo cus on describing the basics the two imp ortant counting function classes P and GapP and several imp ortant language classes dened in terms of these functions We will describ e the basic relationships and closure prop erties among these classes We fo cus in detail on p erhaps the two most imp ortant results in count ing complexity To das To d result showing that one can reduce any language in the p olynomialtime hierarchy to a P function and Beigel Reingold and Spielmans BRS theorem that PP is closed under union 1 Department of Computer Science University of Chicago E th St Chicago IL Email fortnowcsuchicagoedu Supp orted in part by NSF grant CCR Lance Fortnow We prove To das Theorem in three stages First we give a new alge braic pro of of an imp ortant lemma by Valiant and Vazirani VV that randomly reduces a P function to its sign We apply this lemma dur ing the pro of of a result of To da and Ogihara TO that shows that the p olynomialtime hierarchy is in some sense probabilistically low for GapP functions We then use this result with a variant of To das original pro of to show how to reduce the p olynomialtime hierarchy deterministically to a single GapP question This survey should give the reader a taste of the p ower of counting func tions We hop e that the reader will take the concepts herein and learn more ab out counting complexity and apply the techniques not only to counting complexity but also to other areas in computational complexity theory and theoretical computer science Preliminaries We assume that the reader has familiarity with the basic notions of Turing machines nondeterminism and basic complexity classes such as P and NP The reader can nd these notions in a basic undergraduate textb o ok such as that of Hop croft and Ullman HU Fix f g We consider strings over b oth as concatenations of characters and as representations of integers We dene the onetoone cor resp ondence as follows The string represents zero The string x represents the p ositive integer whose binary description is x The string x represents the negative integer whose absolute value has binary description x Note that for a numb er y the length of the representation of y jy j is roughly the logarithm of the absolute value of y We order strings lexicographically ie x y if jxj jy j or jxj jy j and x precedes y in dictionary order Note that we may have x y as strings but x y as integers Let hx x i for k b e a standard tuple function ie oneto k one easily computable and invertible in all arguments When we write a function as f x x this should b e interpreted as f hx x i k k Let M b e a nondeterministic Turing machine that runs in time b ounded j by some p olynomial n for inputs of length n Since most of the denitions in this pap er are based on this typ e of machine we give such machines the name NP machines However we will usually b e interested more in the Counting Complexity numb er of accepting and p ossibly rejecting paths of M rather than whether the machine accepts M represent the NP machine that simulates M but reverses M s Let decision to accept and reject In other words the accepting paths of M x are the rejecting paths of M x and vice versa Let M x b e the numb er of accepting paths of M We then have that M x is the numb er of rejecting paths We also dene the Gap of M by M x M x M x j jxj M x and M x all are b ounded by Note that M x Let FP represent the class of p olynomialtime computable functions We will also consider oracle Turing machines An oracle Turing machine M using oracle A may write a string y on a sp ecial oracle tap e and then enter a sp ecial query state M will then enter a yes state if y is in A and a no state if y is not in A A We dene the relativized class NP as the set of languages that are accepted by p olynomialtime nondeterministic Turing machines with access C A to oracle A For a class C we dene NP as the union of all NP for A Ak in C We use the notation NP to mean that the NP machine can make only at most k queries to the oracle A on any computation path We can dene relativized versions of other complexity classes in a similar manner When we relativize PSPACE computation we only allow the PSPACE machine to write questions on the oracle tap e whose lengths are b ounded by a p olynomial We can use oracle Turing machines to dene other complexity classes p p p such as the p olynomialtime hierarchy MS We dene and i i i inductively as follows p p p P p p i NP i p p co i i p p i P i The sup erscript p distinguishes the p olynomialtime hierarchy used in complexity theory from the arithmetic hierarchy from recursion theory see Rog We dene the p olynomialtime hierarchy PH as the union of these classes p p p PSPACE PH i i i i i i p p We say that the p olynomialtime hierarchy is innite if for i i all i Otherwise we say that the hierarchy is nite or that it col lapses Complexity theorists generally b elieve that the hierarchy is innite Lance Fortnow Counting Functions In this section we will discuss the p ower of counting functions We will examine two p owerful function classes in particular P and GapP The classes P read SharpP or Numb erP and GapP are derived directly from lo oking at the numb er of accepting paths or the gap b etween accepting and rejecting paths resp ectively Denition The class P consists of the functions f such that there exists an NP machine M such that for al l x f x M x The class GapP consists of the functions f such that there exists an NP machine M such that for al l x f x M x First note that P and GapP functions cannot take on values to o large Lemma If f x is a P or GapP function then there exists a poly pjxj nomial pn such that the absolute value of f x is bounded by for every x Pro of Either f x M x or M x for some NP machine M Let pn b ound the running time of M on strings of length n Note that M x pjxj can have at most computation paths The numb ers of accepting and rejecting paths of M x are b ounded by the numb er of computation paths Lemma follows 2 Note that P functions cannot take on negative values though GapP functions can In fact GapP functions are closed under negation Lemma If f is a GapP function then f is also a GapP function Pro of Let M b e an NP machine such that f M Then f M 2 How ab out the relationship b etween P and GapP and the p olynomial time computable functions FP First we show that every P function is also a GapP function Lemma For every NP machine M there is an NP machine N such that N M Pro of Given an input x the machine N guesses a path p of M x If p is accepting N accepts Otherwise N branches once accepting on one branch and rejecting on the other We have for all x N x N x N x M x N x M x M x M x M x fullling the conclusion of Lemma 2 Counting Complexity Note that a P function can take on only nonnegative values so this class cannot capture all of the FP functions However it do es capture those FP functions that only take on nonnegative values The class GapP encompasses all the FP functions without restriction Theorem If an FP function f always takes on nonnegative values then f is in P Every FP function f is in GapP Pro of Let f b e an FP function Create an NP machine M that on input x guesses jf xj computation paths and accepts on these paths if and only if f x Note that M x f x for all x and M x f x when f x 2 It is p ossible that FP captures all the P or GapP functions or that P captures all the GapP functions that take on only nonnegative values but these prop ositions seem unlikely However from the following lemma we will see that P and GapP have essentially the same computational p ower Lemma For al l functions f the fol lowing are equivalent f GapP f is the dierence of two P functions f is the dierence of a P function and an FP function f is the dierence of an FP function and a P function P Corollary GapP FP Pro of Follows from the third characterization of GapP in Lemma 2 Pro of of Lemma For any M we have M M M by denition so GapP is the dierence of two P functions Let f and g b e P functions We can assume that f M and q n g N where M and N are NP machines and N has
Load more

Recommended publications
  • Notes for Lecture 2
    Notes on Complexity Theory Last updated: December, 2011 Lecture 2 Jonathan Katz 1 Review The running time of a Turing machine M on input x is the number of \steps" M takes before it halts. Machine M is said to run in time T (¢) if for every input x the running time of M(x) is at most T (jxj). (In particular, this means it halts on all inputs.) The space used by M on input x is the number of cells written to by M on all its work tapes1 (a cell that is written to multiple times is only counted once); M is said to use space T (¢) if for every input x the space used during the computation of M(x) is at most T (jxj). We remark that these time and space measures are worst-case notions; i.e., even if M runs in time T (n) for only a fraction of the inputs of length n (and uses less time for all other inputs of length n), the running time of M is still T . (Average-case notions of complexity have also been considered, but are somewhat more di±cult to reason about. We may cover this later in the semester; or see [1, Chap. 18].) Recall that a Turing machine M computes a function f : f0; 1g¤ ! f0; 1g¤ if M(x) = f(x) for all x. We will focus most of our attention on boolean functions, a context in which it is more convenient to phrase computation in terms of languages. A language is simply a subset of f0; 1g¤.
    [Show full text]
  • Interactive Proof Systems and Alternating Time-Space Complexity
    Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement.
    [Show full text]
  • 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]
  • The Shrinking Property for NP and Conp✩
    Theoretical Computer Science 412 (2011) 853–864 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs The shrinking property for NP and coNPI Christian Glaßer a, Christian Reitwießner a,∗, Victor Selivanov b a Julius-Maximilians-Universität Würzburg, Germany b A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences, Russia article info a b s t r a c t Article history: We study the shrinking and separation properties (two notions well-known in descriptive Received 18 August 2009 set theory) for NP and coNP and show that under reasonable complexity-theoretic Received in revised form 11 November assumptions, both properties do not hold for NP and the shrinking property does not hold 2010 for coNP. In particular we obtain the following results. Accepted 15 November 2010 Communicated by A. Razborov P 1. NP and coNP do not have the shrinking property unless PH is finite. In general, Σn and P Πn do not have the shrinking property unless PH is finite. This solves an open question Keywords: posed by Selivanov (1994) [33]. Computational complexity 2. The separation property does not hold for NP unless UP ⊆ coNP. Polynomial hierarchy 3. The shrinking property does not hold for NP unless there exist NP-hard disjoint NP-pairs NP-pairs (existence of such pairs would contradict a conjecture of Even et al. (1984) [6]). P-separability Multivalued functions 4. The shrinking property does not hold for NP unless there exist complete disjoint NP- pairs. Moreover, we prove that the assumption NP 6D coNP is too weak to refute the shrinking property for NP in a relativizable way.
    [Show full text]
  • Nitin Saurabh the Institute of Mathematical Sciences, Chennai
    ALGEBRAIC MODELS OF COMPUTATION By Nitin Saurabh The Institute of Mathematical Sciences, Chennai. A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulllment of the requirements For the Degree of Master of Science of HOMI BHABHA NATIONAL INSTITUTE April 2012 CERTIFICATE Certied that the work contained in the thesis entitled Algebraic models of Computation, by Nitin Saurabh, has been carried out under my supervision and that this work has not been submitted elsewhere for a degree. Meena Mahajan Theoretical Computer Science Group The Institute of Mathematical Sciences, Chennai ACKNOWLEDGEMENTS I would like to thank my advisor Prof. Meena Mahajan for her invaluable guidance and continuous support since my undergraduate days. Her expertise and ideas helped me comprehend new techniques. Her guidance during the preparation of this thesis has been invaluable. I also thank her for always being there to discuss and clarify any matter. I am extremely grateful to all the faculty members of theory group at IMSc and CMI for their continuous encouragement and giving me an opportunity to learn from them. I would like to thank all my friends, at IMSc and CMI, for making my stay in Chennai a memorable one. Most of all, I take this opportunity to thank my parents, my uncle and my brother. Abstract Valiant [Val79, Val82] had proposed an analogue of the theory of NP-completeness in an entirely algebraic framework to study the complexity of polynomial families. Artihmetic circuits form the most standard model for studying the complexity of polynomial computations. In a note [Val92], Valiant argued that in order to prove lower bounds for boolean circuits, obtaining lower bounds for arithmetic circuits should be a rst step.
    [Show full text]
  • Computational Complexity: a Modern Approach
    i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist.
    [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]
  • UNIONS of LINES in Fn 1. Introduction the Main Problem We
    UNIONS OF LINES IN F n RICHARD OBERLIN Abstract. We show that if a collection of lines in a vector space over a finite field has \dimension" at least 2(d−1)+β; then its union has \dimension" at least d + β: This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of k-planes. 1. Introduction The main problem we will consider here is to give a lower bound for the dimension of the union of a collection of lines in terms of the dimension of the collection of lines, without imposing a structural hy- pothesis on the collection (in contrast to the Kakeya problem where one assumes that the lines are direction-separated, or perhaps satisfy the weaker \Wolff axiom"). Specifically, we are motivated by the following conjecture of D. Ober- lin (hdim denotes Hausdorff dimension). Conjecture 1.1. Suppose d ≥ 1 is an integer, that 0 ≤ β ≤ 1; and that L is a collection of lines in Rn with hdim(L) ≥ 2(d − 1) + β: Then [ (1) hdim( L) ≥ d + β: L2L The bound (1), if true, would be sharp, as one can see by taking L to be the set of lines contained in the d-planes belonging to a β- dimensional family of d-planes. (Furthermore, there is nothing to be gained by taking 1 < β ≤ 2 since the dimension of the set of lines contained in a d + 1-plane is 2(d − 1) + 2.) Standard Fourier-analytic methods show that (1) holds for d = 1, but the conjecture is open for d > 1: As a model problem, one may consider an analogous question where Rn is replaced by a vector space over a finite-field.
    [Show full text]
  • When Subgraph Isomorphism Is Really Hard, and Why This Matters for Graph Databases Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble
    When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble To cite this version: Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble. When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases. Journal of Artificial Intelligence Research, Association for the Advancement of Artificial Intelligence, 2018, 61, pp.723 - 759. 10.1613/jair.5768. hal-01741928 HAL Id: hal-01741928 https://hal.archives-ouvertes.fr/hal-01741928 Submitted on 26 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Artificial Intelligence Research 61 (2018) 723-759 Submitted 11/17; published 03/18 When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases Ciaran McCreesh [email protected] Patrick Prosser [email protected] University of Glasgow, Glasgow, Scotland Christine Solnon [email protected] INSA-Lyon, LIRIS, UMR5205, F-69621, France James Trimble [email protected] University of Glasgow, Glasgow, Scotland Abstract The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph.
    [Show full text]
  • LSPACE VS NP IS AS HARD AS P VS NP 1. Introduction in Complexity
    LSPACE VS NP IS AS HARD AS P VS NP FRANK VEGA Abstract. The P versus NP problem is a major unsolved problem in com- puter science. This consists in knowing the answer of the following question: Is P equal to NP? Another major complexity classes are LSPACE, PSPACE, ESPACE, E and EXP. Whether LSPACE = P is a fundamental question that it is as important as it is unresolved. We show if P = NP, then LSPACE = NP. Consequently, if LSPACE is not equal to NP, then P is not equal to NP. According to Lance Fortnow, it seems that LSPACE versus NP is easier to be proven. However, with this proof we show this problem is as hard as P versus NP. Moreover, we prove the complexity class P is not equal to PSPACE as a direct consequence of this result. Furthermore, we demonstrate if PSPACE is not equal to EXP, then P is not equal to NP. In addition, if E = ESPACE, then P is not equal to NP. 1. Introduction In complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem [6]. A functional problem F is defined as a binary relation (x; y) 2 R over strings of an arbitrary alphabet Σ: R ⊂ Σ∗ × Σ∗: A Turing machine M solves F if for every input x such that there exists a y satisfying (x; y) 2 R, M produces one such y, that is M(x) = y [6].
    [Show full text]
  • Delegating Computation: Interactive Proofs for Muggles∗
    Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 108 (2017) Delegating Computation: Interactive Proofs for Muggles∗ Shafi Goldwasser Yael Tauman Kalai Guy N. Rothblum MIT and Weizmann Institute Microsoft Research Weizmann Institute [email protected] [email protected] [email protected] Abstract In this work we study interactive proofs for tractable languages. The (honest) prover should be efficient and run in polynomial time, or in other words a \muggle".1 The verifier should be super-efficient and run in nearly-linear time. These proof systems can be used for delegating computation: a server can run a computation for a client and interactively prove the correctness of the result. The client can verify the result's correctness in nearly-linear time (instead of running the entire computation itself). Previously, related questions were considered in the Holographic Proof setting by Babai, Fortnow, Levin and Szegedy, in the argument setting under computational assumptions by Kilian, and in the random oracle model by Micali. Our focus, however, is on the original inter- active proof model where no assumptions are made on the computational power or adaptiveness of dishonest provers. Our main technical theorem gives a public coin interactive proof for any language computable by a log-space uniform boolean circuit with depth d and input length n. The verifier runs in time n · poly(d; log(n)) and space O(log(n)), the communication complexity is poly(d; log(n)), and the prover runs in time poly(n). In particular, for languages computable by log-space uniform NC (circuits of polylog(n) depth), the prover is efficient, the verifier runs in time n · polylog(n) and space O(log(n)), and the communication complexity is polylog(n).
    [Show full text]
  • Interactive Proofs
    Interactive proofs April 12, 2014 [72] L´aszl´oBabai. Trading group theory for randomness. In Proc. 17th STOC, pages 421{429. ACM Press, 1985. doi:10.1145/22145.22192. [89] L´aszl´oBabai and Shlomo Moran. Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. J. Comput. System Sci., 36(2):254{276, 1988. doi:10.1016/0022-0000(88)90028-1. [99] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic ex- ponential time has two-prover interactive protocols. In Proc. 31st FOCS, pages 16{25. IEEE Comp. Soc. Press, 1990. doi:10.1109/FSCS.1990.89520. See item 1991.108. [108] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic expo- nential time has two-prover interactive protocols. Comput. Complexity, 1 (1):3{40, 1991. doi:10.1007/BF01200056. Full version of 1990.99. [136] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. In Proc. 34th FOCS, pages 724{733, Palo Alto CA, 1993. IEEE Comp. Soc. Press. doi:10.1109/SFCS.1993.366815. Conference version of item 1997:160. [160] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. System Sci., 54(2):317{331, 1997. doi:10.1006/jcss.1997.1472. Full version of 1993.136. [111] L´aszl´oBabai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. In Proc.
    [Show full text]