6.046J Lecture 17: Complexity and NP-Completeness
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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. -
If Np Languages Are Hard on the Worst-Case, Then It Is Easy to Find Their Hard Instances
IF NP LANGUAGES ARE HARD ON THE WORST-CASE, THEN IT IS EASY TO FIND THEIR HARD INSTANCES Dan Gutfreund, Ronen Shaltiel, and Amnon Ta-Shma Abstract. We prove that if NP 6⊆ BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is hard for it. That is, the algorithm often errs on inputs from this distribution. This is the ¯rst worst-case to average-case reduction for NP of any kind. We stress however, that this does not mean that there exists one ¯xed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way func- tions and cryptography (even if not a su±cient assumption). Neverthe- less, we do show that there is a ¯xed distribution on instances of NP- complete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case). Our results are based on the following lemma that may be of independent interest: Given the description of an e±cient (probabilistic) algorithm that fails to solve SAT in the worst case, we can e±ciently generate at most three Boolean formulae (of increasing lengths) such that the algorithm errs on at least one of them. Keywords. Average-case complexity, Worst-case to average-case re- ductions, Foundations of cryptography, Pseudo classes Subject classi¯cation. 68Q10 (Modes of computation (nondetermin- istic, parallel, interactive, probabilistic, etc.) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Compu- tational di±culty of problems (lower bounds, completeness, di±culty of approximation, etc.) 94A60 Cryptography 2 Gutfreund, Shaltiel & Ta-Shma 1. -
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. -
Computational Complexity and Intractability: an Introduction to the Theory of NP Chapter 9 2 Objectives
1 Computational Complexity and Intractability: An Introduction to the Theory of NP Chapter 9 2 Objectives . Classify problems as tractable or intractable . Define decision problems . Define the class P . Define nondeterministic algorithms . Define the class NP . Define polynomial transformations . Define the class of NP-Complete 3 Input Size and Time Complexity . Time complexity of algorithms: . Polynomial time (efficient) vs. Exponential time (inefficient) f(n) n = 10 30 50 n 0.00001 sec 0.00003 sec 0.00005 sec n5 0.1 sec 24.3 sec 5.2 mins 2n 0.001 sec 17.9 mins 35.7 yrs 4 “Hard” and “Easy” Problems . “Easy” problems can be solved by polynomial time algorithms . Searching problem, sorting, Dijkstra’s algorithm, matrix multiplication, all pairs shortest path . “Hard” problems cannot be solved by polynomial time algorithms . 0/1 knapsack, traveling salesman . Sometimes the dividing line between “easy” and “hard” problems is a fine one. For example, . Find the shortest path in a graph from X to Y (easy) . Find the longest path (with no cycles) in a graph from X to Y (hard) 5 “Hard” and “Easy” Problems . Motivation: is it possible to efficiently solve “hard” problems? Efficiently solve means polynomial time solutions. Some problems have been proved that no efficient algorithms for them. For example, print all permutation of a number n. However, many problems we cannot prove there exists no efficient algorithms, and at the same time, we cannot find one either. 6 Traveling Salesperson Problem . No algorithm has ever been developed with a Worst-case time complexity better than exponential . -
NP-Completeness: Reductions Tue, Nov 21, 2017
CMSC 451 Dave Mount CMSC 451: Lecture 19 NP-Completeness: Reductions Tue, Nov 21, 2017 Reading: Chapt. 8 in KT and Chapt. 8 in DPV. Some of the reductions discussed here are not in either text. Recap: We have introduced a number of concepts on the way to defining NP-completeness: Decision Problems/Language recognition: are problems for which the answer is either yes or no. These can also be thought of as language recognition problems, assuming that the input has been encoded as a string. For example: HC = fG j G has a Hamiltonian cycleg MST = f(G; c) j G has a MST of cost at most cg: P: is the class of all decision problems which can be solved in polynomial time. While MST 2 P, we do not know whether HC 2 P (but we suspect not). Certificate: is a piece of evidence that allows us to verify in polynomial time that a string is in a given language. For example, the language HC above, a certificate could be a sequence of vertices along the cycle. (If the string is not in the language, the certificate can be anything.) NP: is defined to be the class of all languages that can be verified in polynomial time. (Formally, it stands for Nondeterministic Polynomial time.) Clearly, P ⊆ NP. It is widely believed that P 6= NP. To define NP-completeness, we need to introduce the concept of a reduction. Reductions: The class of NP-complete problems consists of a set of decision problems (languages) (a subset of the class NP) that no one knows how to solve efficiently, but if there were a polynomial time solution for even a single NP-complete problem, then every problem in NP would be solvable in polynomial time. -
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. -
NP-Completeness Part I
NP-Completeness Part I Outline for Today ● Recap from Last Time ● Welcome back from break! Let's make sure we're all on the same page here. ● Polynomial-Time Reducibility ● Connecting problems together. ● NP-Completeness ● What are the hardest problems in NP? ● The Cook-Levin Theorem ● A concrete NP-complete problem. Recap from Last Time The Limits of Computability EQTM EQTM co-RE R RE LD LD ADD HALT ATM HALT ATM 0*1* The Limits of Efficient Computation P NP R P and NP Refresher ● The class P consists of all problems solvable in deterministic polynomial time. ● The class NP consists of all problems solvable in nondeterministic polynomial time. ● Equivalently, NP consists of all problems for which there is a deterministic, polynomial-time verifier for the problem. Reducibility Maximum Matching ● Given an undirected graph G, a matching in G is a set of edges such that no two edges share an endpoint. ● A maximum matching is a matching with the largest number of edges. AA maximummaximum matching.matching. Maximum Matching ● Jack Edmonds' paper “Paths, Trees, and Flowers” gives a polynomial-time algorithm for finding maximum matchings. ● (This is the same Edmonds as in “Cobham- Edmonds Thesis.) ● Using this fact, what other problems can we solve? Domino Tiling Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling The Setup ● To determine whether you can place at least k dominoes on a crossword grid, do the following: ● Convert the grid into a graph: each empty cell is a node, and any two adjacent empty cells have an edge between them. -
Notes on Space Complexity of Integration of Computable Real
Notes on space complexity of integration of computable real functions in Ko–Friedman model Sergey V. Yakhontov Abstract x In the present paper it is shown that real function g(x)= 0 f(t)dt is a linear-space computable real function on interval [0, 1] if f is a linear-space computable C2[0, 1] real function on interval R [0, 1], and this result does not depend on any open question in the computational complexity theory. The time complexity of computable real functions and integration of computable real functions is considered in the context of Ko–Friedman model which is based on the notion of Cauchy functions computable by Turing machines. 1 2 In addition, a real computable function f is given such that 0 f ∈ FDSPACE(n )C[a,b] but 1 f∈ / FP if FP 6= #P. 0 C[a,b] R RKeywords: Computable real functions, Cauchy function representation, polynomial-time com- putable real functions, linear-space computable real functions, C2[0, 1] real functions, integration of computable real functions. Contents 1 Introduction 1 1.1 CF computablerealnumbersandfunctions . ...... 2 1.2 Integration of FP computablerealfunctions. 2 2 Upper bound of the time complexity of integration 3 2 3 Function from FDSPACE(n )C[a,b] that not in FPC[a,b] if FP 6= #P 4 4 Conclusion 4 arXiv:1408.2364v3 [cs.CC] 17 Nov 2014 1 Introduction In the present paper, we consider computable real numbers and functions that are represented by Cauchy functions computable by Turing machines [1]. Main results regarding computable real numbers and functions can be found in [1–4]; main results regarding computational complexity of computations on Turing machines can be found in [5]. -
Lecture 9 1 Interactive Proof Systems/Protocols
CS 282A/MATH 209A: Foundations of Cryptography Prof. Rafail Ostrovsky Lecture 9 Lecture date: March 7-9, 2005 Scribe: S. Bhattacharyya, R. Deak, P. Mirzadeh 1 Interactive Proof Systems/Protocols 1.1 Introduction The traditional mathematical notion of a proof is a simple passive protocol in which a prover P outputs a complete proof to a verifier V who decides on its validity. The interaction in this traditional sense is minimal and one-way, prover → verifier. The observation has been made that allowing the verifier to interact with the prover can have advantages, for example proving the assertion faster or proving more expressive languages. This extension allows for the idea of interactive proof systems (protocols). The general framework of the interactive proof system (protocol) involves a prover P with an exponential amount of time (computationally unbounded) and a verifier V with a polyno- mial amount of time. Both P and V exchange multiple messages (challenges and responses), usually dependent upon outcomes of fair coin tosses which they may or may not share. It is easy to see that since V is a poly-time machine (PPT), only a polynomial number of messages may be exchanged between the two. P ’s objective is to convince (prove to) the verifier the truth of an assertion, e.g., claimed knowledge of a proof that x ∈ L. V either accepts or rejects the interaction with the P . 1.2 Definition of Interactive Proof Systems An interactive proof system for a language L is a protocol PV for communication between a computationally unbounded (exponential time) machine P and a probabilistic poly-time (PPT) machine V such that the protocol satisfies the properties of completeness and sound- ness. -
Theory of Computation
A Universal Program (4) Theory of Computation Prof. Michael Mascagni Florida State University Department of Computer Science 1 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) Enumeration Theorem Definition. We write Wn = fx 2 N j Φ(x; n) #g: Then we have Theorem 4.6. A set B is r.e. if and only if there is an n for which B = Wn. Proof. This is simply by the definition ofΦ( x; n). 2 Note that W0; W1; W2;::: is an enumeration of all r.e. sets. 2 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) The Set K Let K = fn 2 N j n 2 Wng: Now n 2 K , Φ(n; n) #, HALT(n; n) This, K is the set of all numbers n such that program number n eventually halts on input n. 3 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) K Is r.e. but Not Recursive Theorem 4.7. K is r.e. but not recursive. Proof. By the universality theorem, Φ(n; n) is partially computable, hence K is r.e. If K¯ were also r.e., then by the enumeration theorem, K¯ = Wi for some i. We then arrive at i 2 K , i 2 Wi , i 2 K¯ which is a contradiction. -
Lecture 10: Space Complexity III
Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Lecture 10: Space Complexity III Arijit Bishnu 27.03.2010 Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). -
CSE 6321 - Solutions to Problem Set 5
CSE 6321 - Solutions to Problem Set 5 1. 2. Solution: n 1 T T 1 T T D-Q = fhA; P0;P1;:::;Pm; b; q0; q1; : : : ; qm; r1; : : : ; rm; ti j (9x 2 R ) 2 x P0x + q0 x ≤ t; 2 x Pix + qi x + ri ≤ 0; Ax = bg. We can show that the decision version of binary integer programming reduces to D-Q in polynomial time as follows: The constraint xi 2 f0; 1g of binary integer programming is equivalent to the quadratic constrain (possible in D-Q) xi(xi − 1) = 0. The rest of the constraints in binary integer programming are linear and can be represented directly in D-Q. More specifically, let hA; b; c; k0i be an instance of D-0-1-IP = fhA; b; c; k0i j (9x 2 f0; 1gn)Ax ≤ b; cT x ≥ 0 kg. We map it to the following instance of D-Q: P0 = 0, q0 = −b, t = −k , k = 0 (no linear equality 1 T T constraint Ax = b), and then use constraints of the form 2 x Pix + qi x + ri ≤ 0 to represent the constraints xi(xi − 1) ≥ 0 and xi(xi − 1) ≤ 0 for i = 1; : : : ; n and more constraints of the same form to represent the 0 constraint Ax ≤ b (using Pi = 0). Then hA; b; c; k i 2 D-0-1-IP iff the constructed instance of D-Q is in D-Q. The mapping is clearly polynomial time computable. (In an earlier statement of ps6, problem 1, I forgot to say that r1; : : : ; rm 2 Q are also part of the input.) 3.