The Journey from NP to TFNP Hardness
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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]. -
Adaptive Garbled RAM from Laconic Oblivious Transfer
Adaptive Garbled RAM from Laconic Oblivious Transfer Sanjam Garg?1, Rafail Ostrovsky??2, and Akshayaram Srinivasan1 1 University of California, Berkeley fsanjamg,[email protected] 2 UCLA [email protected] Abstract. We give a construction of an adaptive garbled RAM scheme. In the adaptive setting, a client first garbles a \large" persistent database which is stored on a server. Next, the client can provide garbling of mul- tiple adaptively and adversarially chosen RAM programs that execute and modify the stored database arbitrarily. The garbled database and the garbled program should reveal nothing more than the running time and the output of the computation. Furthermore, the sizes of the garbled database and the garbled program grow only linearly in the size of the database and the running time of the executed program respectively (up to poly logarithmic factors). The security of our construction is based on the assumption that laconic oblivious transfer (Cho et al., CRYPTO 2017) exists. Previously, such adaptive garbled RAM constructions were only known using indistinguishability obfuscation or in random oracle model. As an additional application, we note that this work yields the first constant round secure computation protocol for persistent RAM pro- grams in the malicious setting from standard assumptions. Prior works did not support persistence in the malicious setting. 1 Introduction Over the years, garbling methods [Yao86,LP09,AIK04,BHR12b,App17] have been extremely influential and have engendered an enormous number of applications ? Research supported in part from DARPA/ARL SAFEWARE Award W911NF15C0210, AFOSR Award FA9550-15-1-0274, AFOSR YIP Award, DARPA and SPAWAR under contract N66001-15-C-4065, a Hellman Award and research grants by the Okawa Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). -
Paradigms of Combinatorial Optimization
W657-Paschos 2.qxp_Layout 1 01/07/2014 14:05 Page 1 MATHEMATICS AND STATISTICS SERIES Vangelis Th. Paschos Vangelis Edited by This updated and revised 2nd edition of the three-volume Combinatorial Optimization series covers a very large set of topics in this area, dealing with fundamental notions and approaches as well as several classical applications of Combinatorial Optimization. Combinatorial Optimization is a multidisciplinary field, lying at the interface of three major scientific domains: applied mathematics, theoretical computer science, and management studies. Its focus is on finding the least-cost solution to a mathematical problem in which each solution is associated with a numerical cost. In many such problems, exhaustive search is not feasible, so the approach taken is to operate within the domain of optimization problems, in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Some common problems involving combinatorial optimization are the traveling salesman problem and the Combinatorial Optimization minimum spanning tree problem. Combinatorial Optimization is a subset of optimization that is related to operations research, algorithm theory, and computational complexity theory. It 2 has important applications in several fields, including artificial intelligence, nd mathematics, and software engineering. Edition Revised and Updated This second volume, which addresses the various paradigms and approaches taken in Combinatorial Optimization, is divided into two parts: Paradigms of - Paradigmatic Problems, which discusses several famous combinatorial optimization problems, such as max cut, min coloring, optimal satisfiability TSP, Paradigms of etc., the study of which has largely contributed to the development, the legitimization and the establishment of Combinatorial Optimization as one of the most active current scientific domains. -
Solving Non-Boolean Satisfiability Problems with Stochastic Local Search
Solving Non-Boolean Satisfiability Problems with Stochastic Local Search: A Comparison of Encodings ALAN M. FRISCH †, TIMOTHY J. PEUGNIEZ, ANTHONY J. DOGGETT and PETER W. NIGHTINGALE§ Artificial Intelligence Group, Department of Computer Science, University of York, York YO10 5DD, UK. e-mail: [email protected], [email protected] Abstract. Much excitement has been generated by the success of stochastic local search procedures at finding solutions to large, very hard satisfiability problems. Many of the problems on which these procedures have been effective are non-Boolean in that they are most naturally formulated in terms of variables with domain sizes greater than two. Approaches to solving non-Boolean satisfiability problems fall into two categories. In the direct approach, the problem is tackled by an algorithm for non-Boolean problems. In the transformation approach, the non-Boolean problem is reformulated as an equivalent Boolean problem and then a Boolean solver is used. This paper compares four methods for solving non-Boolean problems: one di- rect and three transformational. The comparison first examines the search spaces confronted by the four methods then tests their ability to solve random formulas, the round-robin sports scheduling problem and the quasigroup completion problem. The experiments show that the relative performance of the methods depends on the domain size of the problem, and that the direct method scales better as domain size increases. Along the route to performing these comparisons we make three other contri- butions. First, we generalise Walksat, a highly-successful stochastic local search procedure for Boolean satisfiability problems, to work on problems with domains of any finite size. -
Random Θ(Log N)-Cnfs Are Hard for Cutting Planes
Random Θ(log n)-CNFs are Hard for Cutting Planes Noah Fleming Denis Pankratov Toniann Pitassi Robert Robere University of Toronto University of Toronto University of Toronto University of Toronto noahfl[email protected] [email protected] [email protected] [email protected] Abstract—The random k-SAT model is the most impor- lower bounds for random k-SAT formulas in a particular tant and well-studied distribution over k-SAT instances. It is proof system show that any complete and efficient algorithm closely connected to statistical physics and is a benchmark for based on the proof system will perform badly on random satisfiability algorithms. We show that when k = Θ(log n), any Cutting Planes refutation for random k-SAT requires k-SAT instances. Furthermore, since the proof complexity exponential size in the interesting regime where the number of lower bounds hold in the unsatisfiable regime, they are clauses guarantees that the formula is unsatisfiable with high directly connected to Feige’s hypothesis. probability. Remarkably, determining whether or not a random SAT Keywords-Proof complexity; random k-SAT; Cutting Planes; instance from the distribution F(m; n; k) is satisfiable is controlled quite precisely by the ratio ∆ = m=n, which is called the clause density. A simple counting argument shows I. INTRODUCTION that F(m; n; k) is unsatisfiable with high probability for The Satisfiability (SAT) problem is perhaps the most ∆ > 2k ln 2. The famous satisfiability threshold conjecture famous problem in theoretical computer science, and sig- asserts that there is a constant ck such that random k-SAT nificant effort has been devoted to understanding randomly formulas of clause density ∆ are almost certainly satisfiable generated SAT instances. -
The Classes FNP and TFNP
Outline The classes FNP and TFNP C. Wilson1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function Problems What are Function Problems? Function Problems FSAT Defined Total Functions TSP Defined Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function -
Lecture: Complexity of Finding a Nash Equilibrium 1 Computational
Algorithmic Game Theory Lecture Date: September 20, 2011 Lecture: Complexity of Finding a Nash Equilibrium Lecturer: Christos Papadimitriou Scribe: Miklos Racz, Yan Yang In this lecture we are going to talk about the complexity of finding a Nash Equilibrium. In particular, the class of problems NP is introduced and several important subclasses are identified. Most importantly, we are going to prove that finding a Nash equilibrium is PPAD-complete (defined in Section 2). For an expository article on the topic, see [4], and for a more detailed account, see [5]. 1 Computational complexity For us the best way to think about computational complexity is that it is about search problems.Thetwo most important classes of search problems are P and NP. 1.1 The complexity class NP NP stands for non-deterministic polynomial. It is a class of problems that are at the core of complexity theory. The classical definition is in terms of yes-no problems; here, we are concerned with the search problem form of the definition. Definition 1 (Complexity class NP). The class of all search problems. A search problem A is a binary predicate A(x, y) that is efficiently (in polynomial time) computable and balanced (the length of x and y do not differ exponentially). Intuitively, x is an instance of the problem and y is a solution. The search problem for A is this: “Given x,findy such that A(x, y), or if no such y exists, say “no”.” The class of all search problems is called NP. Examples of NP problems include the following two. -
Securely Obfuscating Re-Encryption
Securely Obfuscating Re-Encryption Susan Hohenberger¤ Guy N. Rothblumy abhi shelatz Vinod Vaikuntanathanx December 1, 2008 Abstract We present a positive obfuscation result for a traditional cryptographic functionality. This posi- tive result stands in contrast to well-known impossibility results [3] for general obfuscation and recent impossibility and improbability [13] results for obfuscation of many cryptographic functionalities. Whereas other positive obfuscation results in the standard model apply to very simple point func- tions, our obfuscation result applies to the signi¯cantly more complex and widely-used re-encryption functionality. This functionality takes a ciphertext for message m encrypted under Alice's public key and transforms it into a ciphertext for the same message m under Bob's public key. To overcome impossibility results and to make our results meaningful for cryptographic functionalities, our scheme satis¯es a de¯nition of obfuscation which incorporates more security-aware provisions. ¤Johns Hopkins University, [email protected]. Research partially performed at IBM Zurich Research Laboratory, Switzer- land. yMIT CSAIL, [email protected]. Research supported by NSF grant CNS-0430450 and NSF grant CFF-0635297. zUniversity of Virginia, [email protected]. Research performed at IBM Zurich Research Laboratory, Switzerland. xMIT CSAIL, [email protected] 1 1 Introduction A recent line of research in theoretical cryptography aims to understand whether it is possible to obfuscate programs so that a program's code becomes unintelligible while its functionality remains unchanged. A general method for obfuscating programs would lead to the solution of many open problems in cryptography. Unfortunately, Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan and Yang [3] show that for many notions of obfuscation, a general program obfuscator does not exist|i.e., they exhibit a class of circuits which cannot be obfuscated. -
Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds
comput. complex. c Springer Nature Switzerland AG 2019 DOI 10.1007/s00037-019-00177-4 computational complexity PREDICTION FROM PARTIAL INFORMATION AND HINDSIGHT, WITH APPLICATION TO CIRCUIT LOWER BOUNDS Or Meir and Avi Wigderson Abstract. Consider a random sequence of n bits that has entropy at least n − k, where k n. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate n looks random to an adversary that is allowed to query ≈ k other co- ordinates of the sequence, even if the adversary is non-deterministic. This implies corresponding results for decision trees and certificates for Boolean functions. As an application of this result, we prove a new result on depth-3 cir- cuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensi- tive functions due to Boppana (Circuits Inf Process Lett 63(5):257–261, 1997). An interesting feature of this proof is that it works in the frame- work of Karchmer and Wigderson (SIAM J Discrete Math 3(2):255–265, 1990), and, in particular, it is a “top-down” proof (H˚astad et al. in Computat Complex 5(2):99–112, 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions, which may be of independent interest. Keywords. Certificate complexity, Circuit complexity, Circuit com- plexity lower bounds, Decision tree complexity, Information theoretic, Query complexity, Sensitivity Subject classification. -
The Relative Complexity of NP Search Problems
Journal of Computer and System Sciences 57, 319 (1998) Article No. SS981575 The Relative Complexity of NP Search Problems Paul Beame* Computer Science and Engineering, University of Washington, Box 352350, Seattle, Washington 98195-2350 E-mail: beameÄcs.washington.edu Stephen Cook- Computer Science Department, University of Toronto, Canada M5S 3G4 E-mail: sacookÄcs.toronto.edu Jeff Edmonds Department of Computer Science, York University, Toronto, Ontario, Canada M3J 1P3 E-mail: jeffÄcs.yorku.ca Russell Impagliazzo9 Computer Science and Engineering, UC, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0114 E-mail: russellÄcs.ucsd.edu and Toniann PitassiÄ Department of Computer Science, University of Arizona, Tucson, Arizona 85721-0077 E-mail: toniÄcs.arizona.edu Received January 1998 1. INTRODUCTION Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of In the study of computational complexity, there are many solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, problems that are naturally expressed as problems ``to find'' and a number of them are complete problems. We consider the question but are converted into decision problems to fit into standard of the relative complexity of these search problem classes. We prove complexity classes. For example, a more natural problem several separations which show that in a generic relativized world the than determining whether or not a graph is 3-colorable search classes are distinct and there is a standard search problem in might be that of finding a 3-coloring of the graph if it exists. -
Total Functions in the Polynomial Hierarchy 11
Electronic Colloquium on Computational Complexity, Report No. 153 (2020) TOTAL FUNCTIONS IN THE POLYNOMIAL HIERARCHY ROBERT KLEINBERG∗, DANIEL MITROPOLSKYy, AND CHRISTOS PAPADIMITRIOUy Abstract. We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for “polynomial empty pigeonhole principle”), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FPNPjpoly. Higher in the hierarchy, we identify the constructive version of the Sauer- Shelah lemma and the appropriate generalization of PPP that contains it. The resulting total function hierarchy turns out to be more stable than the polynomial hierarchy: it is known that, under oracles, total functions within FNP may be easy, but total functions a level higher may still be harder than FPNP. 1. Introduction The complexity of total functions has emerged over the past three decades as an intriguing and productive branch of Complexity Theory. Subclasses of TFNP, the set of all total functions in FNP, have been defined and studied: PLS, PPP, PPA, PPAD, PPADS, and CLS. These classes are replete with natural problems, several of which turned out to be complete for the corresponding class, see e.g. -
Optimisation Function Problems FNP and FP
Complexity Theory 99 Complexity Theory 100 Optimisation The Travelling Salesman Problem was originally conceived of as an This is still reasonable, as we are establishing the difficulty of the optimisation problem problems. to find a minimum cost tour. A polynomial time solution to the optimisation version would give We forced it into the mould of a decision problem { TSP { in order a polynomial time solution to the decision problem. to fit it into our theory of NP-completeness. Also, a polynomial time solution to the decision problem would allow a polynomial time algorithm for finding the optimal value, CLIQUE Similar arguments can be made about the problems and using binary search, if necessary. IND. Anuj Dawar May 21, 2007 Anuj Dawar May 21, 2007 Complexity Theory 101 Complexity Theory 102 Function Problems FNP and FP Still, there is something interesting to be said for function problems A function which, for any given Boolean expression φ, gives a arising from NP problems. satisfying truth assignment if φ is satisfiable, and returns \no" Suppose otherwise, is a witness function for SAT. L = fx j 9yR(x; y)g If any witness function for SAT is computable in polynomial time, where R is a polynomially-balanced, polynomial time decidable then P = NP. relation. If P = NP, then for every language in NP, some witness function is A witness function for L is any function f such that: computable in polynomial time, by a binary search algorithm. • if x 2 L, then f(x) = y for some y such that R(x; y); P = NP if, and only if, FNP = FP • f(x) = \no" otherwise.