Complexity Classes Research Articles

Tight lower bounds for dynamic time warping

Dynamic Time Warping (DTW) is a popular similarity measure for aligning and comparing time series. Due to DTW’s high computation time, lower bounds are often employed to screen poor matches. Many alternative lower bounds have been proposed, providing a range of different trade-offs between tightness and computational efficiency. LB_KEOGH provides a useful trade-off in many applications. Two recent lower bounds, LB_IMPROVED and LB_ENHANCED, are substantially tighter than LB_KEOGH. All three have the same worst case computational complexity—linear with respect to series length and constant with respect to window size. We present four new DTW lower bounds in the same complexity class. LB_PETITJEAN is substantially tighter than LB_IMPROVED, with only modest additional computational overhead. LB_WEBB is more efficient than LB_IMPROVED, while often providing a tighter bound. LB_WEBB is always tighter than LB_KEOGH. The parameter free LB_WEBB is usually tighter than LB_ENHANCED. A parameterized variant, LB_Webb_Enhanced, is always tighter than LB_ENHANCED. A further variant, LB_WEBB*, is useful for some constrained distance functions. In extensive experiments, LB_WEBB proves to be very effective for nearest neighbor search.

The complexity of recognizing minimally tough graphs

A graph is called t-tough if the removal of any vertex set S that disconnects the graph leaves at most |S|∕t components. The toughness of a graph is the largest t for which the graph is t-tough. A graph is minimally t-tough if the toughness of the graph is t and the deletion of any edge from the graph decreases the toughness. The complexity class DP is the set of all languages that can be expressed as the intersection of a language in NP and a language in coNP. In this paper, we prove that recognizing minimally t-tough graphs is DP-complete for any positive rational number t. We introduce a new notion called weighted toughness, which has a key role in our proof.

Placing Conditional Disclosure of Secrets in the Communication Complexity Universe

In the conditional disclosure of secrets (CDS) problem (Gertner et al. in J Comput Syst Sci, 2000) Alice and Bob, who hold n-bit inputs x and y respectively, wish to release a common secret z to Carol, who knows both x and y, if and only if the input (x, y) satisfies some predefined predicate f. Alice and Bob are allowed to send a single message to Carol which may depend on their inputs and some shared randomness, and the goal is to minimize the communication complexity while providing information-theoretic security. Despite the growing interest in this model, very few lower-bounds are known. In this paper, we relate the CDS complexity of a predicate f to its communication complexity under various communication games. For several basic predicates our results yield tight, or almost tight, lower-bounds of \(\Omega (n)\) or \(\Omega (n^{1-\epsilon })\), providing an exponential improvement over previous logarithmic lower-bounds. We also define new communication complexity classes that correspond to different variants of the CDS model and study the relations between them and their complements. Notably, we show that allowing for imperfect correctness can significantly reduce communication—a seemingly new phenomenon in the context of information-theoretic cryptography. Finally, our results show that proving explicit super-logarithmic lower-bounds for imperfect CDS protocols is a necessary step towards proving explicit lower-bounds against the communication complexity class \(\text {AM}^{\text {cc}}\), or even \(\text {AM}^{\text {cc}}\cap \text {co-AM}^{\text {cc}}\)—a well known open problem in the theory of communication complexity. Thus imperfect CDS forms a new minimal class which is placed just beyond the boundaries of the “civilized” part of the communication complexity world for which explicit lower-bounds are known.

Checking Sets of Pure Evolving Association Rules

Extracting association rules from large datasets has been widely studied in many variants in the last two decades; they allow to extract relations between values that occur more “often” in a database. With temporal association rules the concept has been declined to temporal databases. In this context the “most frequent” patterns of evolution of one or more attribute values are extracted. In the temporal setting, especially where the interference betweeen temporal patterns cannot be neglected (e.g., in medical domains), there may be the case that we are looking for a set of temporal association rules for which a “significant” portion of the original database represents a consistent model for all of them. In this work, we introduce a simple and intuitive form for temporal association rules, called pure evolving association rules (PE-ARs for short), and we study the complexity of checking a set of PE-ARs over an instance of a temporal relation under approximation (i.e., a percentage of tuples that may be deleted from the original relation). As a by-product of our study we address the complexity class for a general problem on Directed Acyclic Graphs which is theoretically interesting per se.

Yuri Manin on Biomolecules as Quantum Computers

In 1980, Russian mathematician Yuri Manin published Computable and Uncomputable. On pages 14 and 15 of his introduction, Manin suggests that “Molecular biology furnishes examples of the behavior of natural (not engineered by humans) systems which we have to describe in terms initially devised for discrete automata.” Manin then describes the remarkable energy efficiency of naturally occurring biomolecular processes such as DNA replication. He proposes modeling such behaviors in terms of unitary rotations in a finite-dimensional Hilbert space. The decomposition of such systems then corresponds to the tensor product decomposition of the state space, that is, to quantum entanglement. Manin’s initial focus on biological molecules as examples of highly energy-efficient quantum automata is unique among quantum computing’s founding figures since both he and other early leaders quickly moved to the then-new and exciting concept of von Neumann automata. The von Neumann formalism reinterpreted molecular quantum computing in terms of qubits, which made it possible to imagine the power of quantum computing as not much more than a superposition of virtual binary computers. This paper provides the original excerpt of Manin’s molecular computing argument. A useful analytical feature of Manin’s pre-von-Neumann model of quantum computation is its openness to new formalisms that avoid accidentally making classical physics dominant over the quantum world by expressing quantum states only in terms of concepts such as automata that assume extreme classical precision and complexity.

Analysis of the Results of Applying the Bee Colony Method in the Problem of Coloring General Graphs

Purpose of research. We have discovered a wide range of problems that are important in practice and which can be reduced in polynomial time to discrete combinatorial optimization problems, many of which can be solved using graph theory. One of these tasks is finding the chromatic number of a graph and its corresponding coloring. Taking into account the fact that the combinatorial problem of finding the chromatic number of a graph belongs to the complexity class and does not allow obtaining an optimal solution in a rational time for problems of practically important dimension, the search for a suitable heuristic method that allows obtaining high-quality solutions with low costs required for computation is demanded and relevant. The aim of the study is to analyze the results of using the bee colony method in the task at hand. The tasks of this research are: description of algorithmic techniques in a formalized form, which make it possible to apply the bee colony method in the problem to be solved, making modifications to the bee colony method that increase the efficiency of the method, namely the quality of the resulting final colorings, as well as the determination of factors affecting the quality and the time spent in finding solutions. Methods. To conduct research in the selected area, computational experiments were organized based on the use of heuristic methods in the problem under consideration. Meta-optimization of the tuning parameters of the methods and determination of their convergence rate was carried out, as well as a comparison of the quality and time of obtaining solutions. Results. As a result of the study, the convergence rate of the method was found to be higher than that of the random walk method; the dependence of the quality of the resulting final colorings on the graph size N and density d was found. It was found that the chosen method is faster than the method of weighted random enumeration with the variation of vertices according to the minimum of admissible colors on »67% , which currently generates solutions with the lowest chromatic number, while losing quality to it on »7% . A higher rate of convergence was noticed when compared with the method of random walks, the principle of which is the same as that of foraging bees. Conclusion. It was found that the bee colony method finds colorings with the same average chromatic number in fewer iterations than the random walk method, i.e. it has a higher convergence rate, while remaining significantly fast relative to the method of random search with a variation of vertices to reduce the allowed colors.

Implementation of a Hybrid Classical-Quantum Annealing Algorithm for Logistic Network Design

The logistic network design is an abstract optimization problem that, under the assumption of minimal cost, seeks the optimal configuration of the supply chain's infrastructures and facilities based on customer demand. Key economic decisions are taken about the location, number, and size of manufacturing facilities and warehouses based on the optimal solution. Therefore, improvements in the methods to address this question, which is known to be in the NP-hard complexity class, would have relevant financial consequences. Here, we implement in the D-Wave quantum annealer a hybrid classical-quantum annealing algorithm. The cost function with constraints is translated to a spin Hamiltonian, whose ground state encodes the searched result. As a benchmark, we measure the accuracy of results for a set of paradigmatic problems against the optimal published solutions (the error is on average below $1\%$), and the performance is compared against the classical algorithm, showing a remarkable reduction in the number of iterations. This work shows that state-of-the-art quantum annealers may codify and solve relevant supply-chain problems even still far from useful quantum supremacy.

Polynomial-Time Random Oracles and Separating Complexity Classes

Bennett and Gill [1981] showed that P A ≠ NP A ≠ coNP A for a random oracle A , with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem. (1) We first show that P A ≠ NP A for every oracle A that is p-betting-game random. Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation: (2) If P A ≠ NP A relative to every p-random oracle A , then BPP ≠ EXP. (3) If P A ≠ NP A relative to some p-random oracle A , then P ≠ PSPACE. Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PH A is infinite relative to oracles A that are p-betting-game random. Showing that PH A separates at even its first level would also imply an unrelativized complexity class separation: (4) If NP A ≠ coNP A for a p-betting-game measure 1 class of oracles A , then NP ≠ EXP. (5) If PH A is infinite relative to every p-random oracle A , then PH ≠ EXP. We also consider random oracles for time versus space, for example: (6) L A ≠ P A relative to every oracle A that is p-betting-game random.

The Computational Complexity of Understanding Binary Classifier Decisions

For a d-ary Boolean function Φ: {0, 1}d → {0, 1} and an assignment to its variables x = (x1, x2, . . . , xd) we consider the problem of finding those subsets of the variables that are sufficient to determine the function value with a given probability δ. This is motivated by the task of interpreting predictions of binary classifiers described as Boolean circuits, which can be seen as special cases of neural networks. We show that the problem of deciding whether such subsets of relevant variables of limited size k ≤ d exist is complete for the complexity class NPPP and thus, generally, unfeasible to solve. We then introduce a variant, in which it suffices to check whether a subset determines the function value with probability at least δ or at most δ − γ for 0 < γ < δ. This promise of a probability gap reduces the complexity to the class NPBPP. Finally, we show that finding the minimal set of relevant variables cannot be reasonably approximated, i.e. with an approximation factor d1−α for α > 0, by a polynomial time algorithm unless P = NP. This holds even with the promise of a probability gap.

On the Complexity of Finding the Maximum Entropy Compatible Quantum State

Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated by limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes’ principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is complete for the quantum computational complexity class QSZK, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a procedure polynomial in the number of subsystems that constructs the maximum entropy compatible density operator. Moreover, we extend the Chow–Liu algorithm to the same subclass of quantum states.

Chaotic Multi-Objective Simulated Annealing and Threshold Accepting for Job Shop Scheduling Problem

The Job Shop Scheduling Problem (JSSP) has enormous industrial applicability. This problem refers to a set of jobs that should be processed in a specific order using a set of machines. For the single-objective optimization JSSP problem, Simulated Annealing is among the best algorithms. However, in Multi-Objective JSSP (MOJSSP), these algorithms have barely been analyzed, and the Threshold Accepting Algorithm has not been published for this problem. It is worth mentioning that the researchers in this area have not reported studies with more than three objectives, and the number of metrics they used to measure their performance is less than two or three. In this paper, we present two MOJSSP metaheuristics based on Simulated Annealing: Chaotic Multi-Objective Simulated Annealing (CMOSA) and Chaotic Multi-Objective Threshold Accepting (CMOTA). We developed these algorithms to minimize three objective functions and compared them using the HV metric with the recently published algorithms, MOMARLA, MOPSO, CMOEA, and SPEA. The best algorithm is CMOSA (HV of 0.76), followed by MOMARLA and CMOTA (with HV of 0.68), and MOPSO (with HV of 0.54). In addition, we show a complexity comparison of these algorithms, showing that CMOSA, CMOTA, and MOMARLA have a similar complexity class, followed by MOPSO.

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ time under the exponential-time hypothesis.

Total functions in QMA

The complexity class QMA is the quantum analog of the classical complexity class NP. The functional analogs of NP and QMA, called functional NP (FNP) and functional QMA (FQMA), consist in either outputting a (classical or quantum) witness, or outputting NO if there does not exist a witness.The classical complexity class Total Functional NP (TFNP) is the subset of FNP for which it can be shown that the NO outcome never occurs. TFNP includes many natural and important problems. In the present work we introduce the complexity class of Total Functional QMA (TFQMA), the quantum analog of TFNP. We show that FQMA and TFQMA can be defined in such a way that they do not depend on the values of the completeness and soundness probabilities. In so doing we introduce new notions such as the eigenbasis and spectrum of a quantum verification procedure which are of interest by themselves. We then provide examples of problems that lie in TFQMA, coming from areas such as the complexity of k-local Hamiltonians and public key quantum money. In the context of black-box groups, we note that Group Non-Membership, which was known to belong to QMA, in fact belongs to TFQMA. We also provide a simple oracle with respect to which we have a separation between FBQP and TFQMA. In the conclusion we discuss the relation between TFQMA, public key quantum money, and the complexity of quantum states.

Клеточные автоматы и их обобщения в задачах криптографии. Часть 1

The purpose of the article is an analytical review of the application of cellular automata and their generalizations in cryptography. Research method: an analysis of scientific publications on the topic of the article. Results: The review article analyzes the literature devoted to the use of classical cellular automata and their generalizations for the construction of cryptographic algorithms. The article consists of two parts. The first part is devoted to classical cellular automata and symmetric cryptographic algorithms based on them. It briefly discusses the history of the theory of cellular automata and its applications in various scientific disciplines. The review of the works of a number of authors who proposed symmetric cryptographic algorithms and pseudorandom sequence generators based on one-dimensional cellular automata is presented. The security of such cryptographic algorithms turned out to be insufficient. The following is a review of articles devoted to the use of two-dimensional cellular automata for constructing ciphers (this approach gave the best results). Multidimensional cellular automata are also mentioned. The second part of the article will be devoted to a review of works devoted to the use of generalized cellular automata in cryptography – on the basis of such automata, it is possible to create symmetric encryption algorithms and cryptographic hash functions that provide a high level of security and high performance in hardware implementation (for example, on FPGA), as well as having fairly low requirements for hardware resources. In addition, an attention will be paid to interesting connections of generalized cellular automata, in the context of their use in cryptography, with the theory of expander graphs. Attention will also be paid to the security of cryptographic algorithms based on generalized cellular automata. The works devoted to the implementation of various cryptographic algorithms based on generalized cellular automata on FPGA and GPU will be mentioned. In addition, an overview of asymmetric cryptographic algorithms based on cellular automata will be given. The questions about the belonging of some problems on cellular automata and their generalizations to the class of NP-complete problems, as well as to some other complexity classes, will also be considered.

A cross-sectional study on the assessment of the complexity of removable dental prosthesis at a tertiary hospital in Nigeria

Background: Oral rehabilitation using removable dental prosthesis for patients with maxillofacial defects and missing teeth could pose a challenge while trying to achieve prosthetic treatment goals. It is important therefore to determine how complex the case is before proceeding with treatment. Aim: The aim of the study was to assess the complexity of removable dental prosthesis at a tertiary hospital in Nigeria using the Restorative Index of Treatment Need (RIOTN) System. Materials and Methods: This was a descriptive cross-sectional study of patients who sought for removable prostheses. Data were collated by means of an interviewer-administered question which collated data on biodemographic characteristics of the participant, indication for removable prosthesis, tooth to be replaced with removable prosthesis. Following a clinical examination, the RIOTN System was applied to assess the complexity of the treatment needed. Results: Ninety-eight adult patients with age ranging from 18 years to 90 years with a mean age of 45.17 ± 18.06 years participated in the study. Partial dentures were the most prevalent prostheses provided (91.8%). The most prevalent complexity grade recorded was Grade I (84.7%). There was a statistically significant association between complexity and Kennedy's class of saddle (P < 0.0001) as well as the type of support for the removable prosthesis (P < 0.0001). Conclusion: The pattern of the complexity of treatment using removable dental prostheses was dependent on the type of prosthesis, teeth replaced, support, and saddle.

Bosons vs. Fermions – A computational complexity perspective

Recent years have seen a flurry of activity in the fields of quantum computing and quantum complexity theory, which aim to understand the computational capabilities of quantum systems by applying the toolbox of computational complexity theory. This paper explores the conceptually rich and technologically useful connection between the dynamics of free quantum particles and complexity theory. I review results on the computational power of two simple quantum systems, built out of noninteracting bosons (linear optics) or noninteracting fermions. These rudimentary quantum computers display radically different capabilities—while free fermions are easy to simulate on a classical computer, and therefore devoid of nontrivial computational power, a free-boson computer can perform tasks expected to be classically intractable. To build the argument for these results, I introduce concepts from computational complexity theory. I describe some complexity classes, starting with P and NP and building up to the less common #P and polynomial hierarchy, and the relations between them. I identify how probabilities in free-bosonic and free-fermionic systems fit within this classification, which then underpins their difference in computational power. This paper is aimed at graduate or advanced undergraduate students with a Physics background, hopefully serving as a soft introduction to this exciting and highly evolving field.

Methods for optimizing routes in digital logistics

The current problem of digital logistics is investigated - the calculation of optimal routes for freight transportation by computer means to reduce time and distance. Heuristic methods used in logistics for constructing optimal routes are considered. A comparative analysis of ten methods for solving the optimization problem of the “nondeterministic polynomial time” complexity class traveling salesman is carried out. The study performs a comparative analysis of the following methods: “convex hull, cheapest insertion and angle selection”, “greedy”, “greedy-cycle”, “integer-linear-programming”, “or-opt”, “or-zweig”, “remove crossings”, “space filling curve”, “simulated annealing”, “two-opt”. A computational experiment is performed, on the basis of which the accuracy and computational complexity of the considered methods are estimated. The results of the computational experiment show the construction of the optimal route by the “integer-linear-programming” method and the highest computation speed for the “greedy” method. Application of the “integer-linear-programming” method in logistics is the most accurate at the optimal time for calculating efficient routes of freight traffic.

Efficient Solution of Distributed MIP in Control of Networked Systems

AbstractCertain classes of optimization‐based control problems stated for networked systems involving hybrid dynamics and logical constraints can be cast into Mixed‐Integer Programming (MIP) problems. Since these belong to the complexity class NP‐hard, the motivation arises to find approximations of the optimal solution by distributed solution efficiently. For the cases that the cost functional is linear or quadratic and the constraints are linear, this paper proposes an alternative to the standard centralized schemes, by employing dual decomposition into a set of local problems of moderate size which can be solved in parallel. Numerical examples demonstrate that the scheme can efficiently approximate the global solution.

A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation

Holant problems are a family of counting problems parameterised by sets of algebraic-complex valued constraint functions, and defined on graphs. They arise from the theory of holographic algorithms, which was originally inspired by concepts from quantum computation. Here, we employ quantum information theory to explain existing results about holant problems in a concise way and to derive two new dichotomies: one for a new family of problems, which we call Holant$^+$, and, building on this, a full dichotomy for Holant$^c$. These two families of holant problems assume the availability of certain unary constraint functions -- the two pinning functions in the case of Holant$^c$, and four functions in the case of Holant$^+$ -- and allow arbitrary sets of algebraic-complex valued constraint functions otherwise. The dichotomy for Holant$^+$ also applies when inputs are restricted to instances defined on planar graphs. In proving these complexity classifications, we derive an original result about entangled quantum states.

On the Model Checking Problem for Some Extension of CTL*

Sequential reactive systems include programs and devices that work with two streams of data and convert input streams of data into output streams. Such information processing systems include controllers, device drivers, computer interpreters. The result of the operation of such computing systems are infinite sequences of pairs of events of the request-response type, and, therefore, finite transducers are most often used as formal models for them. The behavior of transducers is represented by binary relations on infinite sequences, and so, traditional applied temporal logics (like HML, LTL, CTL, mu-calculus) are poorly suited as specification languages, since omega-languages, not binary relations on omega-words are used for interpretation of their formulae. To provide temporal logics with the ability to define properties of transformations that characterize the behavior ofreactive systems, we introduced new extensions ofthese logics, which have two distinctive features: 1) temporal operators are parameterized, and languages in the input alphabet oftransducers are used as parameters; 2) languages in the output alphabet oftransducers are used as basic predicates. Previously, we studied the expressive power ofnew extensions Reg-LTL and Reg-CTL ofthe well-known temporal logics oflinear and branching time LTL and CTL, in which it was allowed to use only regular languages for parameterization of temporal operators and basic predicates. We discovered that such a parameterization increases the expressive capabilities oftemporal logic, but preserves the decidability of the model checking problem. For the logics mentioned above, we have developed algorithms for the verification of finite transducers. At the next stage of our research on the new extensions of temporal logic designed for the specification and verification of sequential reactive systems, we studied the verification problem for these systems using the temporal logic Reg-CTL*, which is an extension ofthe Generalized Computational Tree Logics CTL*. In this paper we present an algorithm for checking the satisfiability of Reg-CTL* formulae on models of finite state transducers and show that this problem belongs to the complexity class ExpSpace.