Complexity Classes Research Articles

P vs NP: P is Equal to NP: Desired Proof

Computations and computational complexity are fundamental for mathematics and all computer science, including web load time, cryptography (cryptocurrency mining), cybersecurity, artificial intelligence, game theory, multimedia processing, computational physics, biology (for instance, in protein structure prediction), chemistry, and the P vs. NP problem that has been singled out as one of the most challenging open problems in computer science and has great importance as this would essentially solve all the algorithmic problems that we have today if the problem is solved, but the existing complexity is deprecated and does not solve complex computations of tasks that appear in the new digital age as efficiently as it needs. Therefore, we need to realize a new complexity to solve these tasks more rapidly and easily. This paper presents proof of the equality of P and NP complexity classes when the NP problem is not harder to compute than to verify in polynomial time if we forget recursion that takes exponential running time and goes to regress only (every problem in NP can be solved in exponential time, and so it is recursive, this is a key concept that exists, but recursion does not solve the NP problems efficiently). The paper’s goal is to prove the existence of an algorithm solving the NP task in polynomial running time. We get the desired reduction of the exponential problem to the polynomial problem that takes O(log n) complexity.

The Complexity of Counting Problems Over Incomplete Databases

We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query q , we consider the following two problems: Given as input an incomplete database D , (a) return the number of completions of D that satisfy q ; or (b) return the number of valuations of the nulls of D yielding a completion that satisfies q . We obtain dichotomies between #P-hardness and polynomial-time computability for these problems when q is a self-join–free conjunctive query and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in D (what is called Codd tables ); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations: For instance, while the latter is always in #P, we prove that the former is not in #P under some widely believed theoretical complexity assumption. Moreover, we find that both (1) and (2) can reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial-time randomized approximation scheme (FPRAS), in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes.

A Graph Theoretic Approach for Multi-Objective Budget Constrained Capsule Wardrobe Recommendation

Traditionally, capsule wardrobes are manually designed by expert fashionistas through their creativity and technical prowess. The goal is to curate minimal fashion items that can be assembled into several compatible and versatile outfits. It is usually a cost and time intensive process, and hence lacks scalability. Although there are a few approaches that attempt to automate the process, they tend to ignore the price of items or shopping budget. In this article, we formulate this task as a multi-objective budget constrained capsule wardrobe recommendation ( MOBCCWR ) problem. It is modeled as a bipartite graph having two disjoint vertex sets corresponding to top-wear and bottom-wear items, respectively. An edge represents compatibility between the corresponding item pairs. The objective is to find a 1-neighbor subset of fashion items as a capsule wardrobe that jointly maximize compatibility and versatility scores by considering corresponding user-specified preference weight coefficients and an overall shopping budget as a means of achieving personalization. We study the complexity class of MOBCCWR , show that it is NP-Complete, and propose a greedy algorithm for finding a near-optimal solution in real time. We also analyze the time complexity and approximation bound for our algorithm. Experimental results show the effectiveness of the proposed approach on both real and synthetic datasets.

Quantum information distance

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.

Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms , directed homomorphisms , and injective directed homomorphisms , and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: • The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. • All the polynomial families are hard under p -projections.

Query complexity of generalized Simon's problem

Simon's problem plays an important role in the history of quantum algorithms, as it inspired Shor to discover the celebrated quantum algorithm solving integer factorization in polynomial time. Besides, the quantum algorithm for Simon's problem has been recently applied to break symmetric cryptosystems. Generalized Simon's problem, denoted by GSP(p,n,k), is a natural extension of Simon's problem: Given a function f:Zpn→X where X is a finite set and the promise that for any x,y∈Zpn,f(x)=f(y) iff x−y∈S for a subgroup S≤Zpn of rank k<n, the goal is to find S. In this paper we consider the query complexity of the problem, that is, the minimum number of queries to f required to find S. First, it is not difficult to design a quantum algorithm solving the above problem with query complexity of O(n−k). However, so far it is not clear what is the classical query complexity of the problem, and revealing this complexity is necessary for clarifying the computational power gap between quantum and classical computing on the problem.To tackle this problem, we prove that any classical (deterministic or randomized) algorithm for GSP(p,n,k) has to query at least Ω(max⁡{k,pn−k}) values and any classical nonadaptive deterministic algorithm for GSP(p,n,k) has to query at least Ω(max⁡{k,k⋅pn−k}) values. Hence, we clearly show the classical computing model is less powerful than the quantum counterpart, in terms of query complexity for the generalized Simon's problem. Moreover, we obtain an upper bound O(max⁡{k,k⋅pn−k}) on the classical deterministic query complexity of GSP(p,n,k), by devising a subtle classical algorithm based on group theory and the divide-and-conquer approach. Therefore, we have an almost full characterization of the classical deterministic query complexity of the generalized Simon's problem.

Quantum verification of NP problems with single photons and linear optics

Quantum computing is seeking to realize hardware-optimized algorithms for application-related computational tasks. NP (nondeterministic-polynomial-time) is a complexity class containing many important but intractable problems like the satisfiability of potentially conflict constraints (SAT). According to the well-founded exponential time hypothesis, verifying an SAT instance of size n requires generally the complete solution in an O(n)-bit proof. In contrast, quantum verification algorithms, which encode the solution into quantum bits rather than classical bit strings, can perform the verification task with quadratically reduced information about the solution in tilde O(sqrt n ) qubits. Here we realize the quantum verification machine of SAT with single photons and linear optics. By using tunable optical setups, we efficiently verify satisfiable and unsatisfiable SAT instances and achieve a clear completeness-soundness gap even in the presence of experimental imperfections. The protocol requires only unentangled photons, linear operations on multiple modes and at most two-photon joint measurements. These features make the protocol suitable for photonic realization and scalable to large problem sizes with the advances in high-dimensional quantum information manipulation and large scale linear-optical systems. Our results open an essentially new route toward quantum advantages and extend the computational capability of optical quantum computing.

Complexity of stability

Graph parameters such as the clique number and the chromatic number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs in terms of their computational complexity. We show for various central graph parameters that deciding the stability of a given graph is complete for Θ2p, a well-known complexity class in the second level of the polynomial hierarchy.

FV 12. Non-invasive Suppression of Essential Tremor via Phase-Locked Disruption of its Temporal Coherence

Aberrant neural oscillations hallmark the pathophysiology of numerous neurological and psychiatric disorders. Here, we first report a method to accurately track the phase of neural oscillations in real-time by a Hilbert transform that avoids the characteristic Gibbs distortion at the end of the signal, aka endpoint-corrected Hilbert transform (ecHT). The ecHT method maintains the same computational complexity class of the original Hilbert transform allowing implementation in simple digital hardware. We then used the ecHT method to show that the aberrant neural oscillation that hallmarks treatment-resistant essential tremor (ET), the most common adult movement disorder, can be noninvasively supressed via transcranial electrical stimulation at a fixed phase lag over the cerebellar hemisphere ipsilateral to the tremor movement. This was tested in a quadruple-replication randomized way including stimulation at 6 fixed phases, sham and without phase-locking. In a total of 11 subjects, the suppression of ET activity was sustained after the end of the stimulation and was phenomenologically predicted, post-hoc, from the features of the tremor movement before the start of the stimulation. To test for reproducibility, 6 of the original participants werer stimulated exactly the same way three years after the original experiments: the observed, significant stimulation response remained, i.e. responders continued to respond and non-responders did not. Finally, we used a highly-comparative feature extraction (> 8000 features) with statistical learning and neurophysiological computational modelling to show that the suppression of ET activity can be mechanistically attributed to a disruption of the temporal coherence in the tremor movement that can be originated in a higher bursting entropy at the cortico-olivo-cerebello-thalamic circuitry. The suppression of aberrant neural oscillation via phase-locked driven disruption of temporal coherence may represent a powerful neuromodulatory strategy to treat neurological and psychiatric disorders.

Measuring the Overall Complexity of Graphical and Textual IEC 61131-3 Control Software

Software implements a significant proportion of functionality in factory automation. Thus, efficient development and the reuse of software parts, so-called units, enhance competitiveness. Thereby, complex control software units are more difficult to understand, leading to increased development, testing and maintenance costs. However, measuring complexity is challenging due to many different, subjective views on the topic. This paper compares different complexity definitions from literature and considers with a qualitative questionnaire study the complexity perception of domain experts, who confirm the importance of objective measures to compare complexity. The paper proposes a set of metrics that measure various classes of software complexity to identify the most complex software units as a prerequisite for refactoring. The metrics include complexity caused by size, data structure, control flow, information flow and lexical structure. Unlike most literature approaches, the metrics are compliant with graphical and textual languages from the IEC 61131-3 standard. Further, a concept for interpreting the metric results is presented. A comprehensive evaluation with industrial software from two German plant manufacturers validates the metrics' suitability to measure complexity.

Entropy-based economic complexity of China trade flows

The growing interest in the study of economic complexity has been motivated by the intrinsic features of statistical physics. Entropy is one of essential concepts in statistical physics and a particularly active concept for characterizing the complexity of a system. Motived by this, we propose a new method based on a function of the entropy to quantify economic complexity of China trade flows. We focus on regional economy activities, and collect China trade flows according to the database of national export and import items. Compared with classic economic complexity measurements including diversity and ubiquity, eigenvector-based complexity index, fitness and complexity index, Hirschman-Herfindahl index, the proposed method based on entropy can generate a proper ranking list when we take USA as a benchmark. Furthermore, the comparison of the main economic complexity statics also exposes that several of which performance moderate correlations. The stability of the ranking positions from the static evolution of the economic complexity over 8 years discloses that the index of economic complexity performs extremely sensitive depending on the eigenvector of the product network space. The results reveal that we should pay attention to the relationships between economic complexity methods when analysing the structural characteristics of economic entity activities.

An Ising Hamiltonian solver based on coupled stochastic phase-transition nano-oscillators

Combinatorial optimization problems belong to the non-deterministic polynomial time (NP)-hard complexity class, and their computational requirements scale exponentially with problem size. They can be mapped into the problem of finding the ground state of an Ising model, which describes a physical system with converging dynamics. Various platforms, including optical, electronic and quantum approaches, have been explored to accelerate the ground-state search, but improvements in energy efficiencies and computational abilities are still required. Here we report an Ising solver based on a network of electrically coupled phase-transition nano-oscillators (PTNOs) that form a continuous-time dynamical system (CTDS). The bi-stable phases of the injection-locked PTNOs act as artificial Ising spins and the stable points of the CTDS act as the ground-state solution of the problem. We experimentally show that a prototype with eight PTNOs can solve an NP-hard MaxCut problem with high probability of success (96% for 600 annealing cycles). We also show via numerical simulations that our Ising Hamiltonian solver can solve MaxCut problems of 100 nodes with energy efficiency of 1.3 × 107 solutions per second per watt, offering advantages over other approaches including memristor-based Hopfield networks, quantum annealers and photonic Ising solvers. An Ising solver that is based on a network of electrically coupled phase-transition nano-oscillators, which provides a continuous-time dynamical system, can be used to efficiently solve a non-deterministic polynomial time (NP)-hard MaxCut problem.

Complexity of near-optimal robust versions of multilevel optimization problems

Near-optimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of near-optimal robust multilevel problems, where near-optimal robustness is modelled through additional adversarial decision-makers. Near-optimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without near-optimality robustness under general conditions.

Nonuniform families of polynomial-size quantum finite automata and quantum logarithmic-space computation with polynomial-size advice

The state complexity of a finite(-state) automaton intuitively measures the size of the description of the automaton. Sakoda and Sipser [STOC 1972, pp. 275–286] were concerned with nonuniform families of finite automata and they discussed the behaviors of the nonuniform complexity classes defined by such families of finite automata having polynomial-size state complexity. In a similar fashion, we introduce state complexity classes using nonuniform families of quantum finite automata empowered by the flexible use of garbage tapes. We first present general inclusion and separation relationships among state complexity classes of nonuniform families of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata having polynomially many inner states. For two-way quantum finite automata equipped with flexible garbage tapes, we show a close relationship between the state complexity of the nonuniform family of such polynomial-size quantum finite automata and the parameterized complexity class induced by logarithmic-space quantum computation assisted by polynomial-size advice. We further establish a direct connection between space-bounded quantum computation with quantum advice and quantum finite automata whose transitions are dictated by superpositions of transition tables.

The classes PPA-k: Existence from arguments modulo k

The complexity classes PPA-k, k≥2, have recently emerged as the main candidates for capturing the complexity of important problems in fair division, in particular Alon's Necklace-Splitting problem with k thieves. Indeed, the problem with two thieves has been shown complete for PPA = PPA-2. In this work, we present structural results which provide a solid foundation for the further study of these classes. Namely, we investigate the classes PPA-k in terms of (i) equivalent definitions, (ii) inner structure, (iii) relationship to each other and to other TFNP classes, and (iv) closure under Turing reductions.

A two-stage optimization approach for inspection plan formulation of comprehensive inspection train: The China case

The regular and dynamic inspections of high-speed railways (HSR) are carried out by comprehensive inspection train (CIT). Generally, the network of HSR (e.g., the HSR network in China) has some obvious characteristics such as complex network structure, high traffic density and strict maintenance restrictions of CIT, which make the formulation of CIT inspection plan extremely hard. However, formulating and optimizing the CIT inspection plan is one of the key issues to ensure the safety of HSR operations and to improve the efficiency of testing equipment. In this paper, the original model is proposed for the multiple CITs inspection plan formulation problem (MCIT-IPFP) firstly, and it is proved that the problem belongs to the NP-hard class of computational complexity by reduction. By decomposing the problem into two stages, one is used to divide the HSR network evenly for each CIT, and the other one is used to generate the path, the timetable, and maintenance base assignment for each CIT under requirements of reducing the total time cost and the difficulty of solving. In order to improve the efficiency of calculation, a Constructive Heuristic Algorithm with feedback regulation based on the greedy algorithm is designed to generate the task division plan. With the purpose of quickly formulating the inspection plans with low time cost for each CIT with Euler circuit characteristics, a simulated annealing algorithm with efficient neighborhood Euler circuit generation strategy is proposed. Finally, a case study is conducted to test the efficiency of the algorithm by taking the China’s HSR network as the background. Compared with the previous manual inspection plan, the utilization rate of CITs will increase by 30%-50% when using the models and algorithms proposed in this paper.

Sorting by Multi-Cut Rearrangements

A multi-cut rearrangement of a string S is a string S′ obtained from S by an operation called k-cut rearrangement, that consists of (1) cutting S at a given number k of places in S, making S the concatenated string X1·X2·X3·…·Xk·Xk+1, where X1 and Xk+1 are possibly empty, and (2) rearranging the Xis so as to obtain S′=Xπ(1)·Xπ(2)·Xπ(3)·…·Xπ(k+1), π being a permutation on 1,2,…,k+1 satisfying π(1)=1 and π(k+1)=k+1. Given two strings S and T built on the same multiset of characters from an alphabet Σ, the Sorting by Multi-Cut Rearrangements (SMCR) problem asks whether a given number ℓ of k-cut rearrangements suffices to transform S into T. The SMCR problem generalizes several classical genomic rearrangements problems, such as Sorting by Transpositions and Sorting by Block Interchanges. It may also model chromoanagenesis, a recently discovered phenomenon consisting of massive simultaneous rearrangements. In this paper, we study the SMCR problem from an algorithmic complexity viewpoint. More precisely, we investigate its classical and parameterized complexity, as well as its approximability, in the general case or when S and T are permutations.

English

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The algorithm is exponential in complexity, but the well-known algorithms, such as the extended Euclidean algorithm and Buchberger's algorithm, are superexponential. The superexponential complexity of the well-known algorithms is due to their not being "minimal" in a certain sense. Buchberger's algorithm produces a Groebner basis. The proposed original reduction algorithm achieves the required task, via computation of determinants of parametric Sylvester matrices, and produces a Rabin basis, which is shown to be minimal, when two multivariate polynomials are reduced at a time. The minimality of Rabin basis allows us to prove exponential lower bounds for the space complexity of an algebraic proof of certification, for a specific computational problem in the computational complexity class PSPACE, showing that the complexity classes PSPACE and P cannot be the same. By the same reasoning, it follows that Co-NP is not the same as either of the complexity classes NP and P, and that the polynomial time hierarchy does not collapse. It is also shown that the class BPP of languages decidable by bounded error probabilistic algorithms with (probabilistic) polynomial time proofs for the membership of input words is not the same as any one of the complexity classes P, NP and Co-NP. It follows again, from the discussions, that the complexity classes NP and P are not the same, by relativization of BPP, with respect to P and NP.

Efficient Classical Simulation and Benchmarking of Quantum Processes in the Weyl Basis.

One of the crucial steps in building a scalable quantum computer is to identify the noise sources which lead to errors in the process of quantum evolution. Different implementations come with multiple hardware-dependent sources of noise and decoherence making the problem of their detection manyfoldly more complex. We develop a randomized benchmarking algorithm which uses Weyl unitaries to efficiently identify and learn a mixture of error models which occur during the computation. We provide an efficiently computable estimate of the overhead required to compute expectation values on outputs of the noisy circuit relying only on the locality of the interactions and no further assumptions on the circuit structure. The overhead decreases with the noise rate and this enables us to compute analytic noise bounds that imply efficient classical simulability. We apply our methods to ansatz circuits that appear in the variational quantum eigensolver and establish an upper bound on classical simulation complexity as a function of noise, identifying regimes when they become classically efficiently simulatable.

On Fair and Efficient Allocations of Indivisible Goods

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation, and a non-constructive proof of existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO). We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive, and that it can be computed in pseudo-polynomial time. We also consider the class of k-ary instances where k is a constant, i.e., each agent has at most k different values for the goods. We show that for such instances an EF1+fPO allocation can be computed in polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in polynomial time. Next, we consider instances where the number of agents is constant, and show that an EF1+PO (also EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. Further, we show that the problem of computing an EF1+PO allocation polynomial-time reduces to a problem in the complexity class PLS. We also design a polynomial-time algorithm that computes Nash welfare maximizing allocations when there are constantly many agents with constant many different values for the goods.