Complexity Classes Research Articles

The Complexity of Pacing for Second-Price Auctions

Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a pacing multiplier between zero and one, and her bid on each item is multiplicatively scaled down by the pacing multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is known as a pacing equilibrium. In this work, we show that the problem of finding an approximate pacing equilibrium is PPAD-complete for second-price auctions. This resolves an open question of Conitzer et al. [Conitzer V, Kroer C, Sodomka E, Stier-Moses NE (2022a) Multiplicative pacing equilibria in auction markets. Oper. Res. 70(2):963–989]. As a consequence of our hardness result, we show that the tâtonnement-style budget-management dynamics introduced by Borgs et al. [Borgs C, Chayes J, Immorlica N, Jain K, Etesami O, Mahdian M (2007) Dynamics of bid optimization in online advertisement auctions. Proc. 16th Internat. Conf. World Wide Web (ACM, New York), 531–540] are unlikely to converge efficiently for repeated second-price auctions. This disproves a conjecture by Borgs et al. [Borgs C, Chayes J, Immorlica N, Jain K, Etesami O, Mahdian M (2007) Dynamics of bid optimization in online advertisement auctions. Proc. 16th Internat. Conf. World Wide Web (ACM, New York), 531–540], under the assumption that the complexity class PPAD is not equal to P. Our hardness result also implies the existence of a refinement of supply-aware market equilibria which is hard to compute with simple linear utilities. Funding: This work was supported by National Science Foundation (CCF-1703925, IIS-1838154).

A novel homomorphic polynomial public key encapsulation algorithm

Background: One of the primary drivers in development of novel quantum-safe cryptography techniques is the ongoing National Institute of Standards and Technology (NIST) Post-Quantum Cryptography (PQC) competition, which aims to identify quantum-safe algorithms for standardization. Although NIST has recently announced candidates to be standardized, the development of novel PQC algorithms remains desirable to address the challenges of quantum computing. Furthermore, to enhance security and improve performance. Methods: This paper introduces a novel public key encapsulation algorithm that incorporates an additional layer of encryption during key construction procedure, through a hidden ring. This encryption involves modular multiplication over the hidden ring using a homomorphism operator that is closed under addition and scalar multiplication. The homomorphic encryption key is comprised of two values - one used to create the hidden ring and the other to form an encryption operator. This homomorphic encryption can be applied to any polynomials during key construction over a finite field with their coefficients considered private. Particularly, the proposed homomorphic encryption operator can be applied to the public key of the Multivariate Public Key Cryptography schemes (MPKC) to hide the structure of its central map construction. Results: This paper presents a new variant of the MPKC with its public key encrypted using the proposed homomorphic operator. This novel scheme is called the Homomorphic Polynomial Public Key (HPPK) algorithm, which simplifies MPKC central map to two multivariate polynomials constructed from polynomial multiplications. The HPPK algorithm employs a single polynomial vector for the plaintext and a multi-variate noise vector associated with the central map. In contrast, in MPKC, a single multivariate vector is created by segmenting the secret plaintext over a small finite field. The HPPK algorithm is Indistinguishability Under Chosen-Plaintext Attack (IND-CPA) secure, and its classical complexity for cracking is exponential in the size of the prime field GF(p).

CHR++: An efficient CHR system in C++ with don’t know non-determinism

This article introduces CHR++, an implementation of CHR (Constraint Handling Rules) in C++. CHR++ has been designed to be efficient, user-friendly, and to make it easier to model problems of NP (Non-deterministic Polynomial time) complexity class thanks to the support of don’t know non-determinism. CHR++ conforms to the refined semantics of CHR and provides a high-level syntax similar to the historical CHR implementations in Prolog. It is a powerful system that incorporates many of the high-level optimisations known to date. Like implementations made on top of Prolog, it handles don’t know non-determinism through support for the CHR extension. CHR++ is the first non-Prolog CHR system that natively supports don’t know non-determinism. It makes it easier to model constrained decision or optimisation problems and shows a significant gain in efficiency compared with CHR Prolog systems. It equals or exceeds the fastest CHR systems in Java and C, but is able to tackle a whole class of problems unattainable by those systems.

METHODS OF NAVIGATING ALGORITHMIC COMPLEXITY: BIG-OH AND SMALL-OH NOTATIONS

This article provides an in-depth exploration of Big-Oh and small-oh notations, shedding light on their practical implications in the analysis of algorithm complexity. Big-Oh notation offers a valuable tool for estimating an upper bound on the growth rate of an algorithm's running time, whereas small-oh notation delineates a lower limit on this growth rate. The piece delves into a comprehensive examination of various complexity classes that emerge through the application of Big-Oh notation, underscoring the significance of small-oh notation as it complements and enriches complexity analysis. In the realm of programming and computer science, the employment of these notations holds paramount importance. They empower developers and researchers to make informed decisions regarding algorithm selection and optimization. It is crucial to recognize that while complexity analysis is a vital facet of effective programming, ongoing research endeavors may yield more refined methodologies and approaches within this domain. By understanding and harnessing the power of Big-Oh and small-oh notations, professionals can effectively evaluate algorithm efficiency and scalability. This knowledge equips them with the ability to design and implement algorithms that meet specific performance criteria, which is pivotal in the ever-evolving landscape of technology and computation. As pushing the boundaries of what is possible in the field of algorithm design is being continued, these notations remain invaluable tools for navigating the complex terrain of algorithmic analysis and optimization. By embracing Big-Oh and small-oh notations, professionals can finely assess algorithmic efficiency, ensuring they meet performance criteria in the evolving technological landscape. These notations remain indispensable for algorithmic analysis.

A Formal Verification of a SAT Solution by P Systems with Evolution Communication and Separation Rules

In the framework of membrane computing, several interesting results concerning frontiers of efficiency between the complexity classes P and NP have been found by using different ingredients. One of the main characteristics of cell-like membrane systems is their rooted tree-like structure, where a natural parent-children membrane relationship exists, and objects can travel through the membranes. Separation rules are used as a method to obtain an exponential workspace in terms of membranes in polynomial time. Inspired by cell meiosis, objects from the original membrane are distributed between the two new membranes. In this work, P systems with evolutional symport/antiport rules and separation rules are used to give a solution to SAT, a well known NP-complete problem. One of the advantages of this solution is the use of the environment as a passive agent.

The Complexity of the Hausdorff Distance

We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class { forall exists _{<}mathbb {R}} . This implies that the problem is NP-, co-NP-, exists mathbb {R} -, and forall mathbb {R} -hard.

Comparing mathematical modeling approaches for compact objects: vanishing complexity and embedding class one approaches in spherically symmetric systems with static background

This study delves into the realm of solving Einstein’s field equations within the framework of general relativity. In this paper, we present an exact solution derived through the vanishing complexity approach and offer a comparative analysis with the established embedding class one approach. This exploration sheds light on the efficacy and validity of these methods in generating solutions for Einstein’s field equations. Our study involves a comprehensive exploration of several key parameters, encompassing thermodynamic factors, causality conditions, stability evaluations, as well as analyses of the mass function. Based on our findings, it can be suggested that the vanishing complexity approach could serve as a viable alternative method to the embedding class one approach for the derivation of exact solutions of Einstein’s field equations.

The complexity of NISQ

The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class NISQ, which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, NISQ, and fault-tolerant quantum computation to solve some problems based on modifications of Simon’s problems. We then consider the power of NISQ for three well-studied problems. For unstructured search, we prove that NISQ cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that NISQ only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that NISQ is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits.

Predicting Peri-Operative Outcomes in Patients Treated with Percutaneous Thermal Ablation for Small Renal Masses: The SuNS Nephrometry Score.

Our objective was to develop a new, simple, and ablation-specific nephrometry score to predict peri-operative outcomes and to compare its predictive accuracy to PADUA and RENAL scores. Overall, 418 patients were treated with percutaneous thermal ablation (microwave and radiofrequency) between 2008 and 2021. The outcome of interest was trifecta status (achieved vs. not achieved): incomplete ablation or Clavien-Dindo ≥ 3 complications or postoperative estimated glomerular filtration rate decrease ≥ 30%. First, we validated the discrimination ability of the PADUA and RENAL scoring systems. Second, we created and internally validated a novel scoring (SuNS) system, according to multivariable logistic regression models. The predictive accuracy of the model was tested in terms of discrimination and calibration. Overall, 89 (21%) patients did not achieve trifecta. PADUA and RENAL scores showed poor ability to predict trifecta status (c-indexes 0.60 [0.53-0.67] and 0.62 [0.55-0.69], respectively). We, therefore, developed the SuNS model (c-index: 0.74 [0.67-0.79]) based on: (1) contact surface area; (2) nearness to renal sinus or urinary collecting system; (3) tumour diameter. Three complexity classes were created: low (3-4 points; 11% of no trifecta) vs. moderate (5-6 points; 30% of no trifecta) vs. high (7-8 points; 65% of no trifecta) complexity. Limitations include the retrospective and single-institution nature of the study. In conclusion, we developed an immediate, simple, and reproducible ablation-specific nephrometry score (SuNS) that outperformed PADUA and RENAL nephrometry scores in predicting peri-operative outcomes. External validation is required before daily practice implementation.

Felsbauplanung mit dem geotechnischen Werkzeugkasten von morgen: Eurocode 7 – Allgemeine Regeln (EN 1997‐1:2024)

AbstractThe second generation of Eurocode 7 aims at covering ground, including both soil and rock. Current practice of ground engineering forms the basis, and the aim is to include rock engineering, improving ease of use, harmonization of regulations, and covering new developments. This paper deals with EN 1997 part 1 addressing the general rules of geotechnical design, connected to EN 1990, the basis of structural and geotechnical design, and parts 2 and 3 of EN 1997. The design process from ground investigation to the verification of limit states is illustrated with a focus on rock engineering. The following topics are dealt with in more detail: Geotechnical Categories: They are now based on Consequence Classes and newly introduced Geotechnical Complexity Classes. A limit state design to be sufficient has to be combined with other checks depending on these classes. Replacement of the Design Approaches of the current Eurocode 7: Ultimate limit states will be verified by a combination of Verification Cases with partial factors on actions and action effects, and sets of partial factors on ground properties and resistances, respectively. Application of numerical methods, where a ULS‐verification with two sets of partial factors is defined. Use of probabilistic methods in design by calculation, especially appropriate in cases where properties of discontinuities govern the design. Practice of the Observational Method, including design variants and regulations about their choice.

A bulk manifestation of Krylov complexity

There are various definitions of the concept of complexity in Quantum Field Theory as well as for finite quantum systems. For several of them there are conjectured holographic bulk duals. In this work we establish an entry in the AdS/CFT dictionary for one such class of complexity, namely Krylov or K-complexity. For this purpose we work in the double-scaled SYK model which is dual in a certain limit to JT gravity, a theory of gravity in AdS2. In particular, states on the boundary have a clear geometrical definition in the bulk. We use this result to show that Krylov complexity of the infinite-temperature thermofield double state on the boundary of AdS2 has a precise bulk description in JT gravity, namely the length of the two-sided wormhole. We do this by showing that the Krylov basis elements, which are eigenstates of the Krylov complexity operator, are mapped to length eigenstates in the bulk theory by subjecting K-complexity to the bulk-boundary map identifying the bulk/boundary Hilbert spaces. Our result makes extensive use of chord diagram techniques and identifies the Krylov basis of the boundary quantum system with fixed chord number states building the bulk gravitational Hilbert space.

Code-routing: a new attack on position verification

The cryptographic task of position verification attempts to verify one party's location in spacetime by exploiting constraints on quantum information and relativistic causality. A popular verification scheme known as f-routing involves requiring the prover to redirect a quantum system based on the value of a Boolean function f. Cheating strategies for the f-routing scheme require the prover use pre-shared entanglement, and security of the scheme rests on assumptions about how much entanglement a prover can manipulate. Here, we give a new cheating strategy in which the quantum system is encoded into a secret-sharing scheme, and the authorization structure of the secret-sharing scheme is exploited to direct the system appropriately. This strategy completes the f-routing task using O(SPp(f)) EPR pairs, where SPp(f) is the minimal size of a span program over the field Zp computing f. This shows we can efficiently attack f-routing schemes whenever f is in the complexity class ModpL, after allowing for local pre-processing. The best earlier construction achieved the class L, which is believed to be strictly inside of ModpL. We also show that the size of a quantum secret sharing scheme with indicator function fI upper bounds entanglement cost of f-routing on the function fI.

Borel complexity and Ramsey largeness of sets of oracles separating complexity classes

AbstractWe prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that , in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “large” using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating and is not small, and obtain similar results for the separation of from along with the separation of from . We also show that the set of oracles equating with is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy.

A Bayesian Approach with Prior Mixed Strategy Nash Equilibrium for Vehicle Intention Prediction

The state-of-the-art technology in the field of vehicle automation will lead to a mixed traffic environment in the coming years, where connected and automated vehicles have to interact with human-driven vehicles. In this context, it is necessary to have intention prediction models with the capability of forecasting how the traffic scenario is going to evolve with respect to the physical state of vehicles, the possible maneuvers and the interactions between traffic participants within the seconds to come. This article presents a Bayesian approach for vehicle intention forecasting, utilizing a game-theoretic framework in the form of a Mixed Strategy Nash Equilibrium (MSNE) as a prior estimate to model the reciprocal influence between traffic participants. The likelihood is then computed based on the Kullback-Leibler divergence. The game is modeled as a static nonzero-sum polymatrix game with individual preferences, a well known strategic game. Finding the MSNE for these games is in the PPAD cap PLS complexity class, with polynomial-time tractability. The approach shows good results in simulations in the long term horizon (10s), with its computational complexity allowing for online applications.

Satiation in Fisher Markets and Approximation of Nash Social Welfare

We study linear Fisher markets with satiation. In these markets, sellers have earning limits, and buyers have utility limits. Beyond applications in economics, they arise in the context of maximizing Nash social welfare when allocating indivisible items to agents. In contrast to markets with either earning or utility limits, markets with both limits have not been studied before. They turn out to have fundamentally different properties. In general, the existence of competitive equilibria is not guaranteed. We identify a natural property of markets (termed money clearing) that implies existence. We show that the set of equilibria is not always convex, answering a question posed in the literature. We design an FPTAS to compute an approximate equilibrium and prove that the problem of computing an exact equilibrium lies in the complexity class continuous local search ([Formula: see text]; i.e., the intersection of polynomial local search ([Formula: see text]) and polynomial parity arguments on directed graphs ([Formula: see text])). For a constant number of buyers or goods, we give a polynomial-time algorithm to compute an exact equilibrium. We show how (approximate) equilibria can be rounded and provide the first constant-factor approximation algorithm (with a factor of 2.404) for maximizing Nash social welfare when agents have capped linear (also known as budget-additive) valuations. Finally, we significantly improve the approximation hardness for additive valuations to [Formula: see text]. Funding: J. Garg was supported by the National Science Foundation [Grant CCF-1942321 (CAREER)]. M. Hoefer was supported by Deutsche Forschungsgemeinschaft [Grants Ho 3831/5-1, Ho 3831/6-1, and Ho 3831/7-1].

The effect of environmental information on evolution of cooperation in stochastic games

Many human interactions feature the characteristics of social dilemmas where individual actions have consequences for the group and the environment. The feedback between behavior and environment can be studied with the framework of stochastic games. In stochastic games, the state of the environment can change, depending on the choices made by group members. Past work suggests that such feedback can reinforce cooperative behaviors. In particular, cooperation can evolve in stochastic games even if it is infeasible in each separate repeated game. In stochastic games, participants have an interest in conditioning their strategies on the state of the environment. Yet in many applications, precise information about the state could be scarce. Here, we study how the availability of information (or lack thereof) shapes evolution of cooperation. Already for simple examples of two state games we find surprising effects. In some cases, cooperation is only possible if there is precise information about the state of the environment. In other cases, cooperation is most abundant when there is no information about the state of the environment. We systematically analyze all stochastic games of a given complexity class, to determine when receiving information about the environment is better, neutral, or worse for evolution of cooperation.

SPATIAL STRUCTURE OF SMALL CITIES: THE APPROACHES TO TYPOLOGY (BASED ON THE MATERIALS FROM CENTRAL RUSSIA)

The article examines the spatial structure of small towns in Central Russia. The information base of the study covers satellite images of 190 small towns. The paper proposes a concept of morphological structure of the city, which is determined by the nature (shape) of external boundaries, the presence of significant internal barriers leading to the fragmentation of space, and isolated urban areas. A methodology for assessing the complexity of urban space is suggested. It is based on the scoring of the impact of various geographical barriers on the urban environment. The most significant barriers include rivers and railways; the relief features, the location of industrial enterprises and the presence of protected areas were also considered. Special emphasis was given to urban areas isolated from the main built-up territories, i. e. exclaves. Five classes of urban space complexity have been identified, from very simple to very complex. The most complex space is typical of five small towns, which were initially clusters of workers’ settlements. The combination of data on the morpho-logical structure and complexity of space made it possible to identify six final forms of the spatial structure of the city. More often than others, there are cities with a dissected morphological structure in combination with a medium-complex space. One third of the cities under consideration have such parameters. The smallest number of cities has a dissected complex space. The cities tend to increase the complexity of their space in the process of development. The study showed that despite small population, the towns under consideration have very complex spatial structure

Some Complexity Considerations on the Uniqueness of Graph Colouring

For some well-known N P-complete problems, linked to the colourability of a graph, we study the variation which consists in asking about the uniqueness of a solution (up to permutations of the colours). In particular, we show that the decision problems Unique k-Colouring (U-k-COL) with k &gt; 3 and Unique Colouring (U-COL), have equivalent complexities, up to polynomials, as Unique Satisfiability (U-SAT) and Unique Onein-Three Satisfiability (U-1-3-SAT) by establishing polynomial reductions relating these four problems. As a consequence, all are co-N P-hard (or, equivalently, N P-hard with respect to Turing reductions) and belong to the complexity class DP. We also consider the problem Unique Optimal Colouring (U-OCOL) and show that it belongs to L N P (also denoted Θ2).

Towards logical foundations for probabilistic computation

The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called cCPL and iCPL, respectively, admit a natural semantics, based on the Borel σ-algebra of the Cantor space, together with a sound and complete proof system. Our main results consist in relating cCPL and iCPL with some central concepts in the study of probabilistic computation. On the one hand, the validity of cCPL-formulae in prenex form characterizes the corresponding level of Wagner's hierarchy of counting complexity classes, closely related to probabilistic complexity. On the other hand, proofs in iCPL correspond, in the sense of Curry and Howard, to typing derivations for a randomized extension of the λ-calculus, so that counting quantifiers reveal the probability of termination of the underlying probabilistic programs.

A Parameterized Theory of PAC Learning

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP "dichotomy" in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT - the core concept at the center of the parameterized complexity paradigm - and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.