arXiv subjects

Bakh Khoussainov

Publications and source records attributed to Bakh Khoussainov.

16 recordsLinked to original sources

arXiv2026

Almost free algebras: from the word problem to elimination of quantifiers

Term algebras are important objects in computer science and are correspondingly well-studied. A natural generalization is to quotient these algebras by finitely many ground term equations, obtaining what we call almost free algebras. One of the earliest results on almost free algebras is that their word problem is polynomial time decidable. In this paper, we show that other natural problems: finding canonical representatives; computing the cardinality of a congruence class; checking if all congruence classes are infinite; checking if the algebra is finite; checking if two algebras are isomorphic, are all polynomial time decidable. Another famous result regarding term algebras is that they admit quantifier elimination in a suitably expanded language. Following this pattern, we also show that almost free algebras admit quantifier elimination by expanding the language with the standard tester predicates. While this is implied by existing results, we view our main contribution here as providing a different approach, which we posit can be easily extended to a larger class that is not covered by existing works. Finally, we provide an application to the quantifier elimination procedure, constructing examples of non-initial algebras over arbitrary signatures with a polynomial time word problem.

math.LO↗

arXiv2026

Automatic constraint satisfaction problem

We study constraint satisfaction problems (CSPs) where the constraint languages are defined by finite automata, giving rise to automata-based CSPs. The key notion is the concept of Automatic Constraint Satisfaction Problem ($AutCSP$), where constraint languages and instances are specified by finite automata. The $AutCSP$ captures infinite yet finitely describable sets of relations, enabling concise representations of complex constraints. Studying the complexity of the $AutCSP$s illustrates the interplay between classical CSPs, automata, and logic, sharpening the boundary between tractable and intractable constraints. We show that checking whether an operation is a polymorphism of such a language can be done in polynomial time. Building on this, we establish several complexity classification results for the $AutCSP$. In particular, we prove that Schaefer's Dichotomy Theorem extends to the $AutCSP$ over the Boolean domain, and we provide algorithms that decide tractability of some classes of $AutCSP$s over arbitrary finite domains via automatic polymorphisms. An important part of our work is that our polynomial-time algorithms run on $AutCSP$ instances that can be exponentially more succinct than their standard CSP counterparts.

cs.LO↗

arXiv2026

Model theory of term algebras revisited

Building on work of Maltsev on locally free algebras in finite purely functional languages, we revisit the model theory of (absolutely free) term algebras and their completions. Maltsev's analysis yields a natural axiomatization together with quantifier elimination to positive Boolean combinations of special formulas, and shows that the complete extensions are parametrized exactly by the number $k\in\{0,1,\dots,ω\}$ of indecomposable elements; for $1\le k\leω$ the standard model is the free term algebra on $k$ generators. We give a new, quantifier-elimination--free proof of completeness using Ehrenfeucht--Fraïssé games, and we establish several further structural properties of the standard models and theories. In particular, for $1\le k\leω$ we prove first-order rigidity and atomicity of the standard model. For every $0\le k\leω$ we show that the corresponding theory does not have the finite cover property and weakly eliminates imaginaries. We also provide new proofs of stability-theoretic features previously obtained by Belegradek: the theories are stable but not superstable, normal (hence $1$-based), and have trivial forking; consequently, no infinite group is interpretable in any model. Finally, we analyze model completeness and show that $T_0$ is the model companion of the theory of locally free algebras, while the theories with $k\ge 1$ are not model complete.

math.LO↗

arXiv2025

Strategyproofness and Monotone Allocation of Auction in Social Networks

Strategyproofness in network auctions requires that bidders not only report their valuations truthfully, but also do their best to invite neighbours from the social network. In contrast to canonical auctions, where the value-monotone allocation in Myerson's Lemma is a cornerstone, a general principle of allocation rules for strategyproof network auctions is still missing. We show that, due to the absence of such a principle, even extensions to multi-unit network auctions with single-unit demand present unexpected difficulties, and all pioneering researches fail to be strategyproof. For the first time in this field, we identify two categories of monotone allocation rules on networks: Invitation-Depressed Monotonicity (ID-MON) and Invitation-Promoted Monotonicity (IP-MON). They encompass all existing allocation rules of network auctions as specific instances. For any given ID-MON or IP-MON allocation rule, we characterize the existence and sufficient conditions for the strategyproof payment rules, and show that among all such payment rules, the revenue-maximizing one exists and is computationally feasible. With these results, the obstacle of combinatorial network auction with single-minded bidders is now resolved.

cs.GT↗

arXiv2024

Balancing Efficiency with Equality: Auction Design with Group Fairness Concerns

The issue of fairness in AI arises from discriminatory practices in applications like job recommendations and risk assessments, emphasising the need for algorithms that do not discriminate based on group characteristics. This concern is also pertinent to auctions, commonly used for resource allocation, which necessitate fairness considerations. Our study examines auctions with groups distinguished by specific attributes, seeking to (1) define a fairness notion that ensures equitable treatment for all, (2) identify mechanisms that adhere to this fairness while preserving incentive compatibility, and (3) explore the balance between fairness and seller's revenue. We introduce two fairness notions-group fairness and individual fairness-and propose two corresponding auction mechanisms: the Group Probability Mechanism, which meets group fairness and incentive criteria, and the Group Score Mechanism, which also encompasses individual fairness. Through experiments, we validate these mechanisms' effectiveness in promoting fairness and examine their implications for seller revenue.

cs.GT↗

arXiv2024

Meta-mechanisms for Combinatorial Auctions over Social Networks

Recently there has been a large amount of research designing mechanisms for auction scenarios where the bidders are connected in a social network. Different from the existing studies in this field that focus on specific auction scenarios e.g. single-unit auction and multi-unit auction, this paper considers the following question: is it possible to design a scheme that, given a classical auction scenario and a mechanism $\tilde{\mathcal{M}}$ suited for it, produces a mechanism in the network setting that preserves the key properties of $\tilde{\mathcal{M}}$? To answer this question, we design meta-mechanisms that provide a uniform way of transforming mechanisms from classical models to mechanisms over networks and prove that the desirable properties are preserved by our meta-mechanisms. Our meta-mechanisms provide solutions to combinatorial auction scenarios in the network setting: (1) combinatorial auction with single-minded buyers and (2) combinatorial auction with general monotone valuation. To the best of our knowledge, this is the first work that designs combinatorial auctions over a social network.

cs.GT↗

arXiv2024

Deciding regular games: a playground for exponential time algorithms

Regular games form a well-established class of games for analysis and synthesis of reactive systems. They include coloured Muller games, McNaughton games, Muller games, Rabin games, and Streett games. These games are played on directed graphs $\mathcal G$ where Player 0 and Player 1 play by generating an infinite path $ρ$ through the graph. The winner is determined by specifications put on the set $X$ of vertices in $ρ$ that occur infinitely often. These games are determined, enabling the partitioning of $\mathcal G$ into two sets $W_0$ and $W_1$ of winning positions for Player 0 and Player 1, respectively. Numerous algorithms exist that decide specific instances of regular games, e.g., Muller games, by computing $W_0$ and $W_1$. In this paper we aim to find general principles for designing uniform algorithms that decide all regular games. For this we utilise various recursive and dynamic programming algorithms that leverage standard notions such as subgames and traps. Importantly, we show that our techniques improve or match the performances of existing algorithms for many instances of regular games.

cs.GT↗

arXiv2024

Facility Location Games Beyond Single-Peakedness: the Entrance Fee Model

The facility location game has been studied extensively in mechanism design. In the classical model, each agent's cost is solely determined by her distance to the nearest facility. In this paper, we introduce a novel model where each facility charges an entrance fee. Thus, the cost of each agent is determined by both the distance to the facility and the entrance fee of the facility. In our model, the entrance fee function is allowed to be an arbitrary function, causing agents' preferences may no longer be single-peaked anymore: This departure from the classical model introduces additional challenges. We systematically delve into the intricacies of the model, designing strategyproof mechanisms with favorable approximation ratios. Additionally, we complement these ratios with nearly-tight impossibility results. Specifically, for one-facility and two-facility games, we provide upper and lower bounds for the approximation ratios given by deterministic and randomized mechanisms with respect to utilitarian and egalitarian objectives.

cs.GT↗

arXiv2023

Two new algorithms for solving Müller games and their applications

Müller games form a well-established class of games for model checking and verification. These games are played on directed graphs $\mathcal G$ where Player 0 and Player 1 play by generating an infinite path through the graph. The winner is determined by the set $X$ consisting of all vertices in the path that occur infinitely often. If $X$ belongs to $Ω$, a specified collection of subsets of $\mathcal G$, then Player 0 wins. Otherwise, Player 1 claims the win. These games are determined, enabling the partitioning of $\mathcal G$ into two sets $W_0$ and $W_1$ of winning positions for Player 0 and Player 1, respectively. Numerous algorithms exist that decide Müller games $\mathcal G$ by computing the sets $W_0$ and $W_1$. In this paper, we introduce two novel algorithms that outperform all previously known methods for deciding explicitly given Müller games, especially in the worst-case scenarios. The previously known algorithms either reduce Müller games to other known games (e.g. safety games) or recursively change the underlying graph $\mathcal G$ and the collection of sets in $Ω$. In contrast, our approach does not employ these techniques but instead leverages subgames, the sets within $Ω$, and their interactions. This distinct methodology sets our algorithms apart from prior approaches for deciding Müller games. Additionally, our algorithms offer enhanced clarity and ease of comprehension. Importantly, our techniques are applicable not only to Müller games but also to improving the performance of existing algorithms that handle other game classes, including coloured Müller games, McNaughton games, Rabin games, and Streett games.

cs.GT↗

arXiv2023

Multi-unit Auction over a Social Network

Diffusion auction is an emerging business model where a seller aims to incentivise buyers in a social network to diffuse the auction information thereby attracting potential buyers. We focus on designing mechanisms for multi-unit diffusion auctions. Despite numerous attempts at this problem, existing mechanisms either fail to be incentive compatible (IC) or achieve only an unsatisfactory level of social welfare (SW). Here, we propose a novel graph exploration technique to realise multi-item diffusion auction. This technique ensures that potential competition among buyers stay ``localised'' so as to facilitate truthful bidding. Using this technique, we design multi-unit diffusion auction mechanisms MUDAN and MUDAN-$m$. Both mechanisms satisfy, among other properties, IC and $1/m$-weak efficiency. We also show that they achieve optimal social welfare for the class of rewardless diffusion auctions. While MUDAN addresses the bottleneck case when each buyer demands only a single item, MUDAN-$m$ handles the more general, multi-demand setting. We further demonstrate that these mechanisms achieve near-optimal social welfare through experiments.

cs.GT↗

arXiv2023

Characterizations of Network Auctions and Generalizations of VCG

With the growth of networks, promoting products through social networks has become an important problem. For auctions in social networks, items are needed to be sold to agents in a network, where each agent can bid and also diffuse the sale information to her neighbors. Thus, the agents' social relations are intervened with their bids in the auctions. In network auctions, the classical VCG mechanism fails to retain key properties. In order to better understand network auctions, in this paper, we characterize network auctions for the single-unit setting with respect to weak budget balance, individual rationality, incentive compatibility, efficiency, and other properties. For example, we present sufficient conditions for mechanisms to be efficient and (weakly) incentive compatible. With the help of these properties and new concepts such as rewards, participation rewards, and so on, we show how to design efficient mechanisms to satisfy incentive compatibility as much as possible, and incentive compatibility mechanisms to maximize the revenue. Our results provide insights into understanding auctions in social networks.

cs.GT↗

arXiv2023

Connectivity in the presence of an opponent

The paper introduces two player connectivity games played on finite bipartite graphs. Algorithms that solve these connectivity games can be used as subroutines for solving Müller games. Müller games constitute a well established class of games in model checking and verification. In connectivity games, the objective of one of the players is to visit every node of the game graph infinitely often. The first contribution of this paper is our proof that solving connectivity games can be reduced to the incremental strongly connected component maintenance (ISCCM) problem, an important problem in graph algorithms and data structures. The second contribution is that we non-trivially adapt two known algorithms for the ISCCM problem to provide two efficient algorithms that solve the connectivity games problem. Finally, based on the techniques developed, we recast Horn's polynomial time algorithm that solves explicitly given Müller games and provide an alternative proof of its correctness. Our algorithms are more efficient than that of Horn's algorithm. Our solution for connectivity games is used as a subroutine in the algorithm.

cs.GT↗

arXiv2023

Differentially Private Diffusion Auction: The Single-unit Case

Diffusion auction refers to an emerging paradigm of online marketplace where an auctioneer utilises a social network to attract potential buyers. Diffusion auction poses significant privacy risks. From the auction outcome, it is possible to infer hidden, and potentially sensitive, preferences of buyers. To mitigate such risks, we initiate the study of differential privacy (DP) in diffusion auction mechanisms. DP is a well-established notion of privacy that protects a system against inference attacks. Achieving DP in diffusion auctions is non-trivial as the well-designed auction rules are required to incentivise the buyers to truthfully report their neighbourhood. We study the single-unit case and design two differentially private diffusion mechanisms (DPDMs): recursive DPDM and layered DPDM. We prove that these mechanisms guarantee differential privacy, incentive compatibility and individual rationality for both valuations and neighbourhood. We then empirically compare their performance on real and synthetic datasets.

cs.GT↗

arXiv2023

Learning Density-Based Correlated Equilibria for Markov Games

Correlated Equilibrium (CE) is a well-established solution concept that captures coordination among agents and enjoys good algorithmic properties. In real-world multi-agent systems, in addition to being in an equilibrium, agents' policies are often expected to meet requirements with respect to safety, and fairness. Such additional requirements can often be expressed in terms of the state density which measures the state-visitation frequencies during the course of a game. However, existing CE notions or CE-finding approaches cannot explicitly specify a CE with particular properties concerning state density; they do so implicitly by either modifying reward functions or using value functions as the selection criteria. The resulting CE may thus not fully fulfil the state-density requirements. In this paper, we propose Density-Based Correlated Equilibria (DBCE), a new notion of CE that explicitly takes state density as selection criterion. Concretely, we instantiate DBCE by specifying different state-density requirements motivated by real-world applications. To compute DBCE, we put forward the Density Based Correlated Policy Iteration algorithm for the underlying control problem. We perform experiments on various games where results demonstrate the advantage of our CE-finding approach over existing methods in scenarios with state-density concerns.

cs.AI↗

arXiv2023

MSDC: Exploiting Multi-State Power Consumption in Non-intrusive Load Monitoring based on A Dual-CNN Model

Non-intrusive load monitoring (NILM) aims to decompose aggregated electrical usage signal into appliance-specific power consumption and it amounts to a classical example of blind source separation tasks. Leveraging recent progress on deep learning techniques, we design a new neural NILM model Multi-State Dual CNN (MSDC). Different from previous models, MSDC explicitly extracts information about the appliance's multiple states and state transitions, which in turn regulates the prediction of signals for appliances. More specifically, we employ a dual-CNN architecture: one CNN for outputting state distributions and the other for predicting the power of each state. A new technique is invented that utilizes conditional random fields (CRF) to capture state transitions. Experiments on two real-world datasets REDD and UK-DALE demonstrate that our model significantly outperform state-of-the-art models while having good generalization capacity, achieving 6%-10% MAE gain and 33%-51% SAE gain to unseen appliances.

cs.LG↗

arXiv2019

Computability Aspects of Differential Games in Euclidian Spaces

We study computability-theoretic aspects of differential games. Our focus is on pursuit and evasion games played in Euclidean spaces in the tradition of Rado's "Lion versus Man" game. In some ways, these games can be viewed as continuous versions of reachability games. We prove basic undecidability of differential games, and study natural classes of pursuit-evasion games in Euclidean spaces where the winners can win via computable strategies. The winning strategy for Man found by Besicovitch for the traditional "Lion versus Man" is not computable. We show how to modify it to yield a computable non-deterministic winning strategy, and raise the question whether Man can win in a computable and deterministic way.

math.OC↗