arXiv subjects

Alexander Mayorov

Publications and source records attributed to Alexander Mayorov.

3 recordsLinked to original sources

arXiv2026

A Counterexample to EFX $n \ge 3$ Agents, $m \ge n + 5$ Items, Submodular Valuations via SAT-Solving

The existence of EFX allocations is a central open problem in discrete fair division. An allocation is EFX (envy-free up to any good) if no agent envies another agent after the removal of any single good from the other agent's bundle. We resolve this longstanding question by providing the \textbf{first-ever counterexample} to the existence of EFX allocations for agents with monotone valuations, which in turn immediately implies a counterexample for submodular valuations. Specifically, we show that EFX allocations need not exist for instances with $n \ge 3$ agents and $m \ge n+5$ goods. In contrast, we prove that every instance with three agents and seven goods admits an EFX allocation. Both results are obtained via SAT solving. We encode the negation of EFX existence as a SAT instance: satisfiability yields a counterexample, while unsatisfiability establishes universal existence. The correctness of the encoding is formally verified in Lean. Finally, we establish positive guarantees for fair allocations with three agents and an arbitrary number of goods. Although EFX allocations may fail to exist, we prove that every instance with three agents and monotone valuations admits at least one of two natural relaxations of EFX: tEFX, or EF1 and EEFX.

cs.GT↗

arXiv2023

Variable Independence in Linear Real Arithmetic

Variable independence and decomposability are algorithmic techniques for simplifying logical formulas by tearing apart connections between free variables. These techniques were originally proposed to speed up query evaluation in constraint databases, in particular by representing the query as a Boolean combination of formulas with no interconnected variables. They also have many other applications in SMT, string analysis, databases, automata theory and other areas. However, the precise complexity of variable independence and decomposability has been left open especially for the quantifier-free theory of linear real arithmetic (LRA), which is central in database applications. We introduce a novel characterization of formulas admitting decompositions and use it to show that it is coNP-complete to decide variable decomposability over LRA. As a corollary, we obtain that deciding variable independence is in $ \Sigma_2^p $. These results substantially improve the best known double-exponential time algorithms for variable decomposability and independence. In many practical applications, it is crucial to be able to efficiently eliminate connections between variables whenever possible. We design and implement an algorithm for this problem, which is optimal in theory, exponentially faster compared to the current state-of-the-art algorithm and efficient on various microbenchmarks. In particular, our algorithm is the first one to overcome a fundamental barrier between non-discrete and discrete first-order theories. Formulas arising in practice often have few or even no free variables that are perfectly independent. In this case, our algorithm can compute a best-possible approximation of a decomposition, which can be used to optimize database queries by exploiting partial variable independence, which is present in almost every logical formula or database query constraint.

cs.LO↗

arXiv2018

Gate-Defined Quantum Confinement in InSe-based van der Waals Heterostructures

Indium selenide, a post-transition metal chalcogenide, is a novel two-dimensional (2D) semiconductor with interesting electronic properties. Its tunable band gap and high electron mobility have already attracted considerable research interest. Here we demonstrate strong quantum confinement and manipulation of single electrons in devices made from few-layer crystals of InSe using electrostatic gating. We report on gate-controlled quantum dots in the Coulomb blockade regime as well as one-dimensional quantization in point contacts, revealing multiple plateaus. The work represents an important milestone in the development of quality devices based on 2D materials and makes InSe a prime candidate for relevant electronic and optoelectronic applications.

cond-mat.mes-hall↗