arXiv subjects

Nikita Gaevoy

Publications and source records attributed to Nikita Gaevoy.

3 recordsLinked to original sources

arXiv2026

The Power of Small Symmetries

Resolution with symmetries is a natural extension of the Resolution proof system that allows to use symmetries of the formula to simplify the proof. Symmetries can be global (applied to the whole input formula), local (applied to a subformula), or dynamic (applied to newly derived clauses as well). The framework of Resolution with (global) symmetries was introduced by Krishnamurthy (1985) and further extended by Arai and Urquhart (2000) to local symmetries. Later, Szeider (2005) generalized this approach to homomorphisms and introduced the notion of Resolution with dynamic symmetries. While proving superpolynomial proof-size lower bounds for Resolution with dynamic symmetries remains an open problem already for two decades, the power of proof systems with global and local symmetries is well studied: exponential lower bounds have been proven for these proof systems, as well as exponential separations between all of them. However, these systems are too general to reflect practical applications since it is computationally too hard to find and efficiently exploit arbitrary symmetries. In this work, we introduce the notion of small symmetries: symmetries that can operate on a limited number of variables at the same time. Resolution with small symmetries gives hopes both for practical applications and for theoretical study of dynamic symmetries. We show that proof systems with both local and global small symmetries form strict hierarchies w.r.t. the size of symmetries. We prove exponential separations between proof systems with symmetries of different sizes and types. It turns out that even lower levels of these hierarchies are exponentially separated from Resolution and stronger proof systems, such as constant-depth Frege. As a byproduct of our constructions, we obtain an exponential separation between the classical systems SRCI and SRII that was not known before.

cs.CC↗

arXiv2021

Hard satisfiable formulas for DPLL algorithms using heuristics with small memory

DPLL algorithm for solving the Boolean satisfiability problem (SAT) can be represented in the form of a procedure that, using heuristics $A$ and $B$, select the variable $x$ from the input formula $φ$ and the value $b$ and runs recursively on the formulas $φ[x := b]$ and $φ[x := 1 - b]$. Exponential lower bounds on the running time of DPLL algorithms on unsatisfiable formulas follow from the lower bounds for tree-like resolution proofs. Lower bounds on satisfiable formulas are also known for some classes of DPLL algorithms such as "myopic" and "drunken" algorithms. All lower bounds are made for the classes of DPLL algorithms that limit heuristics access to the formula. In this paper we consider DPLL algorithms with heuristics that have unlimited access to the formula but use small memory. We show that for any pair of heuristics with small memory there exists a family of satisfiable formulas $Φ_n$ such that a DPLL algorithm that uses these heuristics runs in exponential time on the formulas $Φ_n$.

cs.CC↗

arXiv2019

New Competitiveness Bounds for the Shared Memory Switch

We consider one of the simplest and best known buffer management architectures: the shared memory switch with multiple output queues and uniform packets. It was one of the first models studied by competitive analysis, with the Longest Queue Drop (LQD) buffer management policy shown to be at least $\sqrt{2}$- and at most $2$-competitive; a general lower bound of $4/3$ has been proven for all deterministic online algorithms. Closing the gap between $\sqrt{2}$ and $2$ has remained an open problem in competitive analysis for more than a decade, with only marginal success in reducing the upper bound of $2$. In this work, we first present a simplified proof for the $\sqrt{2}$ lower bound for LQD and then, using a reduction to the continuous case, improve the general lower bound for all deterministic online algorithms from $\frac 43$ to $\sqrt{2}$. Then, we proceed to improve the lower bound of $\sqrt{2}$ specifically for LQD, showing that LQD is at least $1.44546086$-competitive. We are able to prove the bound by presenting an explicit construction of the optimal clairvoyant algorithm which then allows for two different ways to prove lower bounds: by direct computer simulations and by proving lower bounds via linear programming. The linear programming approach yields a lower bound for LQD of $1.4427902$ (still larger than $\sqrt{2}$).

cs.NI↗