计算复杂性
We study algorithmic complexity of solving subtraction games in a~fixed dimension with a finite difference set. We prove that there exists a game in this class such that any algorithm solving the game runs in exponential time. Also we prove…
The GKS game was formulated by Justin Gilmer, Michal Koucky, and Michael Saks in their research of the sensitivity conjecture. Mario Szegedy invented a protocol for the game with the cost of $O(n^{0.4732})$. Then a protocol with the cost of…
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $\sigma$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering…
We build on a working program initiated by Pudl\'ak [Pud17] and construct an oracle relative to which each $\mathrm{coNP}$-complete set has $\mathrm{P}$-optimal proof systems and $\mathrm{NP}\cap\mathrm{coNP}$ does not have complete…
As one step in a working program initiated by Pudl\'ak [Pud17] we construct an oracle relative to which $\mathrm{P}\ne\mathrm{NP}$ and all non-empty sets in $\mathrm{NP}\cup\mathrm{coNP}$ have $\mathrm{P}$-optimal proof systems.
Pudl\'ak [Pud17] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are: - $\mathsf{DisjNP}$: The class of all disjoint…
We study projective dimension, a graph parameter (denoted by pd$(G)$ for a graph $G$), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving lower bounds for pd$(G_f)$ for bipartite graphs $G_f$ associated with a Boolean function…
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph $G$ can be reversibly pebbled in time $t$ and space $s$ if and only if there is a…
We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such…
Structured $d$-DNNFs and SDDs are restricted negation normal form circuits used in knowledge compilation as target languages into which propositional theories are compiled. Structuredness is imposed by so-called vtrees. By definition SDDs…
The relationship between the complexity classes $P$ and $NP$ is an unsolved question in the field of theoretical computer science. In the first part of this paper, a lattice framework is proposed to handle the 3-CNF-SAT problems, known to…
The relationship between the complexity classes P and NP is an unsolved question in the field of theoretical computer science. In this paper, we investigate a descriptor approach based on lattice properties. This paper proposes a new way to…
We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemer\'{e}di graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemer\'{e}di. The graphs induced by this…
In this paper, we study the problem of counting the number of different knapsack solutions with a prescribed cardinality. We present an FPTAS for this problem, based on dynamic programming. We also introduce two different types of semi-fair…
We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is a codeword of the Reed-Muller code of dimension $n$ and order $d$ over the finite field $\mathbb{F}_q$ for prime $q$…
We make progress on some questions related to polynomial approximations of ${\rm AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC, 1991), that any ${\rm AC}^0$ circuit…
We write down an explicit sequence of tensors in $C^m\otimes C^m\otimes C^m$, for all $m$ sufficiently large, having border rank at least $2.02m$, overcoming a longstanding barrier. We obtain our lower bounds via the border substitution…
The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be…
We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We…
We investigate the problem of approximating the product $a^TBc$, where $a,c\in S^{n-1}$ and $B\in O_n$, in models of communication complexity and streaming algorithms. The worst meaningful approximation is to simply decide whether the…