计算复杂性
We present a polynomial-time reduction from solving noisy linear equations over $\mathbb{Z}/q\mathbb{Z}$ in dimension $\Theta(k\log n/\mathsf{poly}(\log k,\log q,\log\log n))$ with a uniformly random coefficient matrix to noisy linear…
A subset $S$ of vertices in a graph $G=(V, E)$ is a Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside…
In this paper, we construct new t-server Private Information Retrieval (PIR) schemes with communication complexity subpolynomial in the previously best known, for all but finitely many t. Our results are based on combining derivatives (in…
We prove an \Omega(n/k+k) communication lower bound on (k-1)-round distributional complexity of the k-step pointer chasing problem under uniform input distribution, improving the \Omega(n/k - k log n) lower bound due to Yehudayoff…
Motivated by phase transitions in combinatorial optimization problems, we define two kinds of geometry-preserving reductions between constraint satisfaction problems and other NP-search problems. We give a couple of examples and…
The study of SAT and its variants has provided numerous NP-complete problems, from which most NP-hardness results were derived. Due to the NP-hardness of SAT, adding constraints to either specify a more precise NP-complete problem or to…
Local search is a powerful heuristic in optimization and computer science, the complexity of which has been studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a…
Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical…
We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as…
We define the Super-Unique-Tarski problem, which is a Tarski instance in which all slices are required to have a unique fixed point. We show that Super-Unique-Tarski lies in UEOPL under promise-preserving reductions.
The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive…
In this paper, we consider classes of decision tables with 0-1-decisions closed relative to removal of attributes (columns) and changing decisions assigned to rows. For tables from an arbitrary closed class, we study the dependence of the…
We study the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the $H$-game. Maker wins if she claims the edges of a perfect matching in the first, and…
We employ techniques from group theory to show that, in many cases, counting problems on graphs are almost as hard to solve in a small number of instances as they are in all instances. Specifically, we show the following results. 1.…
In this work, we show that the class of multivariate degree-$d$ polynomials mapping $\{0,1\}^{n}$ to any Abelian group $G$ is locally correctable with $\widetilde{O}_{d}((\log n)^{d})$ queries for up to a fraction of errors approaching half…
This paper looks at the intersection of algorithmic information theory and physics, namely quantum mechanics, thermodynamics, and black holes. We discuss theorems which characterize the barrier between the quantum world and the classical…
The automatic generation of loop invariants is a fundamental challenge in software verification. While this task is undecidable in general, it is decidable for certain restricted classes of programs. This work focuses on invariant…
We show that for every polynomial q* there exist polynomial-size, constant-query, non-adaptive PCPs for NP which are perfect zero knowledge against (adaptive) adversaries making at most q* queries to the proof. In addition, we construct…
The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about…
We prove an $\Omega(n^{1-1/k} \log k \ /2^k)$ lower bound on the $k$-party number-in-hand communication complexity of collision-finding. This implies a $2^{n^{1-o(1)}}$ lower bound on the size of tree-like cutting-planes proofs of the bit…