Related papers: Algebraic Barriers to Halving Algorithmic Informat…
We address the separability problem for straight-line string constraints. The separability problem for languages of a class C by a class S asks: given two languages A and B in C, does there exist a language I in S separating A and B (i.e.,…
Two-party one-way quantum communication has been extensively studied in the recent literature. We target the size of minimal information that is necessary for a feasible party to finish a given combinatorial task, such as distinction of…
As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms $x=y\setminus z$, $x \neq y\setminus z$, and $z =\{x\}$, where $x,y,z$ stand for variables ranging over the von…
Despite their popularity, many questions about the algebraic constraints imposed by linear structural equation models remain open problems. For causal discovery, two of these problems are especially important: the enumeration of the…
We construct a universal decompressor $U$ for plain Kolmogorov complexity $\mathrm{C}_U$ such that the Halting Problem cannot be decided by any polynomial-time oracle machine with access to the set of random strings $R_{\mathrm{C}_U} = \{x…
Diverse applications of Kolmogorov complexity to learning [CIKK16], circuit complexity [OPS19], cryptography [LP20], average-case complexity [Hir21], and proof search [Kra22] have been discovered in recent years. Since the running time of…
When solving data analysis problems it is important to integrate prior knowledge and/or structural invariances. This paper contributes by a novel framework for incorporating algebraic invariance structure into kernels. In particular, we…
We introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle. For such algorithms, access to the input distribution is…
We present three Lagrangian algebras in the modular 2-category associated to the 3+1D $\mathbb{Z}_2$ topological order and discuss their physical interpretations, connecting algebras with gapped boundary conditions, and interestingly, maps…
We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a…
We investigate the complexity of the separation problem associated to classes of regular languages. For a class C, C-separation takes two regular languages as input and asks whether there exists a third language in C which includes the…
We give lower bounds on the communication complexity of graph problems in the multi-party blackboard model. In this model, the edges of an $n$-vertex input graph are partitioned among $k$ parties, who communicate solely by writing messages…
A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the Boolean satisfiability problem (SAT), and…
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically…
A semilinear relation is a finite union of finite intersections of open and closed half-spaces over, for instance, the reals, the rationals, or the integers. Semilinear relations have been studied in connection with algebraic geometry,…
Binary jumbled pattern matching asks to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of length $i$ and has exactly $j$ 1-bits. This problem naturally generalizes to…
We present a functional programming language for specifying constraints over tree-shaped data. The language allows for Haskell-like algebraic data types and pattern matching. Our constraint compiler CO4 translates these programs into…
String matching is one of the most fundamental problems in computer science. A natural problem is to determine the number of characters that need to be queried (i.e. the decision tree complexity) in a string in order to decide whether this…
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at…
Many combinatorial optimisation problems hide algebraic structures that, once exposed, shrink the search space and improve the chance of finding the global optimal solution. We present a general framework that (i) identifies algebraic…