相关论文: Hierarchies within TFNP: building blocks and colla…

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0,…

We show that every language in NP has a PCP verifier that tosses $O(\log n)$ random coins, has perfect completeness, and a soundness error of at most $1/\text{poly}(n)$, while making at most $O(\text{poly}\log\log n)$ queries into a proof…

The framework of consistent query answers and repairs has been introduced to alleviate the impact of inconsistent data on the answers to a query. A repair is a minimally different consistent instance and an answer is consistent if it is…

P vs NP problem is the most important unresolved problem in the field of computational complexity. Its impact has penetrated into all aspects of algorithm design, especially in the field of cryptography. The security of cryptographic…

A central task in knowledge compilation is to compile a CNF-SAT instance into a succinct representation format that allows efficient operations such as testing satisfiability, counting, or enumerating all solutions. Useful representation…

The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means…

Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph \(G\) and a target class \(\mathcal{C}\), the Minimum Subgraph Complementation problem asks for a minimum-size vertex set…

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational…

We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm.…

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total…

We describe a procedure called panel collapse for replacing a CAT(0) cube complex $\Psi$ by a "lower complexity" CAT(0) cube complex $\Psi_\bullet$ whenever $\Psi$ contains a codimension-$2$ hyperplane that is extremal in one of the…

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are…

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e.,…

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 provide a list of new natural $\mathsf{VNP}$-intermediate polynomial families, based on basic (combinatorial) $\mathsf{NP}$-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in…

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen…

In order to prove that the P of problems is different to the NP class, we consider the satisfability problem of propositional calculus formulae, which is an NP-complete problem. It is shown that, for every search algorithm A, there is a set…

A subset of Q^n is called semilinear (or piecewise linear) if it is Boolean combination of linear half-spaces. We study the computational complexity of the constraint satisfaction problem (CSP) over the rationals when all the constraints…

A phenomenon known as ''Neural Collapse (NC)'' in deep classification tasks, in which the penultimate-layer features and the final classifiers exhibit an extremely simple geometric structure, has recently attracted considerable attention,…

Proving super-polynomial size lower bounds for $\textsf{TC}^0$, the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that $\textsf{NEXP}$…