Related papers: Characterizations and approximability of hard coun…
We prove that SVP$_p$ is NP-hard to approximate within a factor of $2^{\log^{1 - \varepsilon} n}$, for all constants $\varepsilon > 0$ and $p > 2$, under standard deterministic Karp reductions. This result is also the first proof that…
We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard…
We provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q-state Potts model when q>2. Specifically we show that the partition function is hard for the complexity class #RHPi_1…
Classification tasks are usually analysed and improved through new model architectures or hyperparameter optimisation but the underlying properties of datasets are discovered on an ad-hoc basis as errors occur. However, understanding the…
Can a deep neural network be approximated by a small decision tree based on simple features? This question and its variants are behind the growing demand for machine learning models that are *interpretable* by humans. In this work we study…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
Building explainable systems is a critical problem in the field of Natural Language Processing (NLP), since most machine learning models provide no explanations for the predictions. Existing approaches for explainable machine learning…
Probabilistically checkable proofs of proximity (PCPP) are proof systems where the verifier is given a 3SAT formula, but has only oracle access to an assignment and a proof. The verifier accepts a satisfying assignment with a valid proof,…
Bipolar Argumentation Frameworks (BAFs) admit several interpretations of the support relation and diverging definitions of semantics. Recently, several classes of BAFs have been captured as instances of bipolar Assumption-Based…
This paper makes two primary contributions. First, we introduce the concept of counting martingales and use it to define counting measures, counting dimensions, and counting strong dimensions. Second, we apply these new tools to strengthen…
The number partition problem is a well-known problem, which is one of 21 Karp's NP-complete problems \cite{karp}. The partition function is a boolean function that is equivalent to the number partition problem with number range restricted.…
The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G,…
In this work, we study the discrete logarithm problem in the context of TFNP - the complexity class of search problems with a syntactically guaranteed existence of a solution for all instances. Our main results establish that suitable…
In this paper the reason why entropy reduction (negentropy) can be used to measure the complexity of any computation was first elaborated both in the aspect of mathematics and informational physics. In the same time the equivalence of…
This paper deals with the computational complexity of conditions which guarantee that the NP-hard problem of finding the sparsest solution to an underdetermined linear system can be solved by efficient algorithms. In the literature, several…
We consider the Low Rank Approximation problem, where the input consists of a matrix $A \in \mathbb{R}^{n_R \times n_C}$ and an integer $k$, and the goal is to find a matrix $B$ of rank at most $k$ that minimizes $\| A - B \|_0$, which is…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
The Boolean satisfiability problem (SAT) is a well-known example of monotonic reasoning, of intense practical interest due to fast solvers, complemented by rigorous fine-grained complexity results. However, for non-monotonic reasoning,…
Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this…
Math reasoning is becoming an ever increasing area of focus as we scale large language models. However, even the previously-toughest evals like MATH are now close to saturated by frontier models (90.0% for o1-mini and 86.5% for Gemini 1.5…