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In this paper we investigate the complexity of abduction, a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining the world's behavior it aims at finding an explanation for some observed manifestation.…

Instruction following is one of the fundamental capabilities of large language models (LLMs). As the ability of LLMs is constantly improving, they have been increasingly applied to deal with complex human instructions in real-world…

The problem of nonparametric functional data classification and bandwidth selection is considered when the response variable, also called the class label, might be missing but not at random (MNAR). This setup is broadly acknowledged to be…

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were…

While SHAP (SHapley Additive exPlanations) and other feature attribution methods are commonly employed to explain model predictions, their application within information retrieval (IR), particularly for complex outputs such as ranked lists,…

We investigate the computational complexity of the problem of counting the maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over the Boolean domain {0,1}. A satisfying assignment is maximal if any new assignment…

We give efficient algorithms for ranking Lyndon words of length $n$ over an alphabet of size $\sigma$. The rank of a Lyndon word is its position in the sequence of lexicographically ordered Lyndon words of the same length. The outputs are…

A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the…

A new approach to solving a class of rankconstrained semi-definite programming (SDP) problems, which appear in many signal processing applications such as transmit beamspace design in multiple-input multiple-output (MIMO) radar, downlink…

Suppose we are given an $n$-dimensional order-3 symmetric tensor $T \in (\mathbb{R}^n)^{\otimes 3}$ that is the sum of $r$ random rank-1 terms. The problem of recovering the rank-1 components is possible in principle when $r \lesssim n^2$…

For many queries in the Web retrieval setting there is an on-going ranking competition: authors manipulate their documents so as to promote them in rankings. Such competitions can have unwarranted effects not only in terms of retrieval…

This work introduces Bilinear Classes, a new structural framework, which permit generalization in reinforcement learning in a wide variety of settings through the use of function approximation. The framework incorporates nearly all existing…

We consider the general problem of blocking all solutions of some given combinatorial problem with only few elements. For example, the problem of destroying all maximum cliques of a given graph by forbidding only few vertices. Problems of…

Whereas most dimensionality reduction techniques (e.g. PCA, ICA, NMF) for multivariate data essentially rely on linear algebra to a certain extent, summarizing ranking data, viewed as realizations of a random permutation $\Sigma$ on a set…

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active,…

The complexity of representing a polynomial by a Read-Once Oblivious Algebraic Branching Program (ROABP) is highly dependent on the chosen variable ordering. Bhargava et al. prove that finding the optimal ordering is NP-hard, and provide…

Evaluating performance across optimization algorithms on many problems presents a complex challenge due to the diversity of numerical scales involved. Traditional data processing methods, such as hypothesis testing and Bayesian inference,…

Various practical problems within the class $\Sigma_{2}^P$ possess an unambiguity property, meaning that yes-instances correspond with a unique witness. The semantic class containing all unambiguous $\Sigma_{2}^P$ problems is denoted…

Let $X$ be a finite set in $Z^d$. We consider the problem of optimizing linear function $f(x) = c^T x$ on $X$, where $c\in Z^d$ is an input vector. We call it a problem $X$. A problem $X$ is related with linear program $\max\limits_{x \in…

Algorithmic decisions in critical domains such as hiring, college admissions, and lending are often based on rankings. Given the impact of these decisions on individuals, organizations, and population groups, it is essential to understand…