In this paper, by constructing extremely hard examples of CSP (with large domains) and SAT (with long clauses), we prove that such examples cannot be solved without exhaustive search, which implies a weaker conclusion P $\neq$ NP. This constructive approach for proving impossibility results is very different (and missing) from those currently used in computational complexity theory, but is similar to that used by Kurt G\"{o}del in proving his famous logical impossibility results. Just as shown by G\"{o}del's results that proving formal unprovability is feasible in mathematics, the results of this paper show that proving computational hardness is not hard in mathematics. Specifically, proving lower bounds for many problems, such as 3-SAT, can be challenging because these problems have various effective strategies available for avoiding exhaustive search. However, in cases of extremely hard examples, exhaustive search may be the only viable option, and proving its necessity becomes more straightforward. Consequently, it makes the separation between SAT (with long clauses) and 3-SAT much easier than that between 3-SAT and 2-SAT. Finally, the main results of this paper demonstrate that the fundamental difference between the syntax and the semantics revealed by G\"{o}del's results also exists in CSP and SAT.
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It is often very challenging to manually design reward functions for complex, real-world tasks. To solve this, one can instead use reward learning to infer a reward function from data. However, there are often multiple reward functions that fit the data equally well, even in the infinite-data limit. This means that the reward function is only partially identifiable. In this work, we formally characterise the partial identifiability of the reward function given several popular reward learning data sources, including expert demonstrations and trajectory comparisons. We also analyse the impact of this partial identifiability for several downstream tasks, such as policy optimisation. We unify our results in a framework for comparing data sources and downstream tasks by their invariances, with implications for the design and selection of data sources for reward learning.
A parametric class of trust-region algorithms for unconstrained nonconvex optimization is considered where the value of the objective function is never computed. The class contains a deterministic version of the first-order Adagrad method typically used for minimization of noisy function, but also allows the use of (possibly approximate) second-order information when available. The rate of convergence of methods in the class is analyzed and is shown to be identical to that known for first-order optimization methods using both function and gradients values, recovering existing results for purely-first order variants and improving the explicit dependence on problem dimension. This rate is shown to be essentially sharp. A new class of methods is also presented, for which a slightly worse and essentially sharp complexity result holds. Limited numerical experiments show that the new methods' performance may be comparable to that of standard steepest descent, despite using significantly less information, and that this performance is relatively insensitive to noise.
In this paper we deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and Mohar, 2013] that both problems are NP-hard; although they required an unbounded number of high-degree vertices (in the first problem) or an unbounded number of anchors (in the second problem) to prove their result. Somehow surprisingly, only three vertices of degree greater than 3, or only three anchors, are sufficient to maintain hardness of these problems, as we prove here. The new result also improves the previous result on hardness of joint crossing number on surfaces by [Hlin\v{e}n\'y and Salazar, 2015]. Our result is best possible in the anchored case since the anchored crossing number of a pair of planar graphs with two anchors each is trivial, and close to being best possible in the almost planar case since the crossing number is efficiently computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello and Mohar 2011].
Analyzing the spectral behavior of random matrices with dependency among entries is a challenging problem. The adjacency matrix of the random $d$-regular graph is a prominent example that has attracted immense interest. A crucial spectral observable is the extremal eigenvalue, which reveals useful geometric properties of the graph. According to the Alon's conjecture, which was verified by Friedman, the (nontrivial) extremal eigenvalue of the random $d$-regular graph is approximately $2\sqrt{d-1}$. In the present paper, we analyze the extremal spectrum of the random $d$-regular graph (with $d\ge 3$ fixed) equipped with random edge-weights, and precisely describe its phase transition behavior with respect to the tail of edge-weights. In addition, we establish that the extremal eigenvector is always localized, showing a sharp contrast to the unweighted case where all eigenvectors are delocalized. Our method is robust and inspired by a sparsification technique developed in the context of Erd\H{o}s-R\'{e}nyi graphs (Ganguly and Nam, '22), which can also be applied to analyze the spectrum of general random matrices whose entries are dependent.
In this paper paired comparison models with stochastic background are investigated. We focus on the models which allow three options for choice and the parameters are estimated by maximum likelihood method. The existence and uniqueness of the estimator is a key issue of the evaluation. In the case of two options, a necessary and sufficient condition is given by Ford in the Bradley-Terry model. We generalize this statement for the set of strictly log-concave distribution. Although in the case of three options necessary and sufficient condition is not known, there are two different sufficient conditions which are formulated in the literature. In this paper we generalize them, moreover we compare these conditions. Their capacities to indicate the existence of the maximum are analyzed by a large number of computer simulations. These simulations support that the new condition indicates the existence of the maximum much more frequently then the previously known ones,
We consider the problem of ranking n experts based on their performances on d tasks. We make a monotonicity assumption stating that for each pair of experts, one outperforms the other on all tasks. We consider the sequential setting where in each round, the learner has access to noisy evaluations of actively chosen pair of expert-task, given the information available up to the actual round. Given a confidence parameter $\delta$ $\in$ (0, 1), we provide strategies allowing to recover the correct ranking of experts and develop a bound on the total number of queries made by our algorithm that hold with probability at least 1 -- $\delta$. We show that our strategy is adaptive to the complexity of the problem (our bounds are instance dependent), and develop matching lower bounds up to a poly-logarithmic factor. Finally, we adapt our strategy to the relaxed problem of best expert identification and provide numerical simulation consistent with our theoretical results.
A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n} is used to label the vertices of G distinctly. A 2-path (a path with three vertices) in a vertex-numbered graph is said to be valid if the number of its middle vertex is smaller than the numbers of its endpoints. The problem of finding a vertex numbering of a given graph that optimises the number of induced valid 2-paths is studied, which is conjectured to be in the NP-hard class. The reported results for several graph classes show that apparently there are not one or more numbering patterns applicable to different classes of graphs, which requires the development of a specific numbering for each graph class under study.
A family $\mathcal{F}$ of sets satisfies the $(p,q)$-property if among every $p$ members of $\mathcal{F}$, some $q$ can be pierced by a single point. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that for any $p \geq q \geq d+1$, any family $\mathcal{F}$ of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$-property can be pierced by a finite number $c(p,q,d)$ of points. A similar theorem with respect to piercing by $(d-1)$-dimensional flats, called $(d-1)$-transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an $(\aleph_0,k+2)$-theorem with respect to $k$-transversals: Let $\mathcal{F}$ be an infinite family of closed balls in $\mathbb{R}^d$, and let $0 \leq k < d$. If among every $\aleph_0$ elements of $\mathcal{F}$, some $k+2$ can be pierced by a $k$-dimensional flat, then $\mathcal{F}$ can be pierced by a finite number of $k$-dimensional flats. The same result holds also for a wider class of families which consist of \emph{near-balls}, to be defined below. This is the first $(p,q)$-theorem in which the assumption is weakened to an $(\infty,\cdot)$ assumption. Our proofs combine geometric and topological tools.
Companies' adoption of artificial intelligence (AI) is increasingly becoming an essential element of business success. However, using AI poses new requirements for companies and their employees, including transparency and comprehensibility of AI systems. The field of Explainable AI (XAI) aims to address these issues. Yet, the current research primarily consists of laboratory studies, and there is a need to improve the applicability of the findings to real-world situations. Therefore, this project report paper provides insights into employees' needs and attitudes towards (X)AI. For this, we investigate employees' perspectives on (X)AI. Our findings suggest that AI and XAI are well-known terms perceived as important for employees. This recognition is a critical first step for XAI to potentially drive successful usage of AI by providing comprehensible insights into AI technologies. In a lessons-learned section, we discuss the open questions identified and suggest future research directions to develop human-centered XAI designs for companies. By providing insights into employees' needs and attitudes towards (X)AI, our project report contributes to the development of XAI solutions that meet the requirements of companies and their employees, ultimately driving the successful adoption of AI technologies in the business context.
Maximising a cumulative reward function that is Markov and stationary, i.e., defined over state-action pairs and independent of time, is sufficient to capture many kinds of goals in a Markov decision process (MDP). However, not all goals can be captured in this manner. In this paper we study convex MDPs in which goals are expressed as convex functions of the stationary distribution and show that they cannot be formulated using stationary reward functions. Convex MDPs generalize the standard reinforcement learning (RL) problem formulation to a larger framework that includes many supervised and unsupervised RL problems, such as apprenticeship learning, constrained MDPs, and so-called `pure exploration'. Our approach is to reformulate the convex MDP problem as a min-max game involving policy and cost (negative reward) `players', using Fenchel duality. We propose a meta-algorithm for solving this problem and show that it unifies many existing algorithms in the literature.
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