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Quantifying the data uncertainty in learning tasks is often done by learning a prediction interval or prediction set of the label given the input. Two commonly desired properties for learned prediction sets are \emph{valid coverage} and \emph{good efficiency} (such as low length or low cardinality). Conformal prediction is a powerful technique for learning prediction sets with valid coverage, yet by default its conformalization step only learns a single parameter, and does not optimize the efficiency over more expressive function classes. In this paper, we propose a generalization of conformal prediction to multiple learnable parameters, by considering the constrained empirical risk minimization (ERM) problem of finding the most efficient prediction set subject to valid empirical coverage. This meta-algorithm generalizes existing conformal prediction algorithms, and we show that it achieves approximate valid population coverage and near-optimal efficiency within class, whenever the function class in the conformalization step is low-capacity in a certain sense. Next, this ERM problem is challenging to optimize as it involves a non-differentiable coverage constraint. We develop a gradient-based algorithm for it by approximating the original constrained ERM using differentiable surrogate losses and Lagrangians. Experiments show that our algorithm is able to learn valid prediction sets and improve the efficiency significantly over existing approaches in several applications such as prediction intervals with improved length, minimum-volume prediction sets for multi-output regression, and label prediction sets for image classification.

We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $\Delta$, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between $1$ and $1+\Delta$. Our goal is to maximize the number of near exact covered bins. If $\Delta=0$ or $\Delta>0$ is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that $P\neq NP$). However, for the case where $\Delta>0$ is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries, extending a recent line of research on random graph matching with independent edge weights to geometric models. Specifically, given a random permutation $\pi^*$ on $[n]$ and $n$ iid pairs of correlated Gaussian vectors $\{X_{\pi^*(i)}, Y_i\}$ in $\mathbb{R}^d$ with noise parameter $\sigma$, the edge weights are given by $A_{ij}=\kappa(X_i,X_j)$ and $B_{ij}=\kappa(Y_i,Y_j)$ for some link function $\kappa$. The goal is to recover the hidden vertex correspondence $\pi^*$ based on the observation of $A$ and $B$. We focus on the dot-product model with $\kappa(x,y)=\langle x, y \rangle$ and Euclidean distance model with $\kappa(x,y)=\|x-y\|^2$, in the low-dimensional regime of $d=o(\log n)$ wherein the underlying geometric structures are most evident. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $\pi^*$ when $\sigma=o(n^{-2/d})$ and almost perfect recovery with a vanishing fraction of errors when $\sigma=o(n^{-1/d})$. Furthermore, these conditions are shown to be information-theoretically optimal even when the latent coordinates $\{X_i\}$ and $\{Y_i\}$ are observed, complementing the recent results of [DCK19] and [KNW22] in geometric models of the planted bipartite matching problem. As a side discovery, we show that the celebrated spectral algorithm of [Ume88] emerges as a further approximation to the maximum likelihood in the geometric model.

When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.

In Formal Concept Analysis, a base for a finite structure is a set of implications that characterizes all valid implications of the structure. This notion can be adapted to the context of Description Logic, where the base consists of a set of concept inclusions instead of implications. In this setting, concept expressions can be arbitrarily large. Thus, it is not clear whether a finite base exists and, if so, how large concept expressions may need to be. We first revisit results in the literature for mining EL bases from finite interpretations. Those mainly focus on finding a finite base or on fixing the role depth but potentially losing some of the valid concept inclusions with higher role depth. We then present a new strategy for mining EL bases which is adaptable in the sense that it can bound the role depth of concepts depending on the local structure of the interpretation. Our strategy guarantees to capture all EL concept inclusions holding in the interpretation, not only the ones up to a fixed role depth.

For a connected graph $G = (V, E)$ and $s, t \in V$, a non-separating $s$-$t$ path is a path $P$ between $s$ and $t$ such that the set of vertices of $P$ does not separate $G$, that is, $G - V(P)$ is connected. An $s$-$t$ path is non-disconnecting if $G - E(P)$ is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating $s$-$t$ path of length at most $k$ is W[1]-hard parameterized by $k$, while the non-disconnecting counterpart is fixed-parameter tractable parameterized by $k$. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is fixed-parameter tractable parameterized by $k$ on planar graphs and polynomial-time solvable on chordal graphs if $k$ is the shortest path distance between $s$ and $t$.

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\mathrm{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr\"obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters $k$, $h$ respectively in terms of the parameter $\epsilon$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ \epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of $\epsilon$. These estimates and a suitable localization technique, via the second order condition (see \cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations

In this paper we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.

A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by \textsc{LHom}($H$), the instance is a graph $G$, whose every vertex is equipped with a subset of $V(H)$, called list. We ask whether there exists a homomorphism from $G$ to $H$, such that every vertex from $G$ is mapped to a vertex from its list. We study the complexity of the \textsc{LHom}($H$) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the question for what graphs $H$ and for what types of geometric objects, the \textsc{LHom}($H$) problem can be solved in time subexponential in the number of vertices of the instance. We fully resolve this question for string graphs, i.e., intersection graphs of continuous curves in the plane. Quite surprisingly, it turns out that the dichotomy exactly coincides with the analogous dichotomy for graphs excluding a fixed path as an induced subgraph [Okrasa, Rz\k{a}\.zewski, STACS 2021]. Then we turn our attention to subclasses of string graphs, defined as intersections of fat objects. We observe that the (non)existence of subexponential-time algorithms in such classes is closely related to the size $\mathrm{mrc}(H)$ of a maximum reflexive clique in $H$, i.e., maximum number of pairwise adjacent vertices, each of which has a loop. We study the maximum value of $\mathrm{mrc}(H)$ that guarantees the existence of a subexponential-time algorithm for \textsc{LHom}($H$) in intersection graphs of (i) convex fat objects, (ii) fat similarly-sized objects, and (iii) disks. In the first two cases we obtain optimal results, by giving matching algorithms and lower bounds. Finally, we discuss possible extensions of our results to weighted generalizations of \textsc{LHom}($H$).