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A triangle in a hypergraph $\mathcal{H}$ is a set of three distinct edges $e, f, g\in\mathcal{H}$ and three distinct vertices $u, v, w\in V(\mathcal{H})$ such that $\{u, v\}\subseteq e$, $\{v, w\}\subseteq f$, $\{w, u\}\subseteq g$ and $\{u, v, w\}\cap e\cap f\cap g=\emptyset$. Johansson proved in 1996 that $\chi(G)=\mathcal{O}(\Delta/\log\Delta)$ for any triangle-free graph $G$ with maximum degree $\Delta$. Cooper and Mubayi later generalized the Johansson's theorem to all rank $3$ hypergraphs. In this paper we provide a common generalization of both these results for all hypergraphs, showing that if $\mathcal{H}$ is a rank $k$, triangle-free hypergraph, then the list chromatic number \[ \chi_{\ell}(\mathcal{H})\leq \mathcal{O}\left(\max_{2\leq \ell \leq k} \left\{\left( \frac{\Delta_{\ell}}{\log \Delta_{\ell}} \right)^{\frac{1}{\ell-1}} \right\}\right), \] where $\Delta_{\ell}$ is the maximum $\ell$-degree of $\mathcal{H}$. The result is sharp apart from the constant. Moreover, our result implies, generalizes and improves several earlier results on the chromatic number and also independence number of hypergraphs, while its proof is based on a different approach than prior works in hypergraphs (and therefore provides alternative proofs to them). In particular, as an application, we establish a bound on chromatic number of sparse hypergraphs in which each vertex is contained in few triangles, and thus extend results of Alon, Krivelevich and Sudakov, and Cooper and Mubayi from hypergraphs of rank 2 and 3, respectively, to all hypergraphs.

The theory of reinforcement learning currently suffers from a mismatch between its empirical performance and the theoretical characterization of its performance, with consequences for, e.g., the understanding of sample efficiency, safety, and robustness. The linear quadratic regulator with unknown dynamics is a fundamental reinforcement learning setting with significant structure in its dynamics and cost function, yet even in this setting there is a gap between the best known regret lower-bound of $\Omega_p(\sqrt{T})$ and the best known upper-bound of $O_p(\sqrt{T}\,\text{polylog}(T))$. The contribution of this paper is to close that gap by establishing a novel regret upper-bound of $O_p(\sqrt{T})$. Our proof is constructive in that it analyzes the regret of a concrete algorithm, and simultaneously establishes an estimation error bound on the dynamics of $O_p(T^{-1/4})$ which is also the first to match the rate of a known lower-bound. The two keys to our improved proof technique are (1) a more precise upper- and lower-bound on the system Gram matrix and (2) a self-bounding argument for the expected estimation error of the optimal controller.

In 2017, Krenn reported that certain problems related to the perfect matchings and colourings of graphs emerge out of studying the constructability of general quantum states using modern photonic technologies. He realized that if we can prove that the \emph{weighted matching index} of a graph, a parameter defined in terms of perfect matchings and colourings of the graph is at most 2, that could lead to exciting insights on the potential of resources of quantum inference. Motivated by this, he conjectured that the {weighted matching index} of any graph is at most 2. The first result on this conjecture was by Bogdanov, who proved that the \emph{(unweighted) matching index} of graphs (non-isomorphic to $K_4$) is at most 2, thus classifying graphs non-isomorphic to $K_4$ into Type 0, Type 1 and Type 2. By definition, the weighted matching index of Type 0 graphs is 0. We give a structural characterization for Type 2 graphs, using which we settle Krenn's conjecture for Type 2 graphs. Using this characterization, we provide a simple $O(|V||E|)$ time algorithm to find the unweighted matching index of any graph. In view of our work, Krenn's conjecture remains to be proved only for Type 1 graphs. We give upper bounds for the weighted matching index in terms of connectivity parameters for such graphs. Using these bounds, for a slightly simplified version, we settle Krenn's conjecture for the class of graphs with vertex connectivity at most 2 and the class of graphs with maximum degree at most 4. Krenn has been publicizing his conjecture in various ways since 2017. He has even declared a reward for a resolution of his conjecture. We hope that this article will popularize the problem among computer scientists.

In a multiple partners matching problem the agents can have multiple partners up to their capacities. In this paper we consider both the two-sided many-to-many stable matching problem and the one-sided stable fixtures problem under lexicographic preferences. We study strong core and Pareto-optimal solutions for this setting from a computational point of view. First we provide an example to show that the strong core can be empty even under these severe restrictions for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. We also show that for a given matching checking Pareto-optimality and the strong core properties are co-NP-complete problems for the many-to-many problem, and deciding the existence of a complete Pareto-optimal matching is also NP-hard for the fixtures problem. On the positive side, we give efficient algorithms for finding a near feasible strong core solution, where the capacities are only violated by at most one unit for each agent, and also for finding a half-matching in the strong core of fractional matchings. These polynomial time algorithms are based on the Top Trading Cycle algorithm. Finally, we also show that finding a maximum size matching that is Pareto-optimal can be done efficiently for many-to-many problems, which is in contrast with the hardness result for the fixtures problem.

In this work, we examine sampling problems with non-smooth potentials. We propose a novel Markov chain Monte Carlo algorithm for sampling from non-smooth potentials. We provide a non-asymptotical analysis of our algorithm and establish a polynomial-time complexity $\tilde {\cal O}(d\varepsilon^{-1})$ to obtain $\varepsilon$ total variation distance to the target density, better than most existing results under the same assumptions. Our method is based on the proximal bundle method and an alternating sampling framework. This framework requires the so-called restricted Gaussian oracle, which can be viewed as a sampling counterpart of the proximal mapping in convex optimization. One key contribution of this work is a fast algorithm that realizes the restricted Gaussian oracle for any convex non-smooth potential with bounded Lipschitz constant.

A matching $M$ is a $\mathscr{P}$-matching if the subgraph induced by the endpoints of the edges of $M$ satisfies property $\mathscr{P}$. As examples, for appropriate choices of $\mathscr{P}$, the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum $\mathscr{P}$-matching is a knowingly NP-Hard problem, with few exceptions, such as connected matchings, which has the same time complexity as usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent researches in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching $M$ such that the endpoint vertices of its edges induce a connected subgraph and the sum of the edge weights of $M$ is maximum. Unlike the unweighted Connected Matching problem, which is in P for general graphs, we show that Maximum Weight Connected Matching is NP-Hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite, and bounded degree planar graphs, while solvable in linear time for trees and subcubic graphs. When we restrict edge weights to be non negative only, we show that the problem turns to be polynomially solvable for chordal graphs, while it remains NP-Hard for most of the cases when weights can be negative. Our final contributions are on parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. In terms of kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover under standard complexity-theoretical hypotheses.

This paper considers the problem of online clustering with bandit feedback. A set of arms (or items) can be partitioned into various groups that are unknown. Within each group, the observations associated to each of the arms follow the same distribution with the same mean vector. At each time step, the agent queries or pulls an arm and obtains an independent observation from the distribution it is associated to. Subsequent pulls depend on previous ones as well as the previously obtained samples. The agent's task is to uncover the underlying partition of the arms with the least number of arm pulls and with a probability of error not exceeding a prescribed constant $\delta$. The problem proposed finds numerous applications from clustering of variants of viruses to online market segmentation. We present an instance-dependent information-theoretic lower bound on the expected sample complexity for this task, and design a computationally efficient and asymptotically optimal algorithm, namely Bandit Online Clustering (BOC). The algorithm includes a novel stopping rule for adaptive sequential testing that circumvents the need to exactly solve any NP-hard weighted clustering problem as its subroutines. We show through extensive simulations on synthetic and real-world datasets that BOC's performance matches the lower bound asymptotically, and significantly outperforms a non-adaptive baseline algorithm.

We study regression adjustments with additional covariates in randomized experiments under covariate-adaptive randomizations (CARs) when subject compliance is imperfect. We develop a regression-adjusted local average treatment effect (LATE) estimator that is proven to improve efficiency in the estimation of LATEs under CARs. Our adjustments can be parametric in linear and nonlinear forms, nonparametric, and high-dimensional. Even when the adjustments are misspecified, our proposed estimator is still consistent and asymptotically normal, and their inference method still achieves the exact asymptotic size under the null. When the adjustments are correctly specified, our estimator achieves the minimum asymptotic variance. When the adjustments are parametrically misspecified, we construct a new estimator which is weakly more efficient than linearly and nonlinearly adjusted estimators, as well as the one without any adjustments. Simulation evidence and empirical application confirm efficiency gains achieved by regression adjustments relative to both the estimator without adjustment and the standard two-stage least squares estimator.

Given $1\le \ell <k$ and $\delta>0$, let $\textbf{PM}(k,\ell,\delta)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $\delta\binom{n-\ell}{k-\ell}$. For $k\ge 3$, the decision problem in general $k$-uniform hypergraphs, equivalently $\textbf{PM}(k,\ell,0)$, is one of Karp's 21 NP-complete problems. Moreover, a reduction of Szyma\'{n}ska showed that $PM(k, \ell, \delta)$ is NP-complete for $\delta < 1-(1-1/k)^{k-\ell}$. A breakthrough by Keevash, Knox and Mycroft [STOC '13] resolved this problem for $\ell=k-1$ by showing that $PM(k, k-1, \delta)$ is in P for $\delta > 1/k$. Based on their result for $\ell=k-1$, Keevash, Knox and Mycroft conjectured that $PM(k, \ell, \delta)$ is in P for every $\delta > 1-(1-1/k)^{k-\ell}$. In this paper it is shown that this decision problem for perfect matchings can be reduced to the study of the minimum $\ell$-degree condition forcing the existence of fractional perfect matchings. That is, we hopefully solve the "computational complexity" aspect of the problem by reducing it to a well-known extremal problem in hypergraph theory. In particular, together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for $\ell\ge 0.4k$.

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.