In this paper we prove new results concerning pseudo-polynomial time algorithms for the classical scheduling problem of minimizing the weighted number of jobs on a single machine, the so-called $1 \mid \mid \Sigma w_j U_j$ problem. The previously best known pseudo-polynomial algorithm for this problem, due to Lawler and Moore [Management Science'69], dates back to the late 60s and has running time $O(d_{\max}n)$ or $O(wn)$, where $d_{max}$ and $w$ are the maximum due date and sum of weights of the job set respectively. Using the recently introduced "prediction technique" by Bateni et al. [STOC'19], we present an algorithm for the problem running in $\widetilde{O}(d_{\#}(n +dw_{\max}))$ time, where $d_{\#}$ is the number of different due dates in the instance, $d$ is the total sum of the $d_{\#}$ different due dates, and $w_{\max}$ is the maximum weight of any job. This algorithm outperform the algorithm of Lawler and Moore for certain ranges of the above parameters, and provides the first such improvement for over 50 years. We complement this result by showing that $1 \mid \mid \Sigma w_j U_j$ has no $\widetilde{O}(n +w^{1-\varepsilon}_{\max}n)$ nor $\widetilde{O}(n +w_{\max}n^{1-\varepsilon})$ time algorithms assuming $\forall \exists$-SETH conjecture, a recently introduced variant of the well known Strong Exponential Time Hypothesis (SETH).
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We study reinforcement learning for two-player zero-sum Markov games with simultaneous moves in the finite-horizon setting, where the transition kernel of the underlying Markov games can be parameterized by a linear function over the current state, both players' actions and the next state. In particular, we assume that we can control both players and aim to find the Nash Equilibrium by minimizing the duality gap. We propose an algorithm Nash-UCRL based on the principle "Optimism-in-Face-of-Uncertainty". Our algorithm only needs to find a Coarse Correlated Equilibrium (CCE), which is computationally efficient. Specifically, we show that Nash-UCRL can provably achieve an $\tilde{O}(dH\sqrt{T})$ regret, where $d$ is the linear function dimension, $H$ is the length of the game and $T$ is the total number of steps in the game. To assess the optimality of our algorithm, we also prove an $\tilde{\Omega}( dH\sqrt{T})$ lower bound on the regret. Our upper bound matches the lower bound up to logarithmic factors, which suggests the optimality of our algorithm.
The Schrijver graph $S(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ that do not include two consecutive elements modulo $n$, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of $S(n,k)$ is $n-2k+2$ (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of $S(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class $\mathsf{PPA}$. We prove that it can be solved by a randomized algorithm with running time $n^{O(1)} \cdot k^{O(k)}$, hence it is fixed-parameter tractable with respect to the parameter $k$.
Follow-the-Regularized-Lead (FTRL) and Online Mirror Descent (OMD) are regret minimization algorithms for Online Convex Optimization (OCO), they are mathematically elegant but less practical in solving Extensive-Form Games (EFGs). Counterfactual Regret Minimization (CFR) is a technique for approximating Nash equilibria in EFGs. CFR and its variants have a fast convergence rate in practice, but their theoretical results are not satisfactory. In recent years, researchers have been trying to link CFRs with OCO algorithms, which may provide new theoretical results and inspire new algorithms. However, existing analysis is restricted to local decision points. In this paper, we show that CFRs with Regret Matching and Regret Matching+ are equivalent to special cases of FTRL and OMD, respectively. According to these equivalences, a new FTRL and a new OMD algorithm, which can be considered as extensions of vanilla CFR and CFR+, are derived. The experimental results show that the two variants converge faster than conventional FTRL and OMD, even faster than vanilla CFR and CFR+ in some EFGs.
In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
We propose a novel concise function representation for graphical models, a central theoretical framework that provides the basis for many reasoning tasks. We then show how we exploit our concise representation based on deterministic finite state automata within Bucket Elimination (BE), a general approach based on the concept of variable elimination that can be used to solve many inference and optimisation tasks, such as most probable explanation and constrained optimisation. We denote our version of BE as FABE. By using our concise representation within FABE, we dramatically improve the performance of BE in terms of runtime and memory requirements. Results achieved by comparing FABE with state of the art approaches for most probable explanation (i.e., recursive best-first and structured message passing) and constrained optimisation (i.e., CPLEX, GUROBI, and toulbar2) following an established methodology confirm the efficacy of our concise function representation, showing runtime improvements of up to 5 orders of magnitude in our tests.
We study dynamic algorithms for the problem of maximizing a monotone submodular function over a stream of $n$ insertions and deletions. We show that any algorithm that maintains a $(0.5+\epsilon)$-approximate solution under a cardinality constraint, for any constant $\epsilon>0$, must have an amortized query complexity that is $\mathit{polynomial}$ in $n$. Moreover, a linear amortized query complexity is needed in order to maintain a $0.584$-approximate solution. This is in sharp contrast with recent dynamic algorithms of [LMNF+20, Mon20] that achieve $(0.5-\epsilon)$-approximation with a $\mathsf{poly}\log(n)$ amortized query complexity. On the positive side, when the stream is insertion-only, we present efficient algorithms for the problem under a cardinality constraint and under a matroid constraint with approximation guarantee $1-1/e-\epsilon$ and amortized query complexities $\smash{O(\log (k/\epsilon)/\epsilon^2)}$ and $\smash{k^{\tilde{O}(1/\epsilon^2)}\log n}$, respectively, where $k$ denotes the cardinality parameter or the rank of the matroid.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
CP decomposition (CPD) is prevalent in chemometrics, signal processing, data mining and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithm for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in areas like signal processing, data analytics and in various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by considering the critical points of a function different from what is used in the previous work. Computing these critical points in an alternating manner motivates an alternating optimization algorithm which corresponds to alternating least squares algorithm in the matrix case. However, for tensors with order greater than equal to $3$, it minimizes an objective function which is different from the commonly used least squares loss. Alternating optimization of this new objective leads to simple updates to the factor matrices with the same asymptotic computational cost as ALS. We show that a subsweep of this algorithm can achieve a superlinear convergence rate for exact CPD with known rank and verify it experimentally. We then view the algorithm as optimizing a Mahalanobis distance with respect to each factor with ground metric dependent on the other factors. This perspective allows us to generalize our approach to interpolate between updates corresponding to the ALS and the new algorithm to manage the tradeoff between stability and fitness of the decomposition. Our experimental results show that for approximating synthetic and real-world tensors, this algorithm and its variants converge to a better conditioned decomposition with comparable and sometimes better fitness as compared to the ALS algorithm.
We propose a simple modification to the iterative hard thresholding (IHT) algorithm, which recovers asymptotically sparser solutions as a function of the condition number. When aiming to minimize a convex function $f(x)$ with condition number $\kappa$ subject to $x$ being an $s$-sparse vector, the standard IHT guarantee is a solution with relaxed sparsity $O(s\kappa^2)$, while our proposed algorithm, regularized IHT, returns a solution with sparsity $O(s\kappa)$. Our algorithm significantly improves over ARHT which also finds a solution of sparsity $O(s\kappa)$, as it does not require re-optimization in each iteration (and so is much faster), is deterministic, and does not require knowledge of the optimal solution value $f(x^*)$ or the optimal sparsity level $s$. Our main technical tool is an adaptive regularization framework, in which the algorithm progressively learns the weights of an $\ell_2$ regularization term that will allow convergence to sparser solutions. We also apply this framework to low rank optimization, where we achieve a similar improvement of the best known condition number dependence from $\kappa^2$ to $\kappa$.
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