Cai and Hemachandra used iterative constant-setting to prove that Few $\subseteq$ $\oplus$P (and thus that FewP $\subseteq$ $\oplus$P). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed "nongappy"-ness) of the easy-to-find "targets" used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant's unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies that all O(loglogn)-ambiguity NP sets are in the restricted counting class $\rm RC_{PRIMES}$.
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We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lov\'asz-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.
The $k$-center problem is to choose a subset of size $k$ from a set of $n$ points such that the maximum distance from each point to its nearest center is minimized. Let $Q=\{Q_1,\ldots,Q_n\}$ be a set of polygons or segments in the region-based uncertainty model, in which each $Q_i$ is an uncertain point, where the exact locations of the points in $Q_i$ are unknown. The geometric objects segments and polygons can be models of a point set. We define the uncertain version of the $k$-center problem as a generalization in which the objective is to find $k$ points from $Q$ to cover the remaining regions of $Q$ with minimum or maximum radius of the cluster to cover at least one or all exact instances of each $Q_i$, respectively. We modify the region-based model to allow multiple points to be chosen from a region and call the resulting model the aggregated uncertainty model. All these problems contain the point version as a special case, so they are all NP-hard with a lower bound 1.822. We give approximation algorithms for uncertain $k$-center of a set of segments and polygons. We also have implemented some of our algorithms on a data-set to show our theoretical performance guarantees can be achieved in practice.
A major open problem in the area of infinite-duration games is to characterize winning conditions that are determined in finite-memory strategies. Infinite-duration games are usually studied over edge-colored graphs, with winning conditions that are defined in terms of sequences of colors. In this paper, we investigate a restricted class of finite-memory strategies called chromatic finite-memory strategies. While general finite-memory strategies operate with sequences of edges of a game graph, chromatic finite-memory strategies observe only colors of these edges. Recent results in this area show that studying finite-memory determinacy is more tractable when we restrict ourselves to chromatic strategies. On the other hand, as was shown by Le Roux (CiE 2020), determinacy in general finite-memory strategies implies determinacy in chromatic finite-memory strategies. Unfortunately, this result is quite inefficient in terms of the state complexity: to replace a winning strategy with few states of general memory, we might need much more states of chromatic memory. The goal of the present paper is to find out the exact state complexity of this transformation. For every winning condition and for every game graph with $n$ nodes we show the following: if this game graph has a winning strategy with $q$ states of general memory, then it also has a winning strategy with $(q + 1)^n$ states of chromatic memory. We also show that this bound is almost tight. For every $q$ and $n$, we construct a winning condition and a game graph with $n + O(1)$ nodes, where one can win with $q$ states of general memory, but not with $q^n - 1$ states of chromatic memory.
Running machine learning algorithms on large and rapidly growing volumes of data is often computationally expensive, one common trick to reduce the size of a data set, and thus reduce the computational cost of machine learning algorithms, is \emph{probability sampling}. It creates a sampled data set by including each data point from the original data set with a known probability. Although the benefit of running machine learning algorithms on the reduced data set is obvious, one major concern is that the performance of the solution obtained from samples might be much worse than that of the optimal solution when using the full data set. In this paper, we examine the performance loss caused by probability sampling in the context of adaptive submodular maximization. We consider a simple probability sampling method which selects each data point with probability at least $r\in[0,1]$. If we set $r=1$, our problem reduces to finding a solution based on the original full data set. We define sampling gap as the largest ratio between the optimal solution obtained from the full data set and the optimal solution obtained from the samples, over independence systems. Our main contribution is to show that if the sampling probability of each data point is at least $r$ and the utility function is policywise submodular, then the sampling gap is both upper bounded and lower bounded by $1/r$. We show that the property of policywise submodular can be found in a wide range of real-world applications, including pool-based active learning and adaptive viral marketing.
In this paper, we study a distributed learning problem constrained by constant communication bits. Specifically, we consider the distributed hypothesis testing (DHT) problem where two distributed nodes are constrained to transmit a constant number of bits to a central decoder. In such cases, we show that in order to achieve the optimal error exponents, it suffices to consider the empirical distributions of observed data sequences and encode them to the transmission bits. With such a coding strategy, we develop a geometric approach in the distribution spaces and establish an inner bound of error exponent regions. In particular, we show the optimal achievable error exponents and coding schemes for the following cases: (i) both nodes can transmit $\log_23$ bits; (ii) one of the nodes can transmit $1$ bit, and the other node is not constrained; (iii) the joint distribution of the nodes are conditionally independent given one hypothesis. Furthermore, we provide several numerical examples for illustrating the theoretical results. Our results provide theoretical guidance for designing practical distributed learning rules, and the developed approach also reveals new potentials for establishing error exponents for DHT with more general communication constraints.
Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. A strong subgraph $H$ of $D$ is called an $S$-strong subgraph if $S\subseteq V(H)$. A pair of $S$-strong subgraphs $D_1$ and $D_2$ are said to be arc-disjoint if $A(D_1)\cap A(D_2)=\emptyset$. Let $\lambda_S(D)$ be the maximum number of arc-disjoint $S$-strong subgraphs in $D$. The strong subgraph $k$-arc-connectivity is defined as $$\lambda_k(D)=\min\{\lambda_S(D)\mid S\subseteq V(D), |S|=k\}.$$ The parameter $\lambda_k(D)$ could be seen as a generalization of classical edge-connectivity of undirected graphs. In this paper, we study the strong subgraph 2-arc-connectivity of Cartesian product $\lambda_2(G\Box H)$ of two digraphs $G$ and $H$. We prove that $\lambda_2(G\Box H)\ge \lambda_2(G)+\lambda_2(H)-1$ but there is no linear upper bound for $\lambda_2(G\Box H)$ in terms of $\lambda_2(G)$ and $\lambda_2(H).$ We also obtain exact values for $\lambda_2(G\Box H)$, where $G$ and $H$ are digraphs from some digraph families.
The static properties of code repositories, e.g., lines of code, dependents, dependencies, etc. can be readily scraped from code hosting platforms such as GitHub, and from package management systems such as npm for JavaScript; Although no less important, information related to the dynamic properties of programs, e.g., number of tests in a test suite that pass or fail is less readily available. This dynamic information could be immensely useful to researchers conducting corpus analyses, as it would give them the ability to differentiate projects based on properties of the projects that can only be observed by running them. In this paper, we present npm-filter, an automated tool that can download, install, build, test, and run custom user scripts over the source code of JavaScript projects available on npm, the most popular JavaScript package manager. We outline this tool, describe its implementation, and show that npm-filter has already been useful in developing evaluation suites for multiple JavaScript tools.
Promoting behavioural diversity is critical for solving games with non-transitive dynamics where strategic cycles exist, and there is no consistent winner (e.g., Rock-Paper-Scissors). Yet, there is a lack of rigorous treatment for defining diversity and constructing diversity-aware learning dynamics. In this work, we offer a geometric interpretation of behavioural diversity in games and introduce a novel diversity metric based on \emph{determinantal point processes} (DPP). By incorporating the diversity metric into best-response dynamics, we develop \emph{diverse fictitious play} and \emph{diverse policy-space response oracle} for solving normal-form games and open-ended games. We prove the uniqueness of the diverse best response and the convergence of our algorithms on two-player games. Importantly, we show that maximising the DPP-based diversity metric guarantees to enlarge the \emph{gamescape} -- convex polytopes spanned by agents' mixtures of strategies. To validate our diversity-aware solvers, we test on tens of games that show strong non-transitivity. Results suggest that our methods achieve much lower exploitability than state-of-the-art solvers by finding effective and diverse strategies.
Detecting objects in aerial images is challenging for at least two reasons: (1) target objects like pedestrians are very small in pixels, making them hardly distinguished from surrounding background; and (2) targets are in general sparsely and non-uniformly distributed, making the detection very inefficient. In this paper, we address both issues inspired by observing that these targets are often clustered. In particular, we propose a Clustered Detection (ClusDet) network that unifies object clustering and detection in an end-to-end framework. The key components in ClusDet include a cluster proposal sub-network (CPNet), a scale estimation sub-network (ScaleNet), and a dedicated detection network (DetecNet). Given an input image, CPNet produces object cluster regions and ScaleNet estimates object scales for these regions. Then, each scale-normalized cluster region is fed into DetecNet for object detection. ClusDet has several advantages over previous solutions: (1) it greatly reduces the number of chips for final object detection and hence achieves high running time efficiency, (2) the cluster-based scale estimation is more accurate than previously used single-object based ones, hence effectively improves the detection for small objects, and (3) the final DetecNet is dedicated for clustered regions and implicitly models the prior context information so as to boost detection accuracy. The proposed method is tested on three popular aerial image datasets including VisDrone, UAVDT and DOTA. In all experiments, ClusDet achieves promising performance in comparison with state-of-the-art detectors. Code will be available in \url{//github.com/fyangneil}.
We present a novel framework for iterative visual reasoning. Our framework goes beyond current recognition systems that lack the capability to reason beyond stack of convolutions. The framework consists of two core modules: a local module that uses spatial memory to store previous beliefs with parallel updates; and a global graph-reasoning module. Our graph module has three components: a) a knowledge graph where we represent classes as nodes and build edges to encode different types of semantic relationships between them; b) a region graph of the current image where regions in the image are nodes and spatial relationships between these regions are edges; c) an assignment graph that assigns regions to classes. Both the local module and the global module roll-out iteratively and cross-feed predictions to each other to refine estimates. The final predictions are made by combining the best of both modules with an attention mechanism. We show strong performance over plain ConvNets, \eg achieving an $8.4\%$ absolute improvement on ADE measured by per-class average precision. Analysis also shows that the framework is resilient to missing regions for reasoning.
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