We prove complex contraction for zero-free regions of counting weighted set cover problem in which an element can appear in an unbounded number of sets, thus obtaining fully polynomial-time approximation schemes(FPTAS) via Barvinok's algorithmic paradigm\cite{barvinok2016combinatorics}. Relying on the computation tree expansion, our approach does not need proof of correlation decay in the real axis. We directly look in the complex plane for a region that contracts into its interior as the tree recursion procedure goes from leaves to the root. For the class of problems under the framework of weighted set covers, we are able to give a general approach for describing the contraction regions and draw a unified algorithmic conclusion. Several previous results, including counting (weighted-)edge covers, counting bipartite independent sets and counting monotone CNFs can be completely or partially covered by our main theorem. In contrast to the correlation decay method which also depends on tree expansions and needs different potential functions for different problems, our approach is more generic in the sense that our contraction region for different problems shares a common shape in the complex plane.

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, no parts of such partitions can be written as the sum of different integers which are not parts. We address in this paper the algorithmic aspects related to unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given number. We design two algorithms to solve the two mentioned problems and we discuss their complexity.

We prove a bound of $O( k (n+m)\log^{d-1})$ on the number of incidences between $n$ points and $m$ axis parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points. That is, the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work by a factor of $\log^d n$, for $d >2$. We also study the variant of the problem for points and halfspaces, where we use shallow cuttings to get a near linear bound in two and three dimensions.

An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed-Solomon (GRS) codes.

This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.

Motivated by many interesting real-world applications in logistics and online advertising, we consider an online allocation problem subject to lower and upper resource constraints, where the requests arrive sequentially, sampled i.i.d. from an unknown distribution, and we need to promptly make a decision given limited resources and lower bounds requirements. First, with knowledge of the measure of feasibility, i.e., $\alpha$, we propose a new algorithm that obtains $1-O(\frac{\epsilon}{\alpha-\epsilon})$ -competitive ratio for the offline problems that know the entire requests ahead of time. Inspired by the previous studies, this algorithm adopts an innovative technique to dynamically update a threshold price vector for making decisions. Moreover, an optimization method to estimate the optimal measure of feasibility is proposed with theoretical guarantee at the end of this paper. Based on this method, if we tolerate slight violation of the lower bounds constraints with parameter $\eta$, the proposed algorithm is naturally extended to the settings without strong feasible assumption, which cover the significantly unexplored infeasible scenarios.

Reward is the driving force for reinforcement-learning agents. This paper is dedicated to understanding the expressivity of reward as a way to capture tasks that we would want an agent to perform. We frame this study around three new abstract notions of "task" that might be desirable: (1) a set of acceptable behaviors, (2) a partial ordering over behaviors, or (3) a partial ordering over trajectories. Our main results prove that while reward can express many of these tasks, there exist instances of each task type that no Markov reward function can capture. We then provide a set of polynomial-time algorithms that construct a Markov reward function that allows an agent to optimize tasks of each of these three types, and correctly determine when no such reward function exists. We conclude with an empirical study that corroborates and illustrates our theoretical findings.

Colorizing a given gray-level image is an important task in the media and advertising industry. Due to the ambiguity inherent to colorization (many shades are often plausible), recent approaches started to explicitly model diversity. However, one of the most obvious artifacts, structural inconsistency, is rarely considered by existing methods which predict chrominance independently for every pixel. To address this issue, we develop a conditional random field based variational auto-encoder formulation which is able to achieve diversity while taking into account structural consistency. Moreover, we introduce a controllability mecha- nism that can incorporate external constraints from diverse sources in- cluding a user interface. Compared to existing baselines, we demonstrate that our method obtains more diverse and globally consistent coloriza- tions on the LFW, LSUN-Church and ILSVRC-2015 datasets.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.

Collecting training data from the physical world is usually time-consuming and even dangerous for fragile robots, and thus, recent advances in robot learning advocate the use of simulators as the training platform. Unfortunately, the reality gap between synthetic and real visual data prohibits direct migration of the models trained in virtual worlds to the real world. This paper proposes a modular architecture for tackling the virtual-to-real problem. The proposed architecture separates the learning model into a perception module and a control policy module, and uses semantic image segmentation as the meta representation for relating these two modules. The perception module translates the perceived RGB image to semantic image segmentation. The control policy module is implemented as a deep reinforcement learning agent, which performs actions based on the translated image segmentation. Our architecture is evaluated in an obstacle avoidance task and a target following task. Experimental results show that our architecture significantly outperforms all of the baseline methods in both virtual and real environments, and demonstrates a faster learning curve than them. We also present a detailed analysis for a variety of variant configurations, and validate the transferability of our modular architecture.

We prove complex contraction for zero-free regions of counting weighted set cover problem in which an element can appear in an unbounded number of sets, thus obtaining fully polynomial-time approximation schemes(FPTAS) via Barvinok's algorithmic paradigm\cite{barvinok2016combinatorics}. Relying on the computation tree expansion, our approach does not need proof of correlation decay in the real axis. We directly look in the complex plane for a region that contracts into its interior as the tree recursion procedure goes from leaves to the root. For the class of problems under the framework of weighted set covers, we are able to give a general approach for describing the contraction regions and draw a unified algorithmic conclusion. Several previous results, including counting (weighted-)edge covers, counting bipartite independent sets and counting monotone CNFs can be completely or partially covered by our main theorem. In contrast to the correlation decay method which also depends on tree expansions and needs different potential functions for different problems, our approach is more generic in the sense that our contraction region for different problems shares a common shape in the complex plane.

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, no parts of such partitions can be written as the sum of different integers which are not parts. We address in this paper the algorithmic aspects related to unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given number. We design two algorithms to solve the two mentioned problems and we discuss their complexity.

We prove a bound of $O( k (n+m)\log^{d-1})$ on the number of incidences between $n$ points and $m$ axis parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points. That is, the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work by a factor of $\log^d n$, for $d >2$. We also study the variant of the problem for points and halfspaces, where we use shallow cuttings to get a near linear bound in two and three dimensions.

An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed-Solomon (GRS) codes.

This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.

Motivated by many interesting real-world applications in logistics and online advertising, we consider an online allocation problem subject to lower and upper resource constraints, where the requests arrive sequentially, sampled i.i.d. from an unknown distribution, and we need to promptly make a decision given limited resources and lower bounds requirements. First, with knowledge of the measure of feasibility, i.e., $\alpha$, we propose a new algorithm that obtains $1-O(\frac{\epsilon}{\alpha-\epsilon})$ -competitive ratio for the offline problems that know the entire requests ahead of time. Inspired by the previous studies, this algorithm adopts an innovative technique to dynamically update a threshold price vector for making decisions. Moreover, an optimization method to estimate the optimal measure of feasibility is proposed with theoretical guarantee at the end of this paper. Based on this method, if we tolerate slight violation of the lower bounds constraints with parameter $\eta$, the proposed algorithm is naturally extended to the settings without strong feasible assumption, which cover the significantly unexplored infeasible scenarios.

Reward is the driving force for reinforcement-learning agents. This paper is dedicated to understanding the expressivity of reward as a way to capture tasks that we would want an agent to perform. We frame this study around three new abstract notions of "task" that might be desirable: (1) a set of acceptable behaviors, (2) a partial ordering over behaviors, or (3) a partial ordering over trajectories. Our main results prove that while reward can express many of these tasks, there exist instances of each task type that no Markov reward function can capture. We then provide a set of polynomial-time algorithms that construct a Markov reward function that allows an agent to optimize tasks of each of these three types, and correctly determine when no such reward function exists. We conclude with an empirical study that corroborates and illustrates our theoretical findings.

Colorizing a given gray-level image is an important task in the media and advertising industry. Due to the ambiguity inherent to colorization (many shades are often plausible), recent approaches started to explicitly model diversity. However, one of the most obvious artifacts, structural inconsistency, is rarely considered by existing methods which predict chrominance independently for every pixel. To address this issue, we develop a conditional random field based variational auto-encoder formulation which is able to achieve diversity while taking into account structural consistency. Moreover, we introduce a controllability mecha- nism that can incorporate external constraints from diverse sources in- cluding a user interface. Compared to existing baselines, we demonstrate that our method obtains more diverse and globally consistent coloriza- tions on the LFW, LSUN-Church and ILSVRC-2015 datasets.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.

Collecting training data from the physical world is usually time-consuming and even dangerous for fragile robots, and thus, recent advances in robot learning advocate the use of simulators as the training platform. Unfortunately, the reality gap between synthetic and real visual data prohibits direct migration of the models trained in virtual worlds to the real world. This paper proposes a modular architecture for tackling the virtual-to-real problem. The proposed architecture separates the learning model into a perception module and a control policy module, and uses semantic image segmentation as the meta representation for relating these two modules. The perception module translates the perceived RGB image to semantic image segmentation. The control policy module is implemented as a deep reinforcement learning agent, which performs actions based on the translated image segmentation. Our architecture is evaluated in an obstacle avoidance task and a target following task. Experimental results show that our architecture significantly outperforms all of the baseline methods in both virtual and real environments, and demonstrates a faster learning curve than them. We also present a detailed analysis for a variety of variant configurations, and validate the transferability of our modular architecture.

We prove complex contraction for zero-free regions of counting weighted set cover problem in which an element can appear in an unbounded number of sets, thus obtaining fully polynomial-time approximation schemes(FPTAS) via Barvinok's algorithmic paradigm\cite{barvinok2016combinatorics}. Relying on the computation tree expansion, our approach does not need proof of correlation decay in the real axis. We directly look in the complex plane for a region that contracts into its interior as the tree recursion procedure goes from leaves to the root. For the class of problems under the framework of weighted set covers, we are able to give a general approach for describing the contraction regions and draw a unified algorithmic conclusion. Several previous results, including counting (weighted-)edge covers, counting bipartite independent sets and counting monotone CNFs can be completely or partially covered by our main theorem. In contrast to the correlation decay method which also depends on tree expansions and needs different potential functions for different problems, our approach is more generic in the sense that our contraction region for different problems shares a common shape in the complex plane.

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, no parts of such partitions can be written as the sum of different integers which are not parts. We address in this paper the algorithmic aspects related to unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given number. We design two algorithms to solve the two mentioned problems and we discuss their complexity.

We prove a bound of $O( k (n+m)\log^{d-1})$ on the number of incidences between $n$ points and $m$ axis parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points. That is, the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work by a factor of $\log^d n$, for $d >2$. We also study the variant of the problem for points and halfspaces, where we use shallow cuttings to get a near linear bound in two and three dimensions.

An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed-Solomon (GRS) codes.

This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.

Motivated by many interesting real-world applications in logistics and online advertising, we consider an online allocation problem subject to lower and upper resource constraints, where the requests arrive sequentially, sampled i.i.d. from an unknown distribution, and we need to promptly make a decision given limited resources and lower bounds requirements. First, with knowledge of the measure of feasibility, i.e., $\alpha$, we propose a new algorithm that obtains $1-O(\frac{\epsilon}{\alpha-\epsilon})$ -competitive ratio for the offline problems that know the entire requests ahead of time. Inspired by the previous studies, this algorithm adopts an innovative technique to dynamically update a threshold price vector for making decisions. Moreover, an optimization method to estimate the optimal measure of feasibility is proposed with theoretical guarantee at the end of this paper. Based on this method, if we tolerate slight violation of the lower bounds constraints with parameter $\eta$, the proposed algorithm is naturally extended to the settings without strong feasible assumption, which cover the significantly unexplored infeasible scenarios.

Reward is the driving force for reinforcement-learning agents. This paper is dedicated to understanding the expressivity of reward as a way to capture tasks that we would want an agent to perform. We frame this study around three new abstract notions of "task" that might be desirable: (1) a set of acceptable behaviors, (2) a partial ordering over behaviors, or (3) a partial ordering over trajectories. Our main results prove that while reward can express many of these tasks, there exist instances of each task type that no Markov reward function can capture. We then provide a set of polynomial-time algorithms that construct a Markov reward function that allows an agent to optimize tasks of each of these three types, and correctly determine when no such reward function exists. We conclude with an empirical study that corroborates and illustrates our theoretical findings.

Colorizing a given gray-level image is an important task in the media and advertising industry. Due to the ambiguity inherent to colorization (many shades are often plausible), recent approaches started to explicitly model diversity. However, one of the most obvious artifacts, structural inconsistency, is rarely considered by existing methods which predict chrominance independently for every pixel. To address this issue, we develop a conditional random field based variational auto-encoder formulation which is able to achieve diversity while taking into account structural consistency. Moreover, we introduce a controllability mecha- nism that can incorporate external constraints from diverse sources in- cluding a user interface. Compared to existing baselines, we demonstrate that our method obtains more diverse and globally consistent coloriza- tions on the LFW, LSUN-Church and ILSVRC-2015 datasets.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.

Collecting training data from the physical world is usually time-consuming and even dangerous for fragile robots, and thus, recent advances in robot learning advocate the use of simulators as the training platform. Unfortunately, the reality gap between synthetic and real visual data prohibits direct migration of the models trained in virtual worlds to the real world. This paper proposes a modular architecture for tackling the virtual-to-real problem. The proposed architecture separates the learning model into a perception module and a control policy module, and uses semantic image segmentation as the meta representation for relating these two modules. The perception module translates the perceived RGB image to semantic image segmentation. The control policy module is implemented as a deep reinforcement learning agent, which performs actions based on the translated image segmentation. Our architecture is evaluated in an obstacle avoidance task and a target following task. Experimental results show that our architecture significantly outperforms all of the baseline methods in both virtual and real environments, and demonstrates a faster learning curve than them. We also present a detailed analysis for a variety of variant configurations, and validate the transferability of our modular architecture.