Matching problems have been widely studied in the research community, especially Ad-Auctions with many applications ranging from network design to advertising. Following the various advancements in machine learning, one natural question is whether classical algorithms can benefit from machine learning and obtain better-quality solutions. Even a small percentage of performance improvement in matching problems could result in significant gains for the studied use cases. For example, the network throughput or the revenue of Ad-Auctions can increase remarkably. This paper presents algorithms with machine learning predictions for the Online Bounded Allocation and the Online Ad-Auctions problems. We constructed primal-dual algorithms that achieve competitive performance depending on the quality of the predictions. When the predictions are accurate, the algorithms’ performance surpasses previous performance bounds, while when the predictions are misleading, the algorithms maintain standard worst-case performance guarantees. We provide supporting experiments on generated data for our theoretical findings.

In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guarantees. Here, we focus on minimizing regret for online arriving non-monotone DR-submodular functions over down-closed and general convex sets. First, we present an online algorithm that achieves a 1/e-approximation ratio with the regret of O(T^{3/4}) for maximizing DR-submodular functions over any down-closed convex set. Note that, the approximation ratio of 1/e matches the best-known guarantee for the offline version of the problem. Next, we give an online algorithm that achieves an approximation guarantee (depending on the search space) for the problem of maximizing non-monotone continuous DR-submodular functions over a general convex set (not necessarily down-closed). To best of our knowledge, no prior algorithm with approximation guarantee was known for non-monotone DR-submodular maximization in the online setting. Finally we run experiments to verify the performance of our algorithms on problems arising in machine learning domain with the real-world datasets.

Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.

In this paper, we present primal-dual algorithms for online problems with non-convex objectives. Problems with convex objectives have been extensively studied in recent years where the analyses rely crucially on the convexity and the Fenchel duality. However, problems with non-convex objectives resist against current approaches and non-convexity represents a strong barrier in optimization in general and in the design of online algorithms in particular. In our approach, we consider configuration linear programs with the multilinear extension of the objectives. We follow the multiplicative weight update framework in which a novel point is that the primal update is defined based on the gradient of the multilinear extension. We introduce new notions, namely (local) smoothness, in order to characterize the competitive ratios of our algorithms. The approach leads to competitive algorithms for several problems with convex/non-convex objectives.

We consider a scheduling problem on unrelated machines with precedence constraints. There are m unrelated machines and n jobs and every job has to be processed non-preemptively in some machine. Moreover, jobs have precedence constraints; specifically, a precedence constraint j ≺ j' requires that job j' can only be started whenever job j has been completed. The objective is to minimize the total completion time. The problem has been widely studied in more restricted machine environments such as identical or related machines. However, for unrelated machines, much less is known. In the paper, we study the problem where the precedence constraints form a forest of arborescences. We present a O((log n)² / (log log n)³)-approximation algorithm - that improves the best-known guarantee of O((log n)² / log log n) due to Kumar et al. a decade ago. The analysis relies on a dual-fitting method in analyzing the Lagrangian function of non-convex programs.

We consider online bandit learning in which at every time step, an algorithm has to make a decision and then observe only its reward. The goal is to design efficient (polynomial-time) algorithms that achieve a total reward approximately close to that of the best fixed decision in hindsight. In this paper, we introduce a new notion of $(\lambda,\mu)$-concave functions and present a bandit learning algorithm that achieves a performance guarantee which is characterized as a function of the concavity parameters $\lambda$ and $\mu$. The algorithm is based on the mirror descent algorithm in which the update directions follow the gradient of the multilinear extensions of the reward functions. The regret bound induced by our algorithm is $\widetilde{O}(\sqrt{T})$ which is nearly optimal. We apply our algorithm to auction design, specifically to welfare maximization, revenue maximization, and no-envy learning in auctions. In welfare maximization, we show that a version of fictitious play in smooth auctions guarantees a competitive regret bound which is determined by the smooth parameters. In revenue maximization, we consider the simultaneous second-price auctions with reserve prices in multi-parameter environments. We give a bandit algorithm which achieves the total revenue at least $1/2$ times that of the best fixed reserve prices in hindsight. In no-envy learning, we study the bandit item selection problem where the player valuation is submodular and provide an efficient $1/2$-approximation no-envy algorithm.

We consider the problem of scheduling jobs to minimize the maximum weighted flow-time on a set of related machines. When jobs can be preempted this problem is well-understood; for example, there exists a constant competitive algorithm using speed augmentation. When jobs must be scheduled non-preemptively, only hardness results are known. In this paper, we present the first online guarantees for the non-preemptive variant. We present the first constant competitive algorithm for minimizing the maximum weighted flow-time on related machines by relaxing the problem and assuming that the online algorithm can reject a small fraction of the total weight of jobs. This is essentially the best result possible given the strong lower bounds on the non-preemptive problem without rejection.

The efficiency of a game is typically quantified by the price of anarchy (PoA), defined as the worst ratio of the value of an equilibrium - solution of the game - and that of an optimal outcome. Given the tremendous impact of tools from mathematical programming in the design of algorithms and the similarity of the price of anarchy and different measures such as the approximation and competitive ratios, it is intriguing to develop a duality-based method to characterize the efficiency of games. In the paper, we present an approach based on linear programming duality to study the efficiency of games. We show that the approach provides a general recipe to analyze the efficiency of games and also to derive concepts leading to improvements. The approach is particularly appropriate to bound the PoA. Specifically, in our approach the dual programs naturally lead to competitive PoA bounds that are (almost) optimal for several classes of games. The approach indeed captures the smoothness framework and also some current non-smooth techniques/concepts. We show the applicability to the wide variety of games and environments, from congestion games to Bayesian welfare, from full-information settings to incomplete-information ones.

Non-linear, especially convex, objective functions have been extensively studied in recent years in which approaches relies crucially on the convexity property of cost functions. In this paper, we present primal-dual approaches based on configuration linear programs to design competitive online algorithms for problems with arbitrarily-grown objective. This approach is particularly appropriate for non-linear (non-convex) objectives in online setting. We first present a simple greedy algorithm for a general cost-minimization problem. The competitive ratio of the algorithm is characterized by the mean of a notion, called smoothness, which is inspired by a similar concept in the context of algorithmic game theory. The algorithm gives optimal (up to a constant factor) competitive ratios while applying to different contexts such as network routing, vector scheduling, energy-efficient scheduling and non-convex facility location. Next, we consider the online $0-1$ covering problems with non-convex objective. Building upon the resilient ideas from the primal-dual framework with configuration LPs, we derive a competitive algorithm for these problems. Our result generalizes the online primal-dual algorithm developed recently by Azar et al. for convex objectives with monotone gradients to non-convex objectives. The competitive ratio is now characterized by a new concept, called local smoothness --- a notion inspired by the smoothness. Our algorithm yields tight competitive ratio for the objectives such as the sum of $\ell_{k}$-norms and gives competitive solutions for online problems of submodular minimization and some natural non-convex minimization under covering constraints.

In this paper, we consider the online problem of scheduling independent jobs non-preemptively so as to minimize the weighted flow-time on a set of unrelated machines. There has been a considerable amount of work on this problem in the preemptive setting where several competitive algorithms are known in the classical competitive model. However, the problem in the non-preemptive setting admits a strong lower bound. Recently, Lucarelli et al. presented an algorithm that achieves a O(1/epsilon^2)-competitive ratio when the algorithm is allowed to reject epsilon-fraction of total weight of jobs and has an epsilon-speed augmentation. They further showed that speed augmentation alone is insufficient to derive any competitive algorithm. An intriguing open question is whether there exists a scalable competitive algorithm that rejects a small fraction of total weights. In this paper, we affirmatively answer this question. Specifically, we show that there exists a O(1/epsilon^3)-competitive algorithm for minimizing weighted flow-time on a set of unrelated machine that rejects at most O(epsilon)-fraction of total weight of jobs. The design and analysis of the algorithm is based on the primal-dual technique. Our result asserts that alternative models beyond speed augmentation should be explored when designing online schedulers in the non-preemptive setting in an effort to find provably good algorithms.

When a computer system schedules jobs there is typically a significant cost associated with preempting a job during execution. This cost can be from the expensive task of saving the memory's state and loading data into and out of memory. There is a need for non-preemptive system schedulers to avoid the costs of preemption on desktops, servers and data centers. Despite this need, there is a gap between theory and practice. Indeed, few non-preemptive online schedulers are known to have strong foundational guarantees. This gap is likely due to strong lower bounds on any online algorithm for popular objectives. Indeed, typical worst case analysis approaches, and even resource augmented approaches such as speed augmentation, result in all algorithms having poor performance guarantees. This paper considers online non-preemptive scheduling problems in the worst-case model where the algorithm is allowed to reject a small fraction of jobs. By rejecting only few jobs, this paper shows that the strong lower bounds can be circumvented. This model can be used to discover scheduling policies with desirable worst-case guarantees. Specifically, the paper presents algorithms for minimizing the total flow-time and minimizing the total weighted flow-time plus energy under the speed-scaling mechanism. The algorithms have a small constant competitive ratio while rejecting only a constant fraction of jobs. Beyond specific results, the paper asserts that alternative models beyond speed augmentation should be explored to aid in the discovery of good schedulers in the face of the requirement of being online and non-preemptive.

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4*2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting.

Resource augmentation is a well-established model for analyzing algorithms, particularly in the online setting. It has been successfully used for providing theoretical evidence for several heuristics in scheduling with good performance in practice. According to this model, the algorithm is applied to a more powerful environment than that of the adversary. Several types of resource augmentation for scheduling problems have been proposed up to now, including speed augmentation, machine augmentation and more recently rejection. In this paper, we present a framework that unifies the various types of resource augmentation. Moreover, it allows generalize the notion of resource augmentation for other types of resources. Our framework is based on mathematical programming and it consists of extending the domain of feasible solutions for the algorithm with respect to the domain of the adversary. This, in turn allows the natural concept of duality for mathematical programming to be used as a tool for the analysis of the algorithm's performance. As an illustration of the above ideas, we apply this framework and we propose a primal-dual algorithm for the online scheduling problem of minimizing the total weighted flow time of jobs on unrelated machines when the preemption of jobs is not allowed. This is a well representative problem for which no online algorithm with performance guarantee is known. Specifically, a strong lower bound of Omega(sqrt{n}) exists even for the offline unweighted version of the problem on a single machine. In this paper, we first show a strong negative result even when speed augmentation is used in the online setting. Then, using the generalized framework for resource augmentation and by combining speed augmentation and rejection, we present an (1+epsilon_s)-speed O(1/(epsilon_s epsilon_r))-competitive algorithm if we are allowed to reject jobs whose total weight is an epsilon_r-fraction of the weights of all jobs, for any epsilon_s > 0 and epsilon_r in (0,1). Furthermore, we extend the idea for analysis of the above problem and we propose an (1+\epsilon_s)-speed epsilon_r-rejection O({k^{(k+3)/k}}/{epsilon_{r}^{1/k}*epsilon_{s}^{(k+2)/k}})-competitive algorithm for the more general objective of minimizing the weighted l_k-norm of the flow times of jobs.

Discrete tomography deals with reconstructing finite spatial objects from their projections. The objects we study in this paper are called tilings or tile-packings, and they consist of a number of disjoint copies of a fixed tile, where a tile is defined as a connected set of grid points. A row projection specifies how many grid points are covered by tiles in a given row; column projections are defined analogously. For a fixed tile, is it possible to reconstruct its tilings from their projections in polynomial time? It is known that the answer to this question is affirmative if the tile is a bar (its width or height is 1), while for some other types of tiles $\mathbb {NP}$ -hardness results have been shown in the literature. In this paper we present a complete solution to this question by showing that the problem remains $\mathbb {NP}$ -hard for all tiles other than bars.