Submodular Maximization Through the Lens of Linear Programming

In the paper, we propose a deterministic approximation algorithm for maximizing a generalized monotone submodular function subject to a matroid constraint. The function is generalized through a curvature parameter \(c\in [0,1]\), and essentially reduces to a submodular function when \(c=1\). Our algorithm employs the deterministic approximation devised by Buchbinder et al. [3] for the \(c=1\) case of the problem as a building block, and eventually attains an approximation ratio of \(\frac{1+g_c(x)+\Delta \cdot \left[ 3+c-(2+c)x-(1+c)g_c(x)\right] }{2+c+(1+c)(1-x)}\) for the curvature parameter \(c\in [0,1]\) and for a calibrating parameter that is any \(x\in [0,1]\). For \(c=1\), the ratio attains 0.5008 by setting \(x=0.9\), coinciding with the renowned performance guarantee of the problem. Moreover, when the submodular set function degenerates to a linear function, our generalized algorithm always produces optimum solutions and thus achieves an approximation ratio 1.

Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (text {MSR}). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the text {MSR} problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.

It is of great importance to design streaming algorithms for submodular maximization, as many applications (e.g., crowdsourcing) have large volume of data satisfying the well-known ''diminishing returns'' property, which cannot be handled by offline algorithms requiring full access to the whole dataset. However, streaming submodular maximization has been less studied than the offline algorithms due to the hardness brought by more stringent requirements on memory consumption. In this paper, we consider the fundamental problem of Submodular Maximization under k -System and d -Knapsack constraints (SMSK), which has only been successfully addressed by offline algorithms in previous studies, and we propose the first streaming algorithm for it with provable performance bounds. Our approach adopts a novel algorithmic framework dubbed MultiplexGreedy , making it also perform well under a single k -system constraint. For the special case of SMSK with only d -knapsack constraints, we further propose a streaming algorithm with better performance ratios than the state-of-the-art algorithms. As the SMSK problem generalizes most of the major problems studied in submodular maximization, our algorithms have wide applications in big data processing.

MEMPSODE: A global optimization software based on hybridization of population-based algorithms and local searches

A modified quasisecant method for global optimization

Landing aircraft safely is an important operation that air traffic controllers have to deal with on a daily basis. For each arriving aircraft a runway and a landing time must be allocated. If these allocations can be done in an efficient way, it could give the airport a competitive advantage. The Aircraft Landing Problem (ALP) aims to minimize the deviation from a preferred target time of each aircraft. It is an NP-hard problem, meaning that we may have to resort to heuristic methods as exact methods may not be suitable, especially as the problem size increases. This paper proposes an iterated local search (ILS) algorithm for the ALP. ILS is a single solution based search methodology that successively invokes a local search procedure to find a local optimum solution. A perturbation operator is used to modify the current solution in order to escape from the local optimum and to provide a new solution for the local search procedure. As different problems and/or instances have different characteristics, the success of the ILS is highly dependent on the local search, the perturbation operator(s) and the perturbation strength. To address these issues, we utilize four perturbation operators and a time varying perturbation strength which changes as the algorithm progresses. A variable neighborhood descent algorithm is used as our local search. The proposed ILS generates high quality solutions for the ALP benchmark instances taken from the scientific literature, demonstrating its efficiency in terms of both solution quality and computational time. Moreover, the proposed ILS produces new best results for some instances.

Due to the pervasive "diminishing returns" property appeared in data-intensive applications, submodular maximization problems have aroused great attention from both the machine learning community and the computation theory community. During the last decades, a lot of algorithms have been proposed for submodular maximization subject to various constraints [4, 6, 8], and these algorithms can be used in numerous applications including sensor placement [9], clustering [5], network design [13], and so on.

Analyzing Residual Random Greedy for monotone submodular maximization

The local search procedure is a method for hybridization and improvement of the main algorithm, when complex problems are solved. It helps to avoid local optimums and to find faster the global one. The theory of intuitionistic fuzzy logic, which is the basis of InterCriteria analysis (ICrA), is used to study the proposed hybrid algorithms for ant colony optimization (ACO). Different algorithms for ICrA implementation are applied on the results obtained by hybrid ACO algorithms for Multiple Knapsack Problem. The hybrid algorithms behavior is compared to the traditional ACO algorithm. Based on the obtained numerical results from the algorithms performance and from the ICrA the efficiency and effectiveness of the proposed hybrid ACO algorithms, combined with appropriate local search procedure, are confirmed.KeywordsLocal searchAnt colony optimizationHybrid algorithmIntercriteria analysis algorithmsKnapsack problemIndex matrixIntuitionistic fuzzy sets

A complete and efficient algorithm is proposed for searching form-closure grasps of n hard fingers on the surface of a three-dimensional object represented by discrete points. Both frictional and frictionless cases are considered. This algorithm starts to search a form-closure grasp from a randomly selected grasp using an efficient local search procedure until encountering a local minimum. The local search procedure employs the powerful ray-shooting technique to search in the direction of reducing the distance between the convex hull corresponding to the grasp and the origin of the wrench space. When the distance reaches a local minimum in the local search procedure, the algorithm decomposes the problem into a few subproblems in subsets of the points according to the existence conditions of form-closure grasps. A search tree whose root represents the original problem is employed to perform the searching process. The subproblems are represented as children of the root node and the same procedure is recursively applied to the children. It is proved that the search tree generates O(KlnK/n) nodes in case a from-closure grasp exists, where K is the number of the local minimum points of the distance in the grasp space and n is the number of fingers. Compared to the exhaustive search, this algorithm is more efficient, and, compared to other heuristic algorithms, the proposed algorithm is complete in the discrete domain. The efficiency of this algorithm is demonstrated by numerical examples.

Performing a local search around solutions found by an evolutionary algorithm is a common practice. Local search is well known to significantly improve the solutions, in particular in the case of combinatorial problems. In this paper a new local search procedure is proposed that uses a locally established utopia point. In the tests in which several instances of the Travelling Salesman Problem (TSP) were solved using an evolutionary algorithm the proposed local search procedure outperformed a local search procedure based on Pareto dominance. Because the local search is focused on improving individual solutions and the multiobjective evolutionary algorithm can improve diversity, various strategies of sharing computational resources between the evolutionary algorithm and the local search are used in this paper. The results attained by the tested methods are compared with respect to computation time, which allows a fair comparison between strategies that distribute computational resources between the evolutionary optimization and the local search in various proportions.

Artificial Neural Networks (ANNs) have been used in a multitude of real-world applications given their predictive capabilities, and algorithms based on gradient descent, such as Backpropagation (BP) and variants, are usually considered for their optimisation. However, these algorithms have been shown to get stuck at local optima, and they require a cautious design of the architecture of the model. This paper proposes a novel memetic training method for simultaneously learning the ANNs structure and weights based on the Coral Reef Optimisation algorithms (CROs), a global-search metaheuristic based on corals’ biology and coral reef formation. Three versions based on the original CRO combined with a Local Search procedure are developed: (1) the basic one, called Memetic CRO; (2) a statistically guided version called Memetic SCRO (M-SCRO) that adjusts the algorithm parameters based on the population fitness; (3) and, finally, an improved Dynamic Statistically-driven version called Memetic Dynamic SCRO (M-DSCRO). M-DSCRO is designed with the idea of improving the M-SCRO version in the evolutionary process, evaluating whether the fitness distribution of the population of ANNs is normal to automatically decide the statistic to be used for assigning the algorithm parameters. Furthermore, all algorithms are adapted to the design of ANNs by means of the most suitable operators. The performance of the different algorithms is evaluated with 40 classification datasets, showing that the proposed M-DSCRO algorithm outperforms the other two versions on most of the datasets. In the final analysis, M-DSCRO is compared against four state-of-the-art methods, demonstrating its superior efficacy in terms of overall accuracy and minority class performance.

An efficient approach for type II robotic assembly line balancing problems

A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1/e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1/2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1/2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1/4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1/6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k/(k-1), such that any algorithm achieving a better than (1-1/nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1/2 (1-1/nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1/2 (for submodular maximization subject to a matroid constraint) and 1-1/nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal.

The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible, and O(1)-approximation mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only O(1)-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). The fact that in our mechanism the agents are not ordered according to their marginal value per cost allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present, for example, at most k agents can be selected. We obtain O(p)-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a p-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about nontrivial approximation guarantees in polynomial time, our results are, asymptotically, the best possible.