Approximation Guarantees for Parallelized Maximization of Monotone Non-submodular Function with a Cardinality Constraint

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+є)-approximate solution under a cardinality constraint, for any constant є>0, must have an amortized query complexity that is 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 [LMN+20, Mon20] that achieve (0.5−є)-approximation with a polylog(n) amortized query complexity.

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.

A number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem. These results hold typically in the value oracle model, where the objective function is accessible via a black box returning $f(S)$ for a given $S$. 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 $(\frac12+\epsilon)$-approximation for unconstrained (nonmonotone) submodular maximization. As a new applica...

Generic submodularity ratio \(\gamma \) is a general measurement to characterize how close a nonnegative monotone set function is to be submodular. In this paper, we make a systematic analysis of greedy algorithms for maximizing a monotone and normalized set function with a generic submodularity ratio \(\gamma \) under Cardinality constraints, Knapsack constraints, Matroid constraints and K-intersection constraints.

Submodular optimization generalizes many classic problems in combinatorial\noptimization and has recently found a wide range of applications in machine\nlearning (e.g., feature engineering and active learning). For many large-scale\noptimization problems, we are often concerned with the adaptivity complexity of\nan algorithm, which quantifies the number of sequential rounds where\npolynomially-many independent function evaluations can be executed in parallel.\nWhile low adaptivity is ideal, it is not sufficient for a distributed algorithm\nto be efficient, since in many practical applications of submodular\noptimization the number of function evaluations becomes prohibitively\nexpensive. Motivated by these applications, we study the adaptivity and query\ncomplexity of adaptive submodular optimization.\n Our main result is a distributed algorithm for maximizing a monotone\nsubmodular function with cardinality constraint $k$ that achieves a\n$(1-1/e-\\varepsilon)$-approximation in expectation. This algorithm runs in\n$O(\\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation\noracle in expectation. The approximation guarantee and query complexity are\noptimal, and the adaptivity is nearly optimal. Moreover, the number of queries\nis substantially less than in previous works. Last, we extend our results to\nthe submodular cover problem to demonstrate the generality of our algorithm and\ntechniques.\n

In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most submodular functions are specified in a set function. Meanwhile, recent advancements have been studied for maximizing a diminishing return submodular (DR-submodular) function on the integer lattice. Because plenty of publications show that the DR-submodular function has wide applications in optimization problems such as sensor placement impose problems, optimal budget allocation, social network, and especially machine learning. In this research, we propose two main streaming algorithms for the problem of maximizing a monotone DR-submodular function under cardinality constraints. Our two algorithms, which are called StrDRS1 and StrDRS2, have (1/2−ϵ), (1−1/e−ϵ) of approximation ratios and O(nϵlog(logBϵ)logk), O(nϵlogB), respectively. We conducted several experiments to investigate the performance of our algorithms based on the budget allocation problem over the bipartite influence model, an instance of the monotone submodular function maximization problem over the integer lattice. The experimental results indicate that our proposed algorithms not only provide solutions with a high value of the objective function, but also outperform the state-of-the-art algorithms in terms of both the number of queries and the running time.

Clique partitioning with value-monotone submodular cost

We consider the problem of maximizing monotone submodular function over the bounded integer lattice with a cardinality constraint. Function [Formula: see text] is submodular over integer lattice if [Formula: see text], [Formula: see text], where ∨ and ∧ represent elementwise maximum and minimum, respectively. Let [Formula: see text], and [Formula: see text], we study the problem of maximizing submodular function [Formula: see text] with constraints [Formula: see text] and [Formula: see text]. A random greedy [Formula: see text]-approximation algorithm and a deterministic [Formula: see text]-approximation algorithm are proposed in this paper. Both algorithms work in value oracle model. In the random greedy algorithm, we assume the monotone submodular function satisfies diminishing return property, which is not an equivalent definition of submodularity on integer lattice. Additionally, our random greedy algorithm makes [Formula: see text] value oracle queries and deterministic algorithm makes [Formula: see text] value oracle queries.

We consider the algorithmic task of computing a maximal autarky for a clause-set F, i.e., a partial assignment which satisfies every clause of F it touches, and where this property is destroyed by adding any non-empty set of further assignments. We employ SAT solvers as oracles here, and we are especially concerned with minimising the number of oracle calls. Using the standard SAT oracle, $$\log _2(n(F))$$ oracle calls suffice, where n(F) is the number of variables, but the drawback is that (translated) cardinality constraints are employed, which makes this approach less efficient in practice. Using an extended SAT oracle, motivated by the capabilities of modern SAT solvers, we show how to compute maximal autarkies with $$2 \sqrt{n(F)}$$ simpler oracle calls, by a novel algorithm, which combines the previous two main approaches, based on the autarky-resolution duality and on SAT translations.

Order continuous of monotone set function and convergence of measurable functions sequence

We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, which is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations, a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the article is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.

A metacurl antenna made of C-type metaatoms (forming a C-type metaline), called the C-metacurl antenna, is investigated. The investigation reveals that the gain for a left-handed circularly polarized wave, LHCP gain, has a maximum value at frequency f_LH and the gain for a right-handed circularly polarized wave, RHCP gain, has a maximum value at different frequency f_RH, where the maximum LHCP gain is smaller than the maximum RHCP gain. Subsequently, a different metaatom that forms the unit cell of an N-type metaline, called the N-type metaatom, is investigated. It is found that the radiation from the N-type metaatom is LHCP. This infers that the maximum LHCP gain for the C-metacurl antenna will increase when some C-type metaatoms are replaced by N-type metaatoms. Based on this inference, a novel metacurl antenna composed of C-type metaatoms and N-type metaatoms, called the compound metacurl antenna, is analyzed. The analysis reveals that replacement with an appropriate number of N-type metaatoms increases the maximum LHCP gain, resulting in a balancing of LHCP gain and RHCP gain. The antenna characteristics under such a gain-balanced situation, including the radiation pattern, axial ratio, and input characteristic (VSWR), are also discussed.

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a 1/2-approximation, and we also present an appropriate extension to the nonmonotone setting. Moreover, we describe a very general local search procedure that applies to a wide range of constraint families and unifies as well as extends previous methods. In our framework, we match known approximation guarantees while disentangling and simplifying previous approaches. Moreover, despite its generality, we are able to show that our local search procedure is slightly faster than previous specialized methods. Furthermore, we negatively answer the open question whether a linear optimization oracle may be enough to obtain strong approximation algorithms for submodular maximization. We do this by providing an example of a constraint family on a ground set of size n for which, if only given a linear optimization oracle, any algorithm for submodular maximization with a polynomial number of calls to the linear optimization oracle has an approximation ratio of only [Formula: see text].

Quasi-submodularity is a unified measurement to characterize submodularity or approximate submodularity of a set function. In this paper, we consider a variant of Minimum Cardinality Cover named Minimum Cardinality Multiple Quasi-submodular Cover, which requires to find the smallest subset of given ground set such that multiple quasi-submodular functions reach a maximum. We design and analyze a greedy approximation algorithm for Minimum Cardinality Multiple Quasi-submodular Cover. As some applications of our results, we achieve a O(ln n)-approximation algorithm for Minimum Resolving Restrained Dominating Set, and a O(ln delta)-approximation algorithm for Minimum Dominating Set in multiplex networks, where $n$ is the node number and delta is the maximum node degree of the input graph.