Monotone Submodular Set Function Research Articles

Output–Input Ratio Maximization for Online Social Networks: Algorithms and Analyses

In the last few decades, profit maximization (PM), which is mainly considered for maximizing net profits, i.e., the difference between gain and cost, has been a prominent issue for online social networks (OSNs). However, the output-to-input ratio, which is an important metric in economics, is also worth studying for OSNs. In this article, we present a novel problem that considers the PM problem from the ratio of gain and cost, known as output-to-input ratio maximization (OIRM). Unfortunately, it is neither submodular nor supermodular. The hill-climbing greedy algorithm for solving this problem is a <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex>1-e^-(1-c_g)</tex> </formula> approximation algorithm, where c_g is the curvature of the monotone submodular set function g. To speed up the hill-climbing greedy algorithm, we propose the threshold decrease algorithm and prove that its approximation ratio is <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex>1-e^-(1-c_g)^2-ε</tex> </formula> . In addition, based on the relationship between classical net PM and OIRM, the algorithms for solving PM can also solve OIRM. Finally, we evaluate the performance of our algorithms using massive experiments on real datasets. To the best of our knowledge, this is the first time to study the OIRM in viral marketing.

A new performance bound for submodular maximization problems and its application to multi-agent optimal coverage problems

Several important problems in multi-agent systems, machine learning, data mining, scheduling and others, may be formulated as set function maximization problems subject to cardinality constraints. In such problems, the set (objective) functions of interest often have monotonicity and submodularity properties. Hence, the class of monotone submodular set function maximization problems has been widely studied in the literature. Owing to its challenging nature, almost all existing solutions for this class of problems are based on greedy algorithms. A seminal work on this topic has exploited the submodularity property to prove a (1-1/e) performance bound for such greedy solutions. More recent literature on this topic has been focused on exploiting different curvature properties to establish improved (tighter) performance bounds. However, such improvements come at the cost of enforcing additional assumptions and increasing computational complexity while facing significant inherent limitations. In this paper, first, a brief review of existing performance bounds is provided. Then, a new performance bound that does not require any additional assumptions and is both practical and computationally inexpensive is proposed. In particular, this new performance bound is established based on a series of upper bounds derived for the objective function that can be computed in parallel with the execution of the greedy algorithm. Finally, to highlight the effectiveness of the proposed performance bound, extensive numerical results obtained from a well-known class of multi-agent coverage problems are provided.

Deterministic approximation algorithm for submodular maximization subject to a matroid constraint

In this paper, we study the generalized submodular maximization problem with a non-negative monotone submodular set function as the objective function and subject to a matroid constraint. The problem is generalized through the curvature parameter α∈[0,1] which measures how far a set function deviates from linearity to submodularity. We propose a deterministic approximation algorithm which uses the approximation algorithm proposed by Buchbinder et al. [2] as a building block and inherits the approximation guarantee for α=1. For general value of the curvature parameter α∈[0,1], we present an approximation algorithm with a factor of 1+hα(y)+Δ⋅[3+α−(2+α)y−(1+α)hα(y)]2+α+(1+α)(1−y), where y∈[0,1] is a predefined parameter for tuning the ratio. In particular, when α=1 we obtain a ratio 0.5008 when setting y=0.9, coinciding with the renowned state-of-art approximate ratio; when α=0 that the object is a linear function, the approximation factor equals one and our algorithm is indeed an exact algorithm that always produces optimum solutions.

Novel algorithms for maximum DS decomposition

DS decomposition plays an important role in set function optimization problem, because there is DS decomposition for any set function. How to design an efficient and effective algorithm to solve maximizing DS decomposition is a heated problem. In this work, we propose a framework called Parameter Conditioned Greedy Algorithm which has a deterministic version and two random versions. In more detail, this framework uses the difference with parameter decomposition function and combines non-negative condition. Besides, if we set the different parameters, the framework can return solution with different approximation ratio. Also, we choose two special case to show our deterministic algorithm gets f(Sk)−(e−1−cg)g(Sk)≥(1−e−1)[f(OPT)−g(OPT)] and f(Sk)−(1−cg)g(Sk)≥(1−e−1)f(OPT)−g(OPT) respectively for cardinality constrained problem, where cg is the curvature of monotone submodular set function. To speed the deterministic algorithm, we introduce a random sample set whose intersection with the optimal solution is as nonempty as possible. Importantly, it also can get the same approximation ratio as deterministic algorithm under expectation. Further, for maximization DS decomposition without constraint, our another random algorithm gets E[f(Sk)−(e−1−cg)g(Sk)]≥(1−e−1)[f(OPT)−g(OPT)] and E[f(Sk)−(1−cg)g(Sk)]≥(1−e−1)f(OPT)−g(OPT) respectively. Because the Parameter Conditioned Algorithm is the general framework, different users can choose the parameters that fit their problem to get a better approximation.

Multi-Objective Multi-Agent Planning for Jointly Discovering and Tracking Mobile Objects

We consider the challenging problem of online planning for a team of agents to autonomously search and track a time-varying number of mobile objects under the practical constraint of detection range limited onboard sensors. A standard POMDP with a value function that either encourages discovery or accurate tracking of mobile objects is inadequate to simultaneously meet the conflicting goals of searching for undiscovered mobile objects whilst keeping track of discovered objects. The planning problem is further complicated by misdetections or false detections of objects caused by range limited sensors and noise inherent to sensor measurements. We formulate a novel multi-objective POMDP based on information theoretic criteria, and an online multi-object tracking filter for the problem. Since controlling multi-agent is a well known combinatorial optimization problem, assigning control actions to agents necessitates a greedy algorithm. We prove that our proposed multi-objective value function is a monotone submodular set function; consequently, the greedy algorithm can achieve a (1-1/e) approximation for maximizing the submodular multi-objective function.

Approximation algorithms for connected maximum coverage problem for the discovery of mutated driver pathways in cancer

This paper addresses the connected maximum coverage problem, motivated by the detection of mutated driver pathways in cancer. The connected maximum coverage problem is NP-hard and therefore approximation algorithms are of interest. We provide here an approximation algorithm for the problem with an approximation bound that strictly improves on previous results. A second approximation algorithm with faster run time, though worse approximation factor, is presented as well. The two algorithms are then applied to submodular maximization over a connected subgraph, with a monotone submodular set function, delivering the same approximation bounds as for the coverage maximization case.

Optimization with Uniform Size Queries

Consider the problem of selecting k items that maximize the value of a monotone submodular set function f, where f can be accessed using value queries. It is well known that polynomially many queries suffice in order to obtain an approximation ratio of $$1 - \frac{1}{e}$$1-1e. We consider a variation on this problem in which the value queries are required to be of size: each queried set, like the desired solution itself, must contain k items. We show that polynomially many queries suffice in order to obtain an approximation ratio of $$\frac{1}{2}$$12, and that an approximation ratio of $$\frac{1 + \epsilon }{2}$$1+∈2 requires a number of queries that is exponential in $$\epsilon k$$∈k. For the interesting special case of coverage functions, we show that an approximation ratio strictly better than $$\frac{1}{2}$$12 is attainable with polynomially many queries. The uniform size requirement is motivated by situations in which a firm may offer a menu of exactly k items to its clients, where k is a parameter determined by external considerations. Performing a query corresponds to physically changing the identities of the items offered, and the reply to the query is deduced by observing the behavior of clients in response to the change. Queries that involve a number of items that differs from k may not be desirable due to these external considerations. In such situations it is natural to ask whether the same approximation ratios that can be guaranteed by general value queries can also be obtained by queries.

Maximizing a Monotone Submodular Function Subject to a Matroid Constraint

Let $f:2^X \rightarrow \cal R_+$ be a monotone submodular set function, and let $(X,\cal I)$ be a matroid. We consider the problem ${\rm max}_{S \in \cal I} f(S)$. It is known that the greedy algorithm yields a $1/2$-approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73–87] for this problem. For certain special cases, e.g., ${\rm max}_{|S| \leq k} f(S)$, the greedy algorithm yields a $(1-1/e)$-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning $f(S)$ for a given set S) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177–188] and for explicitly posed instances assuming $P \neq NP$ [U. Feige, J. ACM, 45 (1998), pp. 634–652]. In this paper, we provide a randomized $(1-1/e)$-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307–328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrák, Proceedings of the $38$th ACM Symposium on Theory of Computing, 2008, pp. 67–74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires $|X|$ to be exponential in the original problem size, we are able to achieve a $(1-1/e-o(1))$-approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.