Maximizing Submodular Set Function With Connectivity Constraint: Theory and Application to Networks

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.

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.

$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\{0, 1, 2, \dots, k\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \{1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, determine whether $f$ is extended to a $k$-submodular function $g : \{0, 1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, where $g$ is called a $k$-submodular relaxation of $f$. We give a polymorphic characterization of those functions which admit a $k$-submodular relaxation, and also give a combinatorial $O((k^n)^2)$-time algorithm to find a $k$-submodular relaxation or establish that a $k$-submodular relaxation does not exist. Our algorithm has interesting properties: (1) If the input function is integer valued, then our algorithm outputs a half-integral relaxation, and (2) if the input function is binary, then our algorithm outputs the unique optimal relaxation. We present applications of our algorithm to valued constraint satisfaction 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.

Submodular maximization has wide applications in machine learning and data mining, where massive datasets have brought the great need for designing efficient and parallelizable algorithms. One measure of the parallelizability of a submodular maximization algorithm is its adaptivity complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an (8+epsilon)-approximation under O(log n) adaptive complexity, which is optimal up to a factor of O(loglog n). Moreover, under slightly larger adaptivity, we also propose approximation algorithms with nearly optimal query complexity of O(n), while achieving better approximation ratios. We show that our algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.

The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c: V -> N, and a weight function w: A -> R^+, a carpool matching is a subset of arcs, M subseteq A, such that every v in V satisfies: (i) d^{in}_M(v) cdot d^{out}_M(v) = 0, (ii) d^{in}_M(v) <= c(v), and (iii) d^{out}_M(v) <= 1. A vertex v for which d^{out}_M(v) = 1 is a passenger, and a vertex for which d^{out}_M(v) = 0 is a driver who has d^{in}_M(v) passengers. In the Maximum Carpool Matching problem the goal is to find a carpool matching M of maximum total weight. The problem arises when designing an online carpool service, such as Zimride, which tries to connect between users based on a similarity function. The problem is known to be NP-hard, even in the unweighted and uncapacitated case. The Maximum Group Carpool Matching problem, is an extension of Maximum Carpool Matching where each vertex represents an unsplittable group of passengers. Formally, each vertex u in V has a size s(u) in N, and the constraint d^{in}_M(v) <= c(v) is replaced with sum_{u:(u,v) in M} s(u) <= c(v). We show that Maximum Carpool Matching can be formulated as an unconstrained submodular maximization problem, thus it admits a 1/2-approximation algorithm. We show that the same formulation does not work for Maximum Group Carpool Matching, nevertheless, we present a local search (1/2 - epsilon)-approximation algorithm for Maximum Group Carpool Matching. For the unweighted variant of both problems when the maximum possible capacity, c_{max}, is bounded by a constant, we provide a local search (1/2 + 1/{2c_{max}} - epsilon)-approximation algorithm. We also show that the problem is APX-hard, even if the maximum degree and c_{max} are at most 3.

Machine learning can provide powerful tools for the design and analysis of multi-agent systems, as well as a novel lens on important questions in the area. In this talk I will focus on the use of machine learning for understanding submodular functions, an important class of discrete functions that model laws of diminishing returns and enjoy many important applications in multi-agent settings. For example, submodular functions are commonly used to model valuation functions for bidders in auctions, the influence of various subsets of agents in social networks, and the benefits of performing different actions in a variety of situations. Traditionally it is assumed that these functions are known to the decision maker; however, for large scale systems, in the age of big data, it is often the case they must be learned from observations. In this talk, I will discuss a recent line of work on studying the learnability of submodular functions, and highlight its applications to the analysis of multi-agent systems. I will discuss both general algorithms for learning such functions, as well as even better guarantees that can be achieved for important classes appearing in multi-agent scenarios that exhibit additional structure. These classes include: probabilistic coverage functions that can be used to model the influence function in classic models of information diffusion in networks; functions with bounded complexity used in modeling bidder valuation functions in auctions, including XOS and gross-substitutes; and classes of functions appearing in cooperative game theory for expressing the values of various types of coalitions. I will additionally discuss a large scale application of our algorithms for learning the influence functions in social networks, that significantly outperforms existing approaches empirically in both synthetic and real world data.

Deterministic approximation algorithm for submodular maximization subject to a matroid constraint

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.

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].

Submodular maximization has attracted extensive attention due to its numerous applications in machine learning and artificial intelligence. Many real-world problems require maximizing multiple submodular objective functions at the same time. In such cases, a common approach is to select a representative subset of Pareto optimal solutions with different trade-offs among multiple objectives. To this end, in this paper, we investigate the regret ratio minimization (RRM) problem in multi-objective submodular maximization, which aims to find at most k solutions to best approximate all Pareto optimal solutions w.r.t. any linear combination of objective functions. We propose a novel HS-RRM algorithm by transforming RRM into HittingSet problems based on the notions of ε-kernel and δ-net, where any α-approximation algorithm for single-objective submodular maximization is used as an oracle. We improve upon the previous best-known bound on the maximum regret ratio (MRR) of the output of HS-RRM and show that the new bound is nearly asymptotically optimal for any fixed number d of objective functions. Experiments on real-world and synthetic data confirm that HS-RRM achieves lower MRRs than existing algorithms.

Region search is an important problem in location-based services due to its wide applications. In this paper, we study the problem of optimal region search with submodular maximization (ORS-SM). This problem considers a region as a connected subgraph. We compute an objective value over the locations in the region using a submodular function and a budget value by summing up the costs of edges in the region, and aim to search the region with the largest objective score under a budget value constraint. ORS-SM supports many applications such as the most diversified region search. We prove that the problem is NP-hard and develop two approximation algorithms with guaranteed error bounds. We conduct experiments on two applications using three real-world datasets. The results demonstrate that our algorithms can achieve high-quality solutions and are faster than a state-of-the-art method by orders of magnitude.

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search &amp; Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001].

Sensor networks consist of many small sensing devices that monitor an environment and communicate using wireless links. The lifetime of these networks is severely curtailed by the limited battery power of the sensors. One line of research in sensor network lifetime management has examined sensor selection techniques, in which applications judiciously choose which sensors' data should be retrieved and are worth the expended energy. In the past, many ad-hoc approaches for sensor selection have been proposed. In this paper, we argue that sensor selection should be based upon a tradeoff between application-perceived benefit and energy consumption of the selected sensor set. We propose a framework wherein the application can specify the utility of measuring data (nearly) concurrently at each set of sensors. The goal is then to select a sequence of sets to measure whose total utility is maximized, while not exceeding the available energy. Alternatively, we may look for the most cost-effective sensor set, maximizing the product of utility and system lifetime. This approach is very generic, and permits us to model many applications of sensor networks. We proceed to study two important classes of utility functions: submodular and supermodular functions. We show that the optimum solution for submodular functions can be found in polynomial time, while optimizing the cost-effectiveness of supermodular functions is NP-hard. For a practically important subclass of supermodular functions, we present an LP-based solution if nodes can send for different amounts of time, and show that we can achieve an O(log n) approximation ratio if each node has to send for the same amount of time. Finally, we study scenarios in which the quality of measurements is naturally expressed in terms of distances from targets. We show that the utility-based approach is analogous to a penalty-based approach in those scenarios, and present preliminary results on some practically important special cases

Promotion of cooperation induced by discriminators in the spatial multi-player donor–recipient game