Submodular function maximization (SFM) has attracted much attention thanks to its applicability to various practical problems. Although most studies have considered SFM with size or budget constraints, more complex constraints often appear in practice. In this paper, we consider a very general class of SFM with such complex constraints (e.g., an s-t path constraint on a given graph). We propose a novel algorithm that takes advantage of zero-suppressed binary decision diagrams, which store all feasible solutions efficiently thus enabling us to circumvent the difficulty of determining feasibility. Theoretically, our algorithm is guaranteed to achieve (1-c)-approximations, where c is the curvature of a submodular function. Experiments show that our algorithm runs much faster than exact algorithms and finds better solutions than those obtained by an existing approximation algorithm in many instances. Notably, our algorithm achieves better than a 90%-approximation in all instances for which optimal values are available.

This paper considers the multiset selection problem with size constraints, which arises in many real-world applications such as budget allocation. Previous studies required the objective function f to be submodular, while we relax this assumption by introducing the notion of the submodularity ratios (denoted by α_f and β_f). We propose an anytime randomized iterative approach POMS, which maximizes the given objective f and minimizes the multiset size simultaneously. We prove that POMS using a reasonable time achieves an approximation guarantee of max{1-1/e^(β_f), (α_f/2)(1-1/e^(α_f))}. Particularly, when f is submdoular, this bound is at least as good as that of the previous greedy-style algorithms. In addition, we give lower bounds on the submodularity ratio for the objectives of budget allocation. Experimental results on budget allocation as well as a more complex application, namely, generalized influence maximization, exhibit the superior performance of the proposed approach.

We consider the following general graph clustering problem: given a complete undirected graph G=(V,E,c) with an edge weight function c:E->Q, we are asked to find a partition C of V that maximizes the sum of edge weights within the clusters in C. Owing to its high generality, this problem has a wide variety of real-world applications, including correlation clustering, group technology, and community detection. In this study, we investigate the design of mathematical programming formulations and constraint satisfaction formulations for the problem. First, we present a novel integer linear programming (ILP) formulation that has far fewer constraints than the standard ILP formulation by Groetschel and Wakabayashi (1989). Second, we propose an ILP-based exact algorithm that solves an ILP problem obtained by modifying our above ILP formulation and then performs simple post-processing to produce an optimal solution to the original problem. Third, we present maximum satisfiability (MaxSAT) counterparts of both our ILP formulation and ILP-based exact algorithm. Computational experiments using well-known real-world datasets demonstrate that our ILP-based approaches and their MaxSAT counterparts are highly effective in terms of both memory efficiency and computation time.

The need for real time analysis of rapidly producing data streams (e.g., video and image streams) motivated the design of streaming algorithms that can efficiently extract and summarize useful information from massive data "on the fly." Such problems can often be reduced to maximizing a submodular set function subject to various constraints. While efficient streaming methods have been recently developed for monotone submodular maximization, in a wide range of applications, such as video summarization, the underlying utility function is non-monotone, and there are often various constraints imposed on the optimization problem to consider privacy or personalization. We develop the first efficient single pass streaming algorithm, Streaming Local Search, that for any streaming monotone submodular maximization algorithm with approximation guarantee α under a collection of independence systems I, provides a constant 1/(1+2/√α+1/α+2d(1+√α)) approximation guarantee for maximizing a non-monotone submodular function under the intersection of I and d knapsack constraints. Our experiments show that for video summarization, our method runs more than 1700 times faster than previous work, while maintaining practically the same performance.

We study large-scale machine learning problems in changing environments where a small part of the dataset is modified, and the effect of the data modification must be monitored in order to know how much the modification changes the optimal model. When the entire dataset is large, even if the amount of the data modification is fairly small, the computational cost for re-training the model would be prohibitively large. In this paper, we propose a novel method, called the optimal solution bounding (OSB), for monitoring such a data modification effect on the optimal model by efficiently evaluating (without actually re-training) it. The proposed method provides bounds on the unknown optimal model with the cost proportional only to the size of the data modification.

We define a class of zero-sum games with combinatorial structure, where the best response problem of one player is to maximize a submodular function. For example, this class includes security games played on networks, as well as the problem of robustly optimizing a submodular function over the worst case from a set of scenarios. The challenge in computing equilibria is that both players' strategy spaces can be exponentially large. Accordingly, previous algorithms have worst-case exponential runtime and indeed fail to scale up on practical instances. We provide a pseudopolynomial-time algorithm which obtains a guaranteed (1 - 1/e)^2-approximate mixed strategy for the maximizing player. Our algorithm only requires access to a weakened version of a best response oracle for the minimizing player which runs in polynomial time. Experimental results for network security games and a robust budget allocation problem confirm that our algorithm delivers near-optimal solutions and scales to much larger instances than was previously possible.

The winner determination problems of many attractive multi-winner voting rules are NP-complete. However, they often admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic programming along the underlying axis. We introduce a new technique: carefully chosen integer linear programming (IP) formulations for certain voting problems admit an LP relaxation which is totally unimodular if preferences are single-peaked, and which thus admits an integral optimal solution. This technique gives efficient algorithms for finding optimal committees under Proportional Approval Voting (PAV) and the Chamberlin-Courant rule with single-peaked preferences, as well as for certain OWA-based rules. For PAV, this is the first technique able to efficiently find an optimal committee when preferences are single-peaked. An advantage of our approach is that no special-purpose algorithm needs to be used to exploit structure in the input preferences: any standard IP solver will terminate in the first iteration if the input is single-peaked, and will continue to work otherwise.

We propose a novel mechanism for solving the assignment problem when we have a two sided matching problem with preferences from one side (the agents/reviewers) over the other side (the objects/papers) and both sides have capacity constraints. The assignment problem is a fundamental in both computer science and economics with application in many areas including task and resource allocation. Drawing inspiration from work in multi-criteria decision making and social choice theory we use order weighted averages (OWAs), a parameterized class of mean aggregators, to propose a novel and flexible class of algorithms for the assignment problem. We show an algorithm for finding an SUM-OWA assignment in polynomial time, in contrast to the NP-hardness of finding an egalitarian assignment. We demonstrate through empirical experiments that using SUM-OWA assignments can lead to high quality and more fair assignments.

The classic bribery problem is to find a minimal subset of voters who need to change their vote to make some preferred candidate win.We find an approximate solution for this problem for a broad family of scoring rules (which includes Borda and t-approval), in the following sense: if there is a strategy which requires bribing k voters, we efficiently find a strategy which requires bribing at most k + Õ(√ k ) voters. Our algorithm is based on a randomized reduction from bribery to coalitional manipulation (UCM). To solve the UCM problem, we apply the Birkhoff-von Neumann (BvN) decomposition to a fractional manipulation matrix. This allows us to limit the size of the possible ballot search space reducing it from exponential to polynomial, while still obtaining good approximation guarantees. Finding the optimal solution in the truncated search space yields a new algorithm for UCM, which is of independent interest.

Voting is almost never done in void, as usually there are some relations between the alternatives on which the voters vote on. These relations shall be taken into consideration when selecting a winning committee of some given multiwinner election. As taking into account all possible relations between the alternatives is generally computationally intractable, in this paper we consider classes of alternatives; intuitively, the number of classes is significantly smaller than the number of alternatives, and thus there is some hope in reaching computational tractability. We model both intraclass relations and interclass relations by functions, which we refer to as synergy functions, and study the computational complexity of identifying the best committee, taking into account those synergy functions. Our model accommodates both positive and negative relations between alternatives; further, our efficient algorithms can also deal with a rich class of diversity wishes, which we show how to model using synergy functions.