Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity

We introduce two new "degree of complementarity" measures: supermodular width and superadditive width. Both are formulated based on natural witnesses of complementarity. We show that both measures are robust by proving that they, respectively, characterize the gap of monotone set functions from being submodular and subadditive. Thus, they define two new hierarchies over monotone set functions, which we will refer to as Supermodular Width (SMW) hierarchy and Superadditive Width (SAW) hierarchy, with foundations - i.e. level 0 of the hierarchies - resting exactly on submodular and subadditive functions, respectively. We present a comprehensive comparative analysis of the SMW hierarchy and the Supermodular Degree (SD) hierarchy, defined by Feige and Izsak. We prove that the SMW hierarchy is strictly more expressive than the SD hierarchy: Every monotone set function of supermodular degree d has supermodular width at most d, and there exists a supermodular-width-1 function over a ground set of m elements whose supermodular degree is m-1. We show that previous results regarding approximation guarantees for welfare and constrained maximization as well as regarding the Price of Anarchy (PoA) of simple auctions can be extended without any loss from the supermodular degree to the supermodular width. We also establish almost matching information-theoretical lower bounds for these two well-studied fundamental maximization problems over set functions. The combination of these approximation and hardness results illustrate that the SMW hierarchy provides not only a natural notion of complementarity, but also an accurate characterization of "near submodularity" needed for maximization approximation. While SD and SMW hierarchies support nontrivial bounds on the PoA of simple auctions, we show that our SAW hierarchy seems to capture more intrinsic properties needed to realize the efficiency of simple auctions. So far, the SAW hierarchy provides the best dependency for the PoA of Single-bid Auction, and is nearly as competitive as the Maximum over Positive Hypergraphs (MPH) hierarchy for Simultaneous Item First Price Auction (SIA). We also provide almost tight lower bounds for the PoA of both auctions with respect to the SAW hierarchy

Constrained Monotone Function Maximization and the Supermodular Degree

The problem of maximizing a constrained monotone set function has many practical applications and generalizes many combinatorial problems. Unfortunately, it is generally not possible to maximize a monotone set function up to an acceptable approximation ratio, even subject to simple constraints. One highly studied approach to cope with this hardness is to restrict the set function. An outstanding disadvantage of imposing such a restriction on the set function is that no result is implied for set functions deviating from the restriction, even slightly. A more flexible approach, studied by Feige and Izsak, is to design an approximation algorithm whose approximation ratio depends on the complexity of the instance, as measured by some complexity measure. Specifically, they introduced a complexity measure called supermodular degree, measuring deviation from submodularity, and designed an algorithm for the welfare maximization problem with an approximation ratio that depends on this measure. In this work, we give the first (to the best of our knowledge) algorithm for maximizing an arbitrary monotone set function, subject to a k-extendible system. This class of constraints captures, for example, the intersection of k-matroids (note that a single matroid constraint is sufficient to capture the welfare maximization problem). Our approximation ratio deteriorates gracefully with the complexity of the set function and k. Our work can be seen as generalizing both the classic result of Fisher, Nemhauser and Wolsey, for maximizing a submodular set function subject to a k-extendible system, and the result of Feige and Izsak for the welfare maximization problem. Moreover, when our algorithm is applied to each one of these simpler cases, it obtains the same approximation ratio as of the respective original work.Comment: 23 page

Weakly Submodular Functions

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the "standard greedy algorithm" achieves a constant approximation ratio where the constant (experimentally) converges to 5.95 as the cardinality constraint increases. For a general matroid constraint, a simple local search algorithm achieves a constant approximation ratio where the constant (analytically) converges to 10.22 as the rank of the matroid increases

A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as $\mathcal{MPH}$ (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, $\mathcal{MPH}$-$m$ (where $m$ is the total number of items) captures all monotone functions. The lowest level, $\mathcal{MPH}$-$1$, captures all monotone submodular functions, and more generally, the class of functions known as $\mathcal{XOS}$. Every monotone function that has a positive hypergraph representation of rank $k$ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in $\mathcal{MPH}$-$k$. Every monotone function that has supermodular degree $k$ (in the sense defined by Feige and Izsak [ITCS 2013]) is in $\mathcal{MPH}$-$(k+1)$. In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of $\mathcal{MPH}$-$k$. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the $\mathcal{MPH}$ hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of $k+1$ if all players hold valuation functions in $\mathcal{MPH}$-$k$. The other is an upper bound of $2k$ on the price of anarchy of simultaneous first price auctions. Being in $\mathcal{MPH}$-$k$ can be shown to involve two requirements -- one is monotonicity and the other is a certain requirement that we refer to as $\mathcal{PLE}$ (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the $\mathcal{PLE}$ hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research

Maximizing Welfare in Social Networks under a Utility Driven Influence Diffusion Model

Motivated by applications such as viral marketing, the problem of influence maximization (IM) has been extensively studied in the literature. The goal is to select a small number of users to adopt an item such that it results in a large cascade of adoptions by others. Existing works have three key limitations. (1) They do not account for economic considerations of a user in buying/adopting items. (2) Most studies on multiple items focus on competition, with complementary items receiving limited attention. (3) For the network owner, maximizing social welfare is important to ensure customer loyalty, which is not addressed in prior work in the IM literature. In this paper, we address all three limitations and propose a novel model called UIC that combines utility-driven item adoption with influence propagation over networks. Focusing on the mutually complementary setting, we formulate the problem of social welfare maximization in this novel setting. We show that while the objective function is neither submodular nor supermodular, surprisingly a simple greedy allocation algorithm achieves a factor of $(1-1/e-\epsilon)$ of the optimum expected social welfare. We develop \textsf{bundleGRD}, a scalable version of this approximation algorithm, and demonstrate, with comprehensive experiments on real and synthetic datasets, that it significantly outperforms all baselines.Comment: 33 page

Cooperative Games with Bounded Dependency Degree

Cooperative games provide a framework to study cooperation among self-interested agents. They offer a number of solution concepts describing how the outcome of the cooperation should be shared among the players. Unfortunately, computational problems associated with many of these solution concepts tend to be intractable---NP-hard or worse. In this paper, we incorporate complexity measures recently proposed by Feige and Izsak (2013), called dependency degree and supermodular degree, into the complexity analysis of cooperative games. We show that many computational problems for cooperative games become tractable for games whose dependency degree or supermodular degree are bounded. In particular, we prove that simple games admit efficient algorithms for various solution concepts when the supermodular degree is small; further, we show that computing the Shapley value is always in FPT with respect to the dependency degree. Finally, we note that, while determining the dependency among players is computationally hard, there are efficient algorithms for special classes of games.Comment: 10 pages, full version of accepted AAAI-18 pape