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

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

Predestined for Greatness? Calvinism’s Role in the Formation of 18th Century American Nationalism

From the Puritans of New England to the Anglicans of the South, a patchwork of competing religious ideals existed in colonial America. Through that diversity, there emerged a singular thread of religious influence which worked, during the Revolution and beyond, to craft a nascent sense of American nationalism: American Calvinism. Beginning with the reformer John Calvin and his disciple, Theodore Beza, in Europe, Calvinism expanded to the New World. There, emboldened by the isolation of the American continent, Calvinism became a rare source of commonality between colonies with disparate religious beliefs. When war raged between Britain and the American colonies in the late eighteenth century, the American Founders utilized Calvinism to rally as many Americans as possible around their national cause. Utilizing the words of the American Founders and peer-reviewed scholarship regarding their outlook, this paper analyzes the history of American Calvinism while focusing on how it crafted a sense of American national identity out of the throes of Revolution. This paper proposes that American Calvinism was unlike its late-medieval European counterpart in its uniqueness. This Americanized branch of Calvinism combined a Lockean focus on individual rights and republicanism with a traditional Calvinist focus on the necessity of morality and formed one of the essential cogs in a new sense of American nationalism. This paper demonstrates that, though not all of the American Founders respected Calvin, or his religious ideals, (Thomas Jefferson was famously critical), they recognized the importance that Calvinism held for the masses of America. Because of this recognition, the paper proposes that the American Founders adapted their speech, mannerisms, and public statements to advance the cause of colonial unity, helping to form a unique version of American nationalism from separate colonies toward the end of the eighteenth century

Special Cases of Carry Propagation

The average time necessary to add numbers by local parallel computation is directly related to the length of the longest carry propagation chain in the sum. The mean length of longest carry propagation chain when adding two independent uniform random n bit numbers is a well studied topic, and useful approximations as well as an exact expression for this value have been found. My thesis searches for similar formulas for mean length of the longest carry propagation chain in sums that arise when a random n-digit number is multiplied by a number of the form 1 + 2d. Letting Cn, d represent the desired mean, my thesis details how to find formulas for Cn,d using probability, generating functions and linear algebra arguments. I also find bounds on Cn,d to prove that Cn,d = log2 n + O(1), and show work towards finding an even more exact approximation for Cn,d

Analytical development of the planetary disturbing function on a digital computer

Fortran II IBM 7094 computer program for analytical development of planetary disturbing functio

Preparing Middle Grades Teachers to Use Drawn Models for Developing Arithmetic with Rational Numbers

We will report on-going efforts to design a research-based content and methods course for future middle grades teachers focused on numbers and operations. A main theme of the course is to solve problems using drawn models and to develop general numeric methods. Attendees will work on activities used in the course that elicit difficulties future teachers experience with this content

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