Campaigning Via LPs: Solving Blotto and Beyond

The competition between the Republican and the Democrat nominees in the U.S presidential election is known as Colonel Blotto in game theory. In the classical Colonel Blotto game -- introduced by Borel in 1921 -- two colonels simultaneously distribute their troops across multiple battlefields. The outcome of each battlefield is determined by a winner-take-all rule, independently of other battlefields. In the original formulation, the goal of each colonel is to win as many battlefields as possible. The Colonel Blotto game and its extensions have been used in a wide range of applications from political campaigns (exemplified by the U.S presidential election) to marketing campaigns, from (innovative) technology competitions, to sports competitions. For almost a century, there have been persistent efforts for finding the optimal strategies of the Colonel Blotto game, however it was left unanswered whether the optimal strategies are polynomially tractable. In this thesis, we present several algorithms for solving Blotto games in polynomial time and will discuss their applications in practice

Modern Large-Scale Algorithms for Classical Graph Problems

Although computing power has advanced at an astonishing rate, it has been far outpaced by the growing scale of data. This has led to an abundance of algorithmic problems where the input tends to be, by orders of magnitude, larger than the memory available on a single machine. The challenges of data processing at this scale are inherently different from those of traditional algorithms. For instance, without having the whole input properly stored in the memory of a single machine, it is unrealistic to assume that any arbitrary location of the input can be accessed at the same cost; an assumption that is essential for traditional algorithms. In this thesis, we focus on modern computational models that capture these challenges more accurately, and devise new algorithms for several classical graph problems. Specifically, we study models of computation that only allow the algorithm to use sublinear resources (such as time, space, or communication). Examples include (i) massively parallel computation (à la MapReduce) algorithms where the workload is distributed among several machines each with sublinear space/communication, (ii) sublinear-time algorithms that take time sublinear in the input size, (iii) streaming algorithms that take only few passes over the input having access to a sublinear space, and (iv) dynamic algorithms that maintain a property of a dynamically changing input using a sublinear time per update. We propose new algorithms for classical graph problems such as maximum/maximal matching, maximal independent set, minimum vertex cover, and graph connectivity in these models that substantially improve upon the state-of-the-art and are in many cases optimal. Many of our algorithms build on model-independent tools and ideas that are of independent interest and lead to improved bounds in more than one of the aforementioned settings