The Computational Complexity of Genetic Diversity

A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition, or whether a single dominant genotype emerges. Classic work by [Kalmus, J. og Genetics, 1945] has established that even in simple diploid species (species with chromosome pairs) diversity can be guaranteed as long as the heterozygous (having different alleles for a gene on two chromosomes) individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g., human blood types) predicting diversity (and its implications) is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus (gene) diploid models, predicting whether diversity survives or not given its fitness landscape is algorithmically intractable. Our hardness results are structurally robust along several dimensions, e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes. Technically, our results exploit connections between game theory, nonlinear dynamical systems, and complexity theory and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games; finding one Nash equilibrium is easy in these games. In the process we characterize stable fixed points of these dynamics using the notions of Nash equilibrium and negative semidefiniteness. This as well as hardness results for decision problems in coordination games may be of independent interest. Finally, we complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. The full version of this paper is available at http://arxiv.org/abs/1411.6322

Combinatorial and exchange markets: Algorithms, complexity, and applications

In today's world, globalization and the Internet have resulted in the creation of enormously many different kinds of marketplaces. The marketplaces naturally tend to find an equilibrium in terms of prices and interaction of agents. Therefore, understanding the equilibria results in better understanding and prediction of the marketplaces. In this thesis, we study two broad classes of equilibria. The first one is called market price equilibria, which can explain and predict prices within a market. The second one is Nash equilibria (NE), which is arguably the most important and well-studied solution concept within game theory. NE helps us to explain and predict the interactions between agents within a market. - Market price equilibria. We introduce a new class of combinatorial markets in which agents have covering constraints over resources required and are interested in delay minimization. Our market model is applicable to several settings including scheduling and communicating over a network. We give a proof of the existence of equilibria and a polynomial time algorithm for finding one, drawing heavily on techniques from LP duality and submodular minimization. Next, we show FIXP-hardness of computing equilibria in Arrow-Debreu exchange markets under Leontief utility functions, and Arrow-Debreu markets under linear utility functions and Leontief production sets, thereby settling these open questions of Vazirani and Yannakakis (2011). As a consequence of the results stated above, and the fact that membership in FIXP has been established for PLC utilities, the entire computational difficulty of Arrow-Debreu markets under PLC utility functions lies in the Leontief utility subcase. Finally, we give a polynomial time algorithm for finding an equilibrium in Arrow-Debreu exchange markets under Leontief utility functions provided the number of agents is a constant. - Nash equilibria. The complexity of 2-player Nash equilibrium is by now well understood. Our contribution is on settling the complexity of finding equilibria with special properties on multi-player games. We show that the following decision versions of 3-Nash are EXISTS-R-complete: checking whether 1. there are two or more equilibria, 2. there exists an equilibrium in which each player gets at least h payoff, where $h$ is a rational number, 3. a given set of strategies are played with non-zero probability, and 5. all the played strategies belong to a given set. EXISTS-R is the class of decision problems which can be reduced in polynomial time to Existential Theory of the Reals. Next, we give a reduction from 3-Nash to symmetric 3-Nash.Ph.D

How Effectively Can We Form Opinions? *

ABSTRACT People make decisions and express their opinions according to their communities. An appropriate idea for controlling the diffusion of an opinion is to find influential people, and employ them to spread the desired opinion. We investigate an influencing problem when individuals' opinions are affected by their friends due to the model of Friedkin and Johnse