Integer Optimization and Computational Algebraic Topology

We present recently discovered connections between integer optimization, or integer programming (IP), and homology. Under reasonable assumptions, these results lead to efficient solutions of several otherwise hard-to-solve problems from computational topology and geometric analysis. The main result equates the total unimodularity of the boundary matrix of a simplicial complex to an algebraic topological condition on the complex (absence of relative torsion), which is often satisfied in real-life applications . When the boundary matrix is totally unimodular, the problem of finding the shortest chain homologous under Z (ring of integers) to a given chain, which is inherently an integer program, can be solved in polynomial time as a linear program. This result is surprising in the backdrop of a previous result, which showed the problem to be NP-hard when the homology is defined over the popularly used field of Z2, consisting of integers 0 and 1. This problem finds applications in several domains including coverage verification in sensor networks and characterizing tunnels in biomolecules. We also present new results on computing the flat norm of currents in the setting of simplicial complexes. Flat norm decomposition is a classical technique from geometric measure theory, and has been applied for several image analysis tasks. Our approach allows one to use flat norm computations in arbitrarily large dimensions - for instance, to denoise high dimensional datasets.https://pdxscholar.library.pdx.edu/systems_science_seminar_series/1056/thumbnail.jp

Topological Features In Cancer Gene Expression Data

We present a new method for exploring cancer gene expression data based on tools from algebraic topology. Our method selects a small relevant subset from tens of thousands of genes while simultaneously identifying nontrivial higher order topological features, i.e., holes, in the data. We first circumvent the problem of high dimensionality by dualizing the data, i.e., by studying genes as points in the sample space. Then we select a small subset of the genes as landmarks to construct topological structures that capture persistent, i.e., topologically significant, features of the data set in its first homology group. Furthermore, we demonstrate that many members of these loops have been implicated for cancer biogenesis in scientific literature. We illustrate our method on five different data sets belonging to brain, breast, leukemia, and ovarian cancers.Comment: 12 pages, 9 figures, appears in proceedings of Pacific Symposium on Biocomputing 201

MOTIVATIONS FOR SOCIAL NETWORK SITE (SNS) GAMING: A USES AND GRATIFICATION & FLOW PERSPECTIVE

The penetration of the internet, smart-phones and tablets has witnessed tremendous increase in the number of people playing online games in the past few years. Social networking site (SNS) games are a subset of digital games. They are platform based, multiplayer and reveal the real identity of the player. These games are hosted on social networks such as Facebook, where in people play with many other players online. The risks associated with social network gaming are addiction, theft, fraud, loneliness, anxiety, aggression, poor academic performance, cognition distortion etc. This study aims to understand the user motivations to continue to play social networking games and spread a word of mouth for these games. To understand this phenomenon, Uses and Gratification theory (U&G) along with flow and immersion have been considered as the antecedents. A total of 242 respondents comprising of 125 participants who play social networking games and 117 participants who do not play social networking games completed the survey. This aided in understanding the participants’ motivations and inhibitions towards playing social networking games. The present findings indicate that gratifications, flow and immersion are significantly related to the continuance motivation, which in turn is significantly related to word of mouth

A Tight Max-Flow Min-Cut Duality Theorem for Non-Linear Multicommodity Flows

The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flow problem, which considers flow of a single commodity. We study a multiple commodity generalization of Max-Flow in which flows are composed of real-valued k-vectors through networks with arc capacities formed by regions in \R^k. Given the absence of a clear notion of ordering in the multicommodity case, we define the generalized max flow as the feasible region of all flow values. We define a collection of concepts and operations on flows and cuts in the multicommodity setting. We study the mutual capacity of a set of cuts, defined as the set of flows that can pass through all cuts in the set. We present a method to calculate the mutual capacity of pairs of cuts, and then generalize the same to a method of calculation for arbitrary sets of cuts. We show that the mutual capacity is exactly the set of feasible flows in the network, and hence is equal to the max flow. Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations. We also study more tractable special cases of the multicommodity max flow problem where the objective is to transport a maximum real or integer multiple of a given vector through the network. We devise an augmenting cycle search algorithm that reduces the optimization problem to one with m constraints in at most \R^{(m-n+1)k} space from one that requires mn constraints in \R^{mk} space for a network with n nodes and m edges. We present efficient algorithms that compute eps-approximations to both the ratio and the integer ratio maximum flow problems