Integer Programming Approaches for Network Optimization and Chance-Constrained Combinatorial Problems

Abstract

Integer Programming (IP), a critical branch of optimization, serves as a powerful and fundamental mathematical tool for formulating and solving discrete decision-making problems that are prevalent across various fields of study and real-world applications. Typically, IP is about ways to solve combinatorial optimization problems with discrete or integer variables. By formulating problems as IP models, decision-makers can derive advanced solution techniques to find optimal or near-optimal solutions within complex environments. In this dissertation, we explore several techniques of IP: modeling, lifting, relaxation, convexification and enumeration, with two significant application domains in the context of network optimization and chance-constrained combinatorial problems. First, we apply IP approaches to studying the spread of influence over networks, a phenomenon with significant implications in diverse areas such as viral marketing, rumor control, epidemiology, social recommendation, etc. Understanding and modeling the spread of influence in social networks are crucial for predicting individual impacts and managing collective dynamics. In this dissertation, we focus on minimizing the spread of influence by removing a subset of nodes from a network. We study the complexity of this problem and develop a novel delayed constraint generation (DCG) algorithm based on an IP model to identify the most critical nodes in the influence propagation process. Furthermore, we derive lifting inequalities for minimal activation sets to enhance our understanding of the dynamics involved. Experiments conducted on the connected Watts-Strogatz small-world networks and real-world networks validate the effectiveness of our methodology. Additionally, we apply techniques to model the spread of infectious diseases over networks through influence maximization. This involves analyzing network structures, modeling connections among individuals with infected probabilities, and incorporating evolving individual behaviors that influence the spread process over time. Simulation results on random networks and a local community network during the COVID-19 pandemic validate the proposed models and their relationships with classic compartmental models. Second, we apply IP approaches to several chance-constrained problems. Uncertainty poses a significant challenge in decision-making processes, especially when the random vector is high-dimensional. Chance-constrained optimization (CCO) has emerged as a powerful paradigm to model uncertainty in optimization problems. In this dissertation, we first consider a variant of the set covering problem with uncertain data, which we refer to as the chance-constrained set multicover problem (CC-SMCP). We develop exact deterministic reformulations and propose an outer-approximation (OA) algorithm for CC-SMCP using some combinatorial methods. Our methods combine enumerative combinatorics, discrete probability distributions and combinatorial optimization, representing a novel and intriguing direction for future research in combinatorial chance-constrained problems. Additionally, we explore strategies to reduce the number of chance constraints by considering vector dominance relations defined in an appropriate partially ordered set. Some theoretical results on sampling-based methods, sample average approximation (SAA) and importance sampling (IS), to approximate the optimal value of CC-SMCP are also presented. Numerical experiments are conducted to validate the effectiveness of our OA method compared to the SAA and IS approaches. In addition to CC-SMCP, we also investigate the maximum probabilistic clique problem (MPCP) and derived its exact MIP reformulation by considering the complement of random graphs. We proposed two exact algorithms, DCG and Branch-and-Bound (BnB), as well as two approximation algorithms, SAA and Reinforcement learning (RL), to address the problem.