Algorithms and Dynamic Data Structures for Basic Graph Optimization Problems

Graph optimization plays an important role in a wide range of areas such as computer graphics, computational biology, networking applications and machine learning. Among numerous graph optimization problems, some basic problems, such as shortest paths, minimum spanning tree, and maximum matching, are the most fundamental ones. They have practical applications in various fields, and are also building blocks of many other algorithms. Improvements in algorithms for these problems can thus have a great impact both in practice and in theory. In this thesis, we study a number of graph optimization problems. The results are mostly about approximation algorithms solving graph problems, or efficient dynamic data structures which can answer graph queries when a number of changes occur. Much of my work focuses on the dynamic subgraph model in which there is a fixed underlying graph and every vertex can be flipped "on" or "off". The queries are based on the subgraph induced by the "on" vertices. Our results make significant improvements to the previous algorithms of these problems. The major results are listed below. • Approximate Matching. We give the first linear time algorithm for computing approximate maximum weighted matching for arbitrarily small approximation ratio. • d-failure Connectivity Oracle. For an undirected graph, we give the first space-efficient data structure that can answer connectivity queries between any pair of vertices avoiding d other failed vertices in time polynomial in d log n. • (Max, Min)-Matrix Multiplication. We give a faster algorithm for the (max, min)-matrix multiplication problem, which has a direct application to the all-pairs bottleneck paths (APBP) problem. • Dual-failure Distance Oracle. For a given directed graph, we construct a data structure of size O(n²) which can efficiently answer distance and shortest path queries in the presence of two node or link failures. • Dynamic Subgraph Connectivity. We give the first subgraph connectivity structure with worst-case sublinear time bounds for both updates and queries. • Bounded-leg Shortest Path. We give an algorithm for preprocessing a directed graph in O(n³) time in order to answer approximate bounded leg distance and bounded leg shortest path queries in merely sub-logarithmic time.