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 and 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. We construct the data structure for a given directed graph of nearly quadratic space 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 nearly cubic time in order to answer approximate bounded-leg distance and bounded-leg shortest path queries in merely sub-logarithmic time.Ph.D.Computer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/89641/1/duanran_1.pd

Parameterized and approximation complexity of the detection pair problem in graphs

We study the complexity of the problem DETECTION PAIR. A detection pair of a graph $G$ is a pair $(W,L)$ of sets of detectors with $W\subseteq V(G)$, the watchers, and $L\subseteq V(G)$, the listeners, such that for every pair $u,v$ of vertices that are not dominated by a watcher of $W$, there is a listener of $L$ whose distances to $u$ and to $v$ are different. The goal is to minimize $|W|+|L|$. This problem generalizes the two classic problems DOMINATING SET and METRIC DIMENSION, that correspond to the restrictions $L=\emptyset$ and $W=\emptyset$, respectively. DETECTION PAIR was recently introduced by Finbow, Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The complexity of monitoring a network with both watchers and listeners. Manuscript, 2015], who proved it to be NP-complete on trees, a surprising result given that both DOMINATING SET and METRIC DIMENSION are known to be linear-time solvable on trees. It follows from an existing reduction by Hartung and Nichterlein for METRIC DIMENSION that even on bipartite subcubic graphs of arbitrarily large girth, DETECTION PAIR is NP-hard to approximate within a sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We show, using a reduction to SET COVER, that DETECTION PAIR is approximable within a factor logarithmic in the number of vertices of the input graph. Our two main results are a linear-time $2$-approximation algorithm and an FPT algorithm for DETECTION PAIR on trees.Comment: 13 page

Distance Oracles for Time-Dependent Networks

We present the first approximate distance oracle for sparse directed networks with time-dependent arc-travel-times determined by continuous, piecewise linear, positive functions possessing the FIFO property. Our approach precomputes $(1+\epsilon)-$approximate distance summaries from selected landmark vertices to all other vertices in the network. Our oracle uses subquadratic space and time preprocessing, and provides two sublinear-time query algorithms that deliver constant and $(1+\sigma)-$approximate shortest-travel-times, respectively, for arbitrary origin-destination pairs in the network, for any constant $\sigma > \epsilon$. Our oracle is based only on the sparsity of the network, along with two quite natural assumptions about travel-time functions which allow the smooth transition towards asymmetric and time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An extended abstract also appeared in the 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014, track-A

Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time

In the bounded-leg shortest path (BLSP) problem, we are given a weighted graph G with nonnegative edge lengths, and we want to answer queries of the form "what\u27s the shortest path from u to v, where only edges of length = f are considered. In this article we give an O~(n^{(omega+3)/2}epsilon^{-3/2}log W) time algorithm to compute a data structure that answers APSP-AF queries in O(log(epsilon^{-1}log (nW))) time and achieves (1+epsilon)-approximation, where omega < 2.373 is the exponent of time complexity of matrix multiplication, W is the upper bound of integer edge lengths, and n is the number of vertices. This is the first truly-subcubic time algorithm for these problems on dense graphs. Our algorithm utilizes the O(n^{(omega+3)/2}) time max-min product algorithm [Duan and Pettie 2009]. Since the all-pair bottleneck path (APBP) problem, which is equivalent to max-min product, can be seen as all-pair reachability for all flow, our approach indeed shows that these problems are almost equivalent in the approximation sense