An O(1)-Approximation for Minimum Spanning Tree Interdiction

Network interdiction problems are a natural way to study the sensitivity of a network optimization problem with respect to the removal of a limited set of edges or vertices. One of the oldest and best-studied interdiction problems is minimum spanning tree (MST) interdiction. Here, an undirected multigraph with nonnegative edge weights and positive interdiction costs on its edges is given, together with a positive budget B. The goal is to find a subset of edges R, whose total interdiction cost does not exceed B, such that removing R leads to a graph where the weight of an MST is as large as possible. Frederickson and Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction, where m is the number of edges. Since then, no further progress has been made regarding approximations, and the question whether MST interdiction admits an O(1)-approximation remained open. We answer this question in the affirmative, by presenting a 14-approximation that overcomes two main hurdles that hindered further progress so far. Moreover, based on a well-known 2-approximation for the metric traveling salesman problem (TSP), we show that our O(1)-approximation for MST interdiction implies an O(1)-approximation for a natural interdiction version of metric TSP

Parameterized Complexity of Edge Interdiction Problems

We study the parameterized complexity of interdiction problems in graphs. For an optimization problem on graphs, one can formulate an interdiction problem as a game consisting of two players, namely, an interdictor and an evader, who compete on an objective with opposing interests. In edge interdiction problems, every edge of the input graph has an interdiction cost associated with it and the interdictor interdicts the graph by modifying the edges in the graph, and the number of such modifications is constrained by the interdictor's budget. The evader then solves the given optimization problem on the modified graph. The action of the interdictor must impede the evader as much as possible. We focus on edge interdiction problems related to minimum spanning tree, maximum matching and shortest paths. These problems arise in different real world scenarios. We derive several fixed-parameter tractability and W[1]-hardness results for these interdiction problems with respect to various parameters. Next, we show close relation between interdiction problems and partial cover problems on bipartite graphs where the goal is not to cover all elements but to minimize/maximize the number of covered elements with specific number of sets. Hereby, we investigate the parameterized complexity of several partial cover problems on bipartite graphs

Data-Oblivious Graph Algorithms in Outsourced External Memory

Motivated by privacy preservation for outsourced data, data-oblivious external memory is a computational framework where a client performs computations on data stored at a semi-trusted server in a way that does not reveal her data to the server. This approach facilitates collaboration and reliability over traditional frameworks, and it provides privacy protection, even though the server has full access to the data and he can monitor how it is accessed by the client. The challenge is that even if data is encrypted, the server can learn information based on the client data access pattern; hence, access patterns must also be obfuscated. We investigate privacy-preserving algorithms for outsourced external memory that are based on the use of data-oblivious algorithms, that is, algorithms where each possible sequence of data accesses is independent of the data values. We give new efficient data-oblivious algorithms in the outsourced external memory model for a number of fundamental graph problems. Our results include new data-oblivious external-memory methods for constructing minimum spanning trees, performing various traversals on rooted trees, answering least common ancestor queries on trees, computing biconnected components, and forming open ear decompositions. None of our algorithms make use of constant-time random oracles.Comment: 20 page

Sensitivity Analysis for Shortest Path Problems and Maximum Capacity Path Problems in Undirected Graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path

Dynamic Algorithms for the Massively Parallel Computation Model

The Massive Parallel Computing (MPC) model gained popularity during the last decade and it is now seen as the standard model for processing large scale data. One significant shortcoming of the model is that it assumes to work on static datasets while, in practice, real-world datasets evolve continuously. To overcome this issue, in this paper we initiate the study of dynamic algorithms in the MPC model. We first discuss the main requirements for a dynamic parallel model and we show how to adapt the classic MPC model to capture them. Then we analyze the connection between classic dynamic algorithms and dynamic algorithms in the MPC model. Finally, we provide new efficient dynamic MPC algorithms for a variety of fundamental graph problems, including connectivity, minimum spanning tree and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2019

A Linear Time Algorithm for Finding Minimum Spanning Tree Replacement Edges

Given an undirected, weighted graph, the minimum spanning tree (MST) is a tree that connects all of the vertices of the graph with minimum sum of edge weights. In real world applications, network designers often seek to quickly find a replacement edge for each edge in the MST. For example, when a traffic accident closes a road in a transportation network, or a line goes down in a communication network, the replacement edge may reconnect the MST at lowest cost. In the paper, we consider the case of finding the lowest cost replacement edge for each edge of the MST. A previous algorithm by Tarjan takes $O(m \alpha(m, n))$ time, where $\alpha(m, n)$ is the inverse Ackermann's function. Given the MST and sorted non-tree edges, our algorithm is the first that runs in $O(m+n)$ time and $O(m+n)$ space to find all replacement edges. Moreover, it is easy to implement and our experimental study demonstrates fast performance on several types of graphs. Additionally, since the most vital edge is the tree edge whose removal causes the highest cost, our algorithm finds it in linear time