Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Node-weighted Steiner tree and group Steiner tree in planar graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ [theta] (logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 [superscript 3] n), or O(log2 [superscript 2] n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group

The Minimum Wiener Connector

The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set $Q\subseteq V$ of query vertices, find a subgraph of $G$ that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless $\mathbf{P} = \mathbf{NP}$. Our main contribution is a constant-factor approximation algorithm running in time $\widetilde{O}(|Q||E|)$. A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set $Q$ a small number of important vertices (i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International Conference on Management of Dat

Algorithms and complexity analyses for some combinational optimization problems

The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design. In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios. In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs. Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost