Results

**1 - 3**of**3**### New Results in the Theory of Approximation -- Fast Graph Algorithms and Inapproximability

, 2013

"... For several basic optimization problems, it is NP-hard to find an exact solution. As a result, understanding the best possible trade-off between the running time of an algorithm and its approximation guarantee, is a fundamental question in theoretical computer science, and the central goal of the th ..."

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For several basic optimization problems, it is NP-hard to find an exact solution. As a result, understanding the best possible trade-off between the running time of an algorithm and its approximation guarantee, is a fundamental question in theoretical computer science, and the central goal of the theory of approximation. There are two aspects to the theory of approximation: (1) efficient approximation algorithms that establish trade-offs between approximation guarantee and running time, and (2) inapproximability results that give evidence against them. In this thesis, we contribute to both facets of the theory of approximation. In the first part of this thesis, we present the first near-linear-time algorithm for Balanced Separator- given a graph, partition its vertices into two roughly equal parts without cutting too many edges- that achieves the best approximation guarantee possible for algorithms in its class. This is a classic graph partitioning problem and has deep connections to several areas of both theory and practice, such as metric embeddings, Markov chains, clustering, etc.

### Capacitated Vehicle Routing with Non-Uniform Speeds

"... The capacitated vehicle routing problem (CVRP) [TV02] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a general-ization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heteroge ..."

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The capacitated vehicle routing problem (CVRP) [TV02] involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a general-ization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles having respective speeds {λi}ki=1, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al. [LST90]. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees [KRY95]. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot.