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Algorithms Theory: Spring 2005 Precept 6 - Min Leaf Spanning Tree & Traveling Salesperson, Exercises of Algorithms and Programming

Aliah UniversityAlgorithms and Programming

This document from cos 423, theory of algorithms, spring 2005, covers two problems: determining the existence of a spanning tree with a specified number of leaf nodes in a connected graph and the polynomial-time equivalence between the traveling salesperson problem and a decision problem regarding a tour of length at most a given upper bound.

Typology: Exercises

2011/2012

Uploaded on 07/16/2012

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santhanakrishnan 🇮🇳

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COS 423 Theory of Algorithms Spring 2005
Precept 6
1. Given a connected graph Gand a parameter k, determine whether there exists a spanning tree with
at most kleaf nodes. A leaf node is a node of a degree one. Show that 3-Sat ≤pMinimumLeafS-
panningTree.
2. Problem 8.27.
3. Given a set of ncities and pairwise distances d(u, v) between them, the traveling salesperson problem
is to find a tour of minimum length. Show that this search problem is polynomial-time equivalent to
the following decision problem.
Tsp. Given a set of ncities, pairwise distances d(u, v) between them, and an upper bound D,isthere
a tour of length at most D?
1
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COS 423 Theory of Algorithms Spring 2005

Precept 6

  1. Given a connected graph G and a parameter k, determine whether there exists a spanning tree with at most k leaf nodes. A leaf node is a node of a degree one. Show that 3-Sat ≤p MinimumLeafS- panningTree.
  2. Problem 8.27.
  3. Given a set of n cities and pairwise distances d(u, v) between them, the traveling salesperson problem is to find a tour of minimum length. Show that this search problem is polynomial-time equivalent to the following decision problem.

Tsp. Given a set of n cities, pairwise distances d(u, v) between them, and an upper bound D, is there a tour of length at most D?

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