Approximation algorithms for TSP with neighborhoods in the plane

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Approximation algorithms for TSP with
neighborhoods in the plane

• R90922026 ???
• R90922038 ???

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The Problem

• TSPN Euclidean TSP with neighborhoods in the
  plane, i.e., dimention 2
• Definition A salesman wants to meet a set of
  potential buyers. Each buyer specifies a region
  of the plane, his neighborhood. The salesman
  wants to find a tour of shortest length that
  visit all buyers neighborhoods.

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Part 1 Introduction
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Introduction

• O(1)-approximation algorithm for TSPN on disks of
  the same size
• A PTAS for disjoint equal disks
• O(1)-approximation algorithm for TSPN on
  connected regions of the same diameter
• O(1)-approximation algorithm running in linear
  time for TSPN on lines

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Part 2 O(1)-approximation for TSPN on disks of
the same size
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Equal Disks

• Assume that all disks are unit
• Simplify the problem to unit disks
• First case on disjoint unit disks
• Simply approximate on centers of disks
• Argue with the boundsTc for center tour, Tr for
  region tour

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Disjoint Unit Disks
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Disjoint Unit Disks (cont.)

• Sweep along center tour with a disk of radius 2
• Ratio lt 3.55 for n large

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Disjoint Unit Disks (cont.)
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Disjoint Unit Disks (cont.)
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Disjoint Unit Disks (cont.)

• Expect ration cannot smaller than 2

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Overlapping Unit Disks

• Algorithm
• Compute maximal independent pairwise-disjoint set
  of disks
• Compute Ci, the approximation of the center tour
  above
• Output R, obtained by going along Ci and
  boundaries of each disks in the set
• Argue with the ratio lt 11.15

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Overlapping Unit Disks (cont.)

• OPT optimal tour of the problem
• OPTi optimal tour of independent set
• R approximated result
• Ci center tour of independent set

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Overlapping Unit Disks (cont.)
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Part 3 A PTAS for disjoint equal disks
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Slide not finished
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Part 4 O(1)-approximation algorithm for TSPN on
connected regions of the same diameter
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Definition 4.1. (diameter of a region)

• The diameter of a region, d, is the distance
  between two points in the region that are
  farthest apart
• In the problem here we deal with connected
  regions of the same diameter. Without loss of
  generality, we assume that all regions have unit
  diamter, d 1.

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Lemma 4.2.(Combination Lemma)

• Given regions that can be partitioned into two
  types, and constants c1, c2 bounding the error
  ratios with which we can approximate optimal
  tours on regions of type 1 and 2, then we can
  approximate the optimal tour on all regions with
  an error ratio bounded by c1 c2 2

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Two types of connected regions

• (1) Those for which the selected diameter is
  almost horizontal, by which we mean its slope is
  between -45 and 45
• (2) Those for which the selected diameter is
  almost vertical , by which we mean all others.
• The paper provide an constant ratio approximation
  algorithm for each of these two region types, and
  then apply Lemma 4.2.

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Fig 4.a. (two types of regions)
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Fact 4.3.

• A tour touching all four sides of a rectangle is
  of length at least twice the diagonal of the
  rectangle.
• See Fig 4.b. to get the idea.

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Fig 4.b.
The shortest tour hits each side is equal to the
angle with which it departs that side, by Snells
law
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Fact 4.4.

• For positive a, b, w, h the following inequality
  holds.

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Definition 4.5.(covering lines)

• A set of lines is a cover of a set of regions if
  each region is intersected by at least one line
  from the covering set. We refer to this set of
  lines as covering lines.

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The algorithm step 1

• W.l.o.g. we use the algorithm for type (1)
• Construct a greedy covering lines of the regions
  by a minimum number of vertical lines.
• The procedure works in a greedy fashion, namely
  the leftmost line is as far right as possible, so
  that it is a right tangent of some region.
• Then representative points of each region are
  arbitrarily selected on the corresponding
  covering lines.

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Fig 4.c.(greedy covering lines representative
points)
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The algorithm step 2

• Proceed according the following three cases.
• Case 1 The greedy cover contains one covering
  line.
• Case 2 The greedy cover contains two covering
  lines.
• Case 3 The greedy cover contains at least three
  covering lines.

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The algorithm step 2 case 1

• One covering line.
• Compute a smallest perimeter rectangle Q of width
  w and height h that touches all regions.
• Add twice the two vertical segments of height h
  which divide its width in three equal parts, to
  get a tour R. Output R.

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Fig 4.d.
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Fig 4.e.
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Fig 4.f.
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Proof of step 2 - case 1
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The algorithm step 2 case 2

• Two covering lines
• Move the rightmost vertical covering line to the
  left as much as possible.
• Set D to be the distance between the two covering
  lines, clearly D gt 0.

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The algorithm step 2 case 2.1

• D gt 3
• Construct rectangle Q of width w D, with its
  vertical sides along the two covering lines, and
  of minimal height h, which includes all
  representative points (on the two covering
  lines). Output the tour R that is the perimeter
  of Q.

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Fig 4.g.
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Proof of step 2 case 2.1

• Case 2.1.a. hlt2

• Case 2.1.b. hgt2

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The algorithm step 2 case 2.2

• D lt 3
• Compute a smallest perimeter rectangle Q with
  width w and height h that touches all regions.
  Add twice the seven vertical segments of height h
  which divide its width into eight equal parts, to
  get a tour R. Output R.

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Fig 4.h.
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Proof of step 2 case 2.2
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The algorithm step 2 case 3

• At least three covering lines
• Construct R, a (1e)-approximation tour of the
  representative points as the output tour.

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Proof of step2 case 3

• Partition the optimal tour OPT into blocks OPTi,
  with i gt 1. OPTi starts at an arbitrary point of
  intersection of OPT with the ith covering line
  from left, and ends at the last intersection of
  OPT with the (i1)th covering line from left.
• Consider the bounding box of OPTi, the smallest
  perimeter aligned rectangle which includes OPTi,
  w for its width and h for its height.

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Proof of step2 case 3.1

• OPTi intersects regions stabbed by two
  consecutive covering lines only l1, l2 say, at
  distance w1.

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Fig 4.i.
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Fig 4.j.
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Proof of step 2 case 3.2

• OPTi intersects regions stabbed by three
  consecutive covering lines only, l1, l2, l3 say
  at distances w1, w2.

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Fig 4.k.
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Fig 4.l.
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Some special cases

• Parallel equal segments
• Convex region

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Part5O(1)-approximation for TSPN on lines
running in linear time
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Lines for Neighborhood

• Infinite lines in the plane
• Find a tour that visits all lines in the plane
• Surprisingly, the problem is not in NP
• Can be computed in linear time O(n6)
• Cause of the high running time, an approximation
  algorithm is expected
• Ratio lt 1.58

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Tour visits all lines
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Polynomial time algorithm

• Convert the problem to watchman route problem
  and solve it in linear time
• Build rectangle covering all intersecting points
  of lines in L
• Grow 2 narrow spikes for every line yielding a
  simple polygon
• Solve the watchman route for 6n4 vertices

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Polynomial time algorithm (cont.)
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Polynomial time algorithm (cont.)
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Polynomial time algorithm (cont.)
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Approximation on TSPN on Lines

• Algorithm Compute a minimum touching circle that
  touches all lines, output the circle as the tour

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Circle touching all lines
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Optimal tour is convex

• Because if it is not convex, convex is better and
  still touching all lines

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Optimal is convex
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Minimum touching circle

• The minimum touching circle is determined by 3
  lines, i.e. inscribed circle of the triangle

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Minimum touching circle
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Minimum touching convex in Triangle

• Case 1 acute triangle
• Case 2 obtuse triangle

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Minimum touching convex in acute triangle
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Minimum touching convex in obtuse triangle
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Algorithm to compute the touching circle
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Algorithm to compute the touching circle

• Solving linear program in fixed dimension is in
  O(n)