Algorithms for Network Optimization Problems

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Algorithms for Network Optimization Problems

• This handout
• Minimum Spanning Tree Problem
• Approximation Algorithms
• Traveling Salesman Problem

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Terminology of Graphs

• A graph (or network) consists of
• a set of points
• a set of lines connecting certain pairs of the
  points.
• The points are called nodes (or vertices).
• The lines are called arcs (or edges or links).
• Example

3
Terminology of Graphs Paths

• A path between two nodes is a sequence of
  distinct nodes and edges connecting these nodes.
• Example

a
b
4
Terminology of Graphs Cycles, Connectivity and
Trees

• A path that begins and ends at the same node is
  called a cycle.
• Example
• Two nodes are connected if there is a path
  between them.
• A graph is connected if every pair of its nodes
  is connected.
• A graph is acyclic if it doesnt have any cycle.
• A graph is called a tree if it is connected and
  acyclic.
• Example

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Minimum Spanning Tree Problem

• Given Graph G(V, E), Vn
• Cost function c E ? R .
• Goal Find a minimum-cost spanning tree for V
• i.e., find a subset of arcs E ? E which
• connects any two nodes of V
• with minimum possible cost.
• Example

Min. span. tree G(V,E)
G(V,E)
Red bold arcs are in E
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Algorithm for solving the Minimum Spanning Tree
Problem

• Initialization Select any node arbitrarily,
• connect to its nearest node.
• Repeat
• Identify the unconnected node
• which is closest to a connected node
• Connect these two nodes
• Until all nodes are connected
• Note Ties for the closest node are broken
  arbitrarily.

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The algorithm applied to our example

• Initialization Select node a to start.
• Its closest node is node b. Connect nodes a
  and b.
• Iteration 1 There are two unconnected node
  closest to a connected node nodes c and d
• (both are 3 units far from node b).
• Break the tie arbitrarily by
• connecting node c to node b.

Red bold arcs are in E thin arcs represent
potential links.
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The algorithm applied to our example

• Iteration 2 The unconnected node closest to a
  connected node is node d (3 far from node b).
  Connect nodes b and d.
• Iteration 3 The only unconnected node left is
  node e. Its closest connected node is node c
• (distance between c and e is 4).
• Connect node e to node c.
• All nodes are connected. The bold
• arcs give a min. spanning tree.

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• Recall Classes of discrete optimization
  problems
• Class 1 problems have polynomial-time algorithms
  for solving the problems optimally.
• Ex. Min. Spanning Tree problem
• Assignment Problem
• For Class 2 problems (NP-hard problems)
• No polynomial-time algorithm is known
• And more likely there is no one.
• Ex. Traveling Salesman Problem
• Coloring problem

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• Three main directions to solve
• NP-hard discrete optimization problems
• Integer programming techniques
• Heuristics
• Approximation algorithms
• We gave examples of the first two methods for
  TSP.
• In this handout,
• an approximation algorithm for TSP.

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Definition of Approximation Algorithms

• Definition An a-approximation algorithm is a
  polynomial-time algorithm which always produces a
  solution of value within a times the value of an
  optimal solution.
• That is, for any instance of the problem
• Zalgo / Zopt ? a , (for a minimization problem)
• where Zalgo is the cost of the algorithm
  output,
• Zopt is the cost of an optimal solution.
• a is called the approximation guarantee (or
  factor) of the algorithm.

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Some Characteristics of Approximation Algorithms

• Time-efficient (sometimes not as efficient as
  heuristics)
• Dont guarantee optimal solution
• Guarantee good solution within some factor of the
  optimum
• Rigorous mathematical analysis to prove the
  approximation guarantee
• Often use algorithms for related problems as
  subroutines
• Next we will give
• an approximation algorithm for TSP.

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An approximation algorithm for TSP

• Given an instance for TSP problem,
• Find a minimum spanning tree (MST) for that
  instance.
• (using the algorithm of the previous handout)
• To get a tour, start from any node and traverse
  the arcs of MST by taking shortcuts when
  necessary.
• Example
• Stage 1 Stage 2

red bold arcs form a tour
start from this node
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Approximation guarantee for the algorithm

• In many situations, it is reasonable to assume
  that triangle inequality holds for the cost
  function c E ? R defined on the arcs of network
  G(V,E)
• cuw ? cuv cvw for any u, v, w ?V
• Theorem
• If the cost function satisfies the triangle
  ineqality,
• then the algorithm for TSP
• is a 2-approximation algorithm.

v
w
u
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Approximation guarantee for the algorithm (proof)

• First lets compare the optimal solutions of MST
  and TSP for any problem instance G(V,E), c E ?
  R .
• Idea Get a tour from Minimum spanning tree
  without increasing its cost too much (at most
  twice in our case).

Optimal MST sol-n
Optimal TSP sol-n
A tree obtained from the tour
()
Cost (Opt. TSP sol-n)
Cost (of this tree)
Cost (Opt. MST sol-n)


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Approximation guarantee for the algorithm (proof)
red bold arcs form a tour

• The algorithm
• takes a minimum spanning tree
• starts from any node
• traverse the MST arcs
• by taking shortcuts when necessary
• to get a tour.
• What is the cost of the tour compared to the cost
  of MST?
• Each tour (bold) arc e is a shortcut
• for a set of tree (thin) arcs f1, , fk
• (or simply coincides with a tree arc)

start from this node
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Approximation guarantee for the algorithm (proof)

• Based on triangle inequality,
• c(e) ? c(f1)c(fk)
• E.g, c15 ? c13 c35
• c23 ? c23
• But each tree (thin) arc
• is shortcut exactly twice. ()
• E.g., tree arc 3-5 is shortcut by tour arcs 1-5
  and 5-6 .
• The following chain of inequalities concludes the
  proof,
• by using the facts we obtained so far

red bold arcs form a tour
start from this node
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Performance of TSP algorithms in practice

• A more sophisticated algorithm (which again uses
  the MST algorithm as a subroutine) guarantees a
  solution within factor of 1.5 of the optimum
  (Christofides).
• For many discrete optimization problems, there
  are benchmarks of instances on which algorithms
  are tested.
• For TSP, such a benchmark is TSPLIB.
• On TSPLIB instances, the Christofides algorithm
  outputs solutions which are on average 1.09 times
  the optimum.
• For comparison, the nearest neighbor algorithm
  outputs solutions which are on average 1.26 times
  the optimum.
• A good approximation factor often leads to good
  performance in practice.