A Hybridised Evolutionary Algorithm for the Multi Criterion Minimum Spanning Tree Problem

1
A Hybridised Evolutionary Algorithm for the Multi
Criterion Minimum Spanning Tree Problem

• Madeleine Davis-Moradkhan Will Browne

2
KEA Aims

• to use local search to find extreme points of the
  Pareto front in order to guide an EA for the
  MCMST problem in order to find an approximate
  Pareto front an order of magnitude faster than
  enumeration or conventional EC approaches whilst
  still sampling many possible solutions on the
  true PF

3
Organization

• Definitions
• Applications Why study MCMST
• Problem and Solution Approaches
• Major Aims
• Proposed Algorithm
• Results
• Conclusions

4
Network, is a collection of objects connected to
each other in some fashion
5
Examples

• People in a Network of friendship
• Servers in the Network of the world web
• Cities in a Network of highways
• Stations in a Network of railroads
• Sites in a Network of pipelines
• Power stations in an electrical Network
• Pins in an electrical circuit

6
Networks are represented by Graphs

• Graph G (N, E)
• N Set of n Nodes (Points)
• E Set of m Edges (Lines) with Costs

7
Spanning Tree T (N, E )

• is a subgraph of G containing all the n nodes of
  G but a subset of the edges without cycles (E
  n -1) such that if any of the edges is deleted
  T would become disconnected, and with any
  additional edge a cycle would be created

8
Optimisation Minimisation

• If costs are associated to the links joining the
  objects in a network.
• The objective would be to find the most
  economical way to join the objects.
• The most economical way of joining the objects is
  a minimum spanning tree (MST).

9
Single-objective Minimisation

• When only one set of cost is associated to each
  edge, the minimum spanning tree can be found
  easily. For example, using Kruskals algorithm
  this MST is found in polynomial time.

10
Solution Approaches

• Find the minimum spanning tree to find the most
  economical solution
• Straight forward for single objective problems,
  but
• Real life problems involve more than one cost and
  have constraints

11
Real Life Problems

• Building a network of roads to connect n towns
  such that the cost of building and the travel
  times are minimized.
• Designing a layout for telecommunication system
  between n locations so as to minimize time of
  communication and maximize reliability.
• Designing an electric network between n
  terminals so as to minimize stray effects and
  maximize reliability.

12
More Real Life Problems

• Building a pipeline network between n cities such
  that the length of pipes is minimized and
  environmental constraints are met.
• Designing a confidential network of
  communication between n clients so as to minimize
  the probability of messages becoming known to
  outsiders.
• Designing an electrical wiring board with n pins
  such that as little wire as possible is used and
  a fixed number of wires meet any pin.

13
Multi-Criterion Optimisation

• Multi Criterion Minimum Spanning Tree (MCMST)
• Combinatorial Optimization Problem in Graphs,
  rarely admits one solution
• NP-Hard (if G is complete it has n n-2 Spanning
  Trees)
• Solution Procedure Heuristics
• Deterministic
• Probabilistic (Evolutionary)

14
Example Equivalent MCMSTs
15
Nondominated MCMSTs The Pareto Front

• MCMST Total Costs
• MCMST0 (19, 11) extreme
• MCMST 1 (24, 8) extreme
• MCMST 2 (21, 10)
• MCMST 3 (21, 10)
• MCMST 4 (22, 9)

16
Scale of the problem

• Complete enumeration is impractical for n gt 10

17
Approximate Pareto Set (APS)

• Find an approximate set of trade-off solutions
• the Approximate Pareto Set consists of all or
  some of the nondominated solutions
• Useful to have as many nondominated solutions as
  possible (e.g. in overlay mesh construction for
  broadcasting and multi-path routing where
  edge-disjoint MCMSTs are required so that a
  faulty route can quickly be replaced by another
  that is just as economical).

18
Supported Efficient Solutions

• Most algorithms use an aggregated objective
  function by assigning weights to every objective
  and find some or all of the supported efficient
  solutions, but do not find the unsupported
  efficient solutions

19
Proposed Algorithm KEA

• Knowledge-Based Evolutionary Algorithm
• Finds both the supported and non-supported
  efficient solutions
• Is scalable to n gt 500
• Is scalable to more than 2 dimensions
• Is validated against an exhaustive search
• is faster and more efficient than NSGA-II in
  terms of spread and number of solutions found

20
The main features of KEA

• include the application of deterministic
  approaches to calculate the extreme points of the
  Pareto front. These are used to produce the
  initial population comprising of an elite set of
  parents. An elitist evolutionary search attempts
  to find the remaining Pareto optimal points by
  applying a knowledge-based mutation operator. The
  domain knowledge is based on the K-best
  approaches in deterministic methods.

21
Phases of KEA

• Phase 1 Find the extreme point MSTs using
  classical single-objective optimisation
• Phase 2 Create the initial elite population, the
  genotypic neighbours of the extreme MSTs, by
  mutating only one edge of the extreme MSTs at a
  time for each new solution
• Phase 3 Evolutionary phase where a
  knowledge-based mutation operator creates as many
  offspring as possible from the single and
  randomly selected parent in each generation

22
Code of KEA
23
Results 1 Validation

• Pareto fronts for a graph with 10 nodes and anti-
  correlated costs, found by EXS (left) and KEA
  (right) are identical.
• coefficient of correlation -0.7, (KEA g 40)
• EXS Exhaustive Search algorithm of Christofides

24
Results 2 Speed
25
Results 3 Efficient Approximate Pareto Sets
Superiority of KEA in spread, front occupation
and dominance

• Fig. 1 Attainment Surface plots for a graph with
  100 nodes and anti-correlated costs where the
  execution time has been limited to that of KEA
  approximate CPU 698.3 sec 12 min.
• Fig. 2 Attainment surface plots for a graph with
  100 nodes and random costs, where the execution
  time has been limited to that of KEA approximate
  CPU
• 515.5 sec 9 min.

26
Three Dimensions

• Best Attainment Surface plot obtained by KEA for
  a Graph with 100 Nodes, Random costs and three
  criteria
• resolution 14,
• g 30,000

27
Conclusions

• Hybridisation is used across the three phases of
  KEA making it fast and scalable.
• KEA does not have the limitations of previous
  algorithms because of its speed, its scalability
  to more than two criteria, and its ability to
  find both the supported and unsupported optimal
  solutions.
• It can be used to determine optimal and near
  optimal solutions to problems in the design,
  operations and management of networks, for
  example, large scale telecommunication networks,
  and spatial networks for commodity distribution.