Heuristics: Representation of Search Space and Neighborhoods 1

Search Problems in General

The primary difference between an optimization problem and a search problem is, that the search problem concentrates on the variables you have to fill in for the optimal function value, whereas the optimization problem only focuses on the optimal value itself.

Search spaces are defined by the following:

Search Space S: Set of valid transition operators: Initial state from set S: Subset of goal states in S:

See below. Function which describes the valid sub-solutions for the successiv steps. The starting point from which the problem begins to unfold. Describes the boundaries of a valid solution.

Search Spaces

There are multiple definitions of the Search Space. You can divide it into infinite and finite search spaces and in solution and problem spaces.

Finite Search Spaces

The finite search space is characterized by its finite amount of nodes and edges. You can use algorithms or heuristics to find the best solution, but if you have a finite but larger search space it is often more efficient to use heuristic due to the shorter runtime.

Infinite Search Spaces

The infinite search space is characterized by its infinite amount of nodes and edges. You can only use heuristics to find the optimal solution. If one would use algorithms in infinite search spaces, it would not terminate, as there are infinite solution possibilites. Heuristics have the advantage of not computing every last possibility, but rather to narrow down to the best solutions. So heuristics should be used for infinite search spaces.

Solution Spaces

In the solution space, the objective function defines an order of preference among all the possible solutions. There are given potential solutions (e.g. valid tours of a TSP) which are represented by the nodes of the graph. The edges between the nodes illustrate the transition from a given solution to a new (neighboring) solution.

As you can see below, the problem space can be portrayed as a undirected graph.

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Problem Spaces

In the problem space the nodes depict the problems and sub-problems. The root of the tree is the initial problem while the interim nodes are the subproblems. The leaves represent the solutions. The edges are the decomposition of a problem into a simpler sub-problem.

As you can see below, the problem space can be portrayed as a spanning tree.

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Neighborhoods in General (Mathematically)

In directed Graphs

Let Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): G= \left ( V, R, \alpha, \omega \right )

be the directed graph 

with the functions Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \alpha: R \rightarrow V

and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \omega: R \rightarrow V
which define starting and end node.

Let Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): v \in V

be a node of $G$, then you could define:

• Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \delta_{G}^{+}(v) = \left \{ r \in R \mid \alpha(r) = v \right \}

set of outgoing arcs of $v$

• Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \delta_{G}^{-}(v) = \left \{ r \in R \mid \omega(r) = v \right \}

set of incoming arcs of $v$

• Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N_{G}^{+}(v) = \left \{ \omega(r)\mid r \in \delta_{G}^{+}(v) \right \}

set of successors of $v$

• Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N_{G}^{-}(v) = \left \{ \alpha(r)\mid r \in \delta_{G}^{-}(v) \right \}

set of predecessors of $v$

Two nodes Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): u,v \in V

are called adjacent, also known as neighbored, if an arc exists with $\alpha (r)=u$ and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \omega(r)=v
or $\alpha (r)=v$ and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \omega(r)=u

. Therefore you could declare the Neighbordhood of $v$ as Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N(v) = N_{G}^{+}(v) + N_{G}^{-}(v)

In undirected Graphs

Let $G=(V,E,\gamma )$ be the undirected graph with the function Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \gamma : E \rightarrow V * V

which defines the two end nodes of the edge.

Two nodes Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): u,v \in V

are called adjacent, also known as neighbored, if an edge exists with $\gamma (e)=\left\{u,v\right\}$.

Therefore you could declare the Neighbordhood of $v$ as Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): N(v) = \{ u \in V \mid \gamma(e) = \{ u, v \}\} .

Conclusion

Two nodes are neighbored if an arc or an edge connects both nodes.

If you want the neighborhood of a set of nodes, you take a look at the neighborhoods of single nodes and sum them up.

Heuristics in reference to Search Space and Neighborhoods

Opening procedures

The opening procedures construct a feasible solution with the help of algorithms.

A few examples for opening procedures are:

• Nearest Neighbor
• Double-sided nearest Neighbor
• Nearest Addition
• Nearest Insert
• Cheapest Insert
• Farthest Insert

Improving procedures

The improving procedures take these feasible solutions and try to further improve them.

A few examples for improving procedures are:

• 2-Opt
• 3-Opt
• R-Opt
• 2-Nodes-Opt

The Nearest-Neighbor-Algorithm

In General

The algorithm belongs to the category of "greedy-algorithms" and has a runtime of ${\mathcal {O}}(n^{2})$.

Let $G=(V,E,\gamma )$ be the undirected and weighted graph and $T=\{marked$ $nodes\}$

1. Set $T=\{starting$ $node\}$
2. Choose the edge $e=\{u,v\}$ with the least weight from Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): v \in V \setminus T

to the node you chose previously Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): u \in T

1. Add $v$ to $T$
2. Repeat step (2) and (3) until $T=\{all$ $nodes\}$
3. Take the edge which connects the starting node with the previously chosen node

Example

The Search Problem in reference to the example

Search Space S: Set of valid transition operators: Initial state from set S: Subset of goal states in S:

All nodes {A,B,C,D,E,F} - therefore it is a finite Problem Space Take the nearest neighbor of the current node, which has not been visited yet. Otherwise if all nodes have been visited, go back to the starting node E. E You have a cycle {E,C,A,B,D,F,E} through all nodes.

Initial state

Here we have a classic (strongly simplified) Traveling Salesman Problem which we want to solve by using the nearest neighbor heuristic. As the nearest neighbor heuristic is just an opening procedure, we will not necessarily get the optimal value of the objective function and therefore not the best travelling route itself. However, the solution we are going to find will be a more or less good foundation from which you could further improve it by using improving procedures. There are no weighted edges between the nodes shown yet, despite their existing.

initial situation

Iteration steps

Here, we randomly pick one of the nodes (in this case node E) to be our starting node from which we will further iterate. The edges from E to any other node are now presented by the lines and the weights as numbers above them. As you can see, the weights differ from edge to edge and because we want to find a preferably short route through all the edges we choose the edge with the smallest weight from E to another node. We are confronted by a special case of the nearest neighbor heuristic: Two edges share the smallest weight. As the algorithm is not particularly foreshadowing we can now choose randomly between the two smallest edges.

after choosing the first node

We chose node C and are now confronted by new edges between C and all other nodes we have not yet visited. The shortest route beginning from C would now be to A.

iteration 2

Repeat the last steps:

iteration 3	iteration 4
iteration 5	iteration 6

This goes on until we reach the final node that has not yet been visited, in this case node F. Now, as we have a Travelling Salesman Problem, we just need to close the cycle by connecting the last node with the starting node.

final solution

Summary

With this, the algorithm from the nearest neighbor heuristic has ended and we found an opening travelling route from which we can further improve our cycle. The last step would be the addition of the weights of the travelled edges so we get the value of our route.

In this case: 2+2+3+5+1+6 = 19

The solution is presented as: Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): E\rightarrow C \rightarrow A \rightarrow B \rightarrow D \rightarrow F\rightarrow E

Sources

Scripts:

• Operations Research - Prof. Dr. Oliver Wendt (SS 2013)
• Introduction to Network Optimization - Dr. Clemens Thielen (SS 2013)

Books:

• Optimierung, Operations Research, Spieltheorie - Karl Heinz Borgwardt
• Lineare Optimierung und Netzwerkoptimierung - Horst W. Hamacher und Kathrin Klamroth
• Einführung in Operations Research - W. Domschke und A. Drexl

Internet:

• Wikipedia-Entry "Repräsentation des Suchraums"
• Wikipedia-Entry "Suchproblem"
• Wikipedia-Entry "Suchraum"
• Wikipedia-Entry "Suchverfahren"
• Wikipedia-Entry "Nearest-Neighbor-Heuristik"
• Wikipedia-Entry "TSP"

About:

Brought to you by Anna Pfeiffer, Markus Kaiser and Viktor Barie

Comment by group 2

In your wiki- entry you described the theoretical backgrounds of "search space" and "neighborhood" very detailed! You pointed out all aspects of this topic pretty well and we couldn´t find any mistakes in your description. We could have done this in a more detailed way, too. In difference to our entry you focused on the nearest neighbor algorithm which is the simplest one of of the search algorithms. We think if we combine both of our entries (your theoretical part + nearest neighbor algorithm + our VNS ) we will cover the whole topic!