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Postman ProblemsAssume and is connected.

Thm 5.23: A connected graph has an Euler tour if and only if every node of has even degree.

Let be the number of extra traversals of edge in a postman tour. Construct the graph by making copies of for each .

5.4 T-joins and Postman Problems

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Postman Problem is equivalent to the problem:

Minimize (5.30)subject to (mod 2), for all

, for all . integer, for all .

There is an optimal solution for which is -valued. (since )

We call a set a postman set of if, for every , is incident with an odd number of edges from iff has odd degree in .

Postman Problem

Given: A graph and such that .Objective: To find a postman set such that is minimum.

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(T-Joins)

Let be a graph, and let such that is even.A T-join of is a set of edges such that

(mod 2), for all

is a -join iff the odd-degree nodes of the subgraph are exactly the elements of .

Optimal T-Join Problem

Given: A graph , a set such that is even, and a cost vector .Objective: Find a -join of such that is minimum.

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Examples: Postman sets: Let is odd} ( need )

Even set:Even set is a set such that every node of has even degree. Let .If costs are nonnegative, is optimal.A set is even iff it can be decomposed into edge-sets of edge-disjoint circuits.Hence is optimal iff has no negative-cost circuit. (can find a negative-cost circuit or determine that none exists.)

-paths:Let , and .Every -join contains the edge-set of an -path (otherwise, the component of the subgraph containing has only one node of odd degree, which is impossible.). So minimal -joins are edge-sets of simple -paths.

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Deleting a simple -path from a (not minimal) -join , we obtain even sets. Hence if the graph contains no negative-cost circuit, optimal -join (and mini-mal) is minimum cost simple -path.(can solve the shortest simple -path problem for undirected graphs when nega-tive edge costs are allowed, but negative cost circuits are not.)

Prop 5.24: Let be a -join of . Then is a -join of if and only if is a -join of .(Pf) ( ) Suppose is a -join and is a -join. Let . Then is even (mod 2)

is an element of neither or both of and . .

() apply "only if" part with replaced by and replaced by .

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Solving the Optimal T-Join Problem

Assume . Then there is always an optimal -join that is minimal. Problems with negative costs can be transformed into nonnegative costs case.

Prop 5.25: Every minimal -join is the union of the edge-sets of edge-disjoint simple paths, which join the nodes in in pairs.(Pf) is a -join.Now, enough to show that any -join contains such a set of edge-disjoint paths. Let , and let be the component of that contains . Then there is a node in . (otherwise is the only node in of odd degree)So there is a simple path such that . Now is a -join, where by Proposition 5.24. Repeat the argument.

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Suppose an optimal -join is expressed as union of edge-sets of paths and is one of these paths, with joining . Then is a minimum cost -path.Suppose there is a -path in that has smaller cost than . By Proposition 5.24, is a -join. Since , its cost is

,a contradiction. Hence

Prop 5.26: Suppose that . Then there is an optimal -join that is the union of edge-disjoint shortest paths joining the nodes of in pairs.

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For any pair of nodes in , let be the cost of a least cost -path in . Let . The minimum cost -join is (assuming )

minimize s.t. is a pairing of the elements of .

Form a complete graph , give edge weight , and find a minimum-weight per-fect matching of . Join the selected pairs in using shortest paths. If some edges overlap (since allowed), take symmetric difference.

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Negative Costs

Given , let . Let be the set of nodes of that have odd degree in the sub-graph . Then is a -join.By proposition 5.24, is a -join is a -join.

.( is the vector defined by .)

is a constant that does not depend on . Hence is an optimal -join w.r.t. cost vector

is an optimal -join w.r.t. cost vector .

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Optimal T-Join Algorithm

Step 1. Identify the set of edges having negative cost, and the set of nodes incident with an odd number of edges from . Replace by and by .

Step 2. Find a least-cost -path w.r.t. cost vector for each pair of nodes from . Let be the cost of .

Step 3. Form a complete graph with having weight for each . Find a minimum-weight perfect matching in .

Step 4. Let be the symmetric difference of the edge-sets of paths for .Step 5. Replace by .

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