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Postman Problems Assume and is connected.

5.4 T-joins and Postman Problems. Postman Problems Assume and is connected. Thm 5.23: A connected graph has an Euler tour if and only if every node of has even degree.

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Postman Problems Assume and is connected.

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  1. 5.4 T-joins and Postman Problems • Postman Problems Assume 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 . 0 1 0 1 0 1 0 0 1 0 , x, and

  2. 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.

  3. (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.

  4. 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 iffhas 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.

  5. 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 minimal) is minimum cost simple -path. (can solve the shortest simple -path problem for undirected graphs when negative 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 . 

  6. 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. 

  7. 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.

  8. 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 perfect matching of . Join the selected pairs in using shortest paths. If some edges overlap (since allowed), take symmetric difference. 2 8 1 1 1 4 3 4 2 2 4 1 3 9

  9. 2 2 8 5 5 6 7 3 6 1 4 1 1 5 4 4 3 3 3 4 2 2 4 7 1 3 9 3 2 8 1 1 1 4 3 4 2 2 4 Optimal -join 1 3 9

  10. Negative Costs • Given , let . Let be the set of nodes of that have odd degree in the subgraph. 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 .

  11. 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 .

  12. 2 2 2 2 -2 2 5 5 -3 3 6 6 1 1 1 1 2 2 2 2 -optimal -join -optimal -join

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