Euler tours, postman tours and mixed graphs

1 © B.Raghavachari & J.Veerasamy, UTD

Euler tours, postman tours and mixed graphs

Jeyakesavan Veerasamy*

* Joint work with Balaji Raghavachari

Samsung Telecommunications America, Inc. and

The University of Texas at Dallas

[email protected]


Approximation Algorithms

• An algorithm for solving a minimization problem that– runs in polynomial time and outputs a feasible solution S

– performance ratio = Cost(S) / Cost(Opt)

• In practice, if problem is NP-hard (intractable), – we can apply exact algorithms (if time permits), or

– apply heuristic algorithms and approximation algorithms, and select best solution obtained.

• Critical part of approximation algorithm analysis is lower-bound analysis of optimal solution

• Such lower bounds can be used in practice to speed up exact algorithms such as branch and bound, and evaluate performance of heuristic algorithms.


Euler Tour Problem

Find a tour that traverses all edges exactly once.

Even degree node

A

Input Output

A graph is Eulerian if and only if it is connected andthe degree of each vertex is even (Euler, 1736!)


Non-Eulerian graph

A

B

Even degree node

Odd degree node

Walk starting at A got stuck at B


Directed Euler Tour Problem

A

Balanced node

Find a tour in a directed graph that traverses all edges exactly once.

A directed graph is Eulerian if and only if it is strongly connected and indegree = outdegree at each vertex


Non-Eulerian Directed Graph

A B

Balanced node

Deficit node

Surplus nodeWalk starting at A got stuck at B


Chinese Postman Problem (CPP)

Find shortest tour visiting all edges at least once.

Even-degree node

Odd-degree node

A

Input Augmented Output

Add a minimum-weight T-join which has odd degree at odd-degree nodes and even-degree at even-degree nodes


Directed CPP

Balanced node

Deficit node

Surplus node

Input Output

Add a minimum-weight subgraph which has outdegree = indegree + surplus at surplus nodesindegree = outdegree + deficit at deficit nodes


Directed CPP: Another example

Balanced node

Deficit node

Surplus node

Output

Input

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© B.Raghavachari & J.Veerasamy, UTD

Mixed Postman Problem (MPP)

Balanced node

Unbalanced node

Solution 1 Solution 2

Includes both undirected and directed edges

Input

NP-hard!

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Applications: City Management

Elm

Commerce

Main

JacksonHouston

Street cleaning, Mail delivery, Snow removal, Courier delivery route

12


Previous work

• Edmonds, Johnson, 1973

• Frederickson, 1979: – Designed two approximation algorithms

– Select the best of the two solutions computed

– Worst-case performance ratio = 5/3

• Christofides, 1984

• Ralphs, 1993: Linear Programming formulation

• Nobert, Picard, 1996: Applied LP-based solution to schedule snow removal in Montreal

• Raghavachari, Veerasamy, 1998: – Improved Frederickson’s algorithm for MPP

– Performance ratio = 3/2

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Properties of Eulerian Graphs• Even-degree condition: degree of each node is even

• Balanced set condition: For any cut (S, V \ S), the difference between number of directed edges crossing cut in each direction is at most the number of undirected edges crossing cut

• Eulerian mixed graphs can be detected in polynomial time

S V \ S

Given a graph satisfying even-degree condition, Frederickson showed how to find minimum-cost augmentation to make it Eulerian.

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Even-degree mixed graphs

• Solve flow problem and obtain IN = OUT at every node• Consider undirected edges and duplicated edges only• Traverse cycles formed by these edges and correct parity Balanced node

Unbalanced node

15


Input G

Frederickson’s Algorithm (Part 1)

Match Flow

H

Evenparity

Solution S1

C(H) C(Opt)

C(S1) C(Opt) + 2C(D) Balanced node

Unbalanced node

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Frederickson’s Algorithm (Part 2)

Input G

Flow Double

C(Opt) C(D) + C(U)

C(S2) C(D) + 2C(U) 2C(Opt) - C(D)

Solution S2

Solution S = min(S1, S2)

C(S) 5/3 C(Opt)

D

U

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Remarks

• Input GFlow

IN=OUT

Directed edges DUndirected edges U

• Cost(Opt) C(D) + C(U)

• D may contain any number of copies of edges in G.

• H is obtained from G by adding a T-join of odd-degree nodes, ignoring the directions.

• If H-G contains only edges of D:

C(S1) 2C(D) + C(U)

C(S) (3/2) C(Opt)

• Flow augmentation cost CIO(G) = C(D) + C(U) - C(G)

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Idea for Improvement

• Modify computation of S1 as follows:– When H is computed, make cost of edges in D to be 0.

– T-join prefers edges of D to U.

• Problem: We may be forced to duplicate some edges of U.

Components of D may have odd-number of odd-degree nodes

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Improved lower bound

Shrink each directed componentto a single node, and compute T-join between odd degree components.

X, subset of U, forms the T-join. Optimal solution also has to pay C(X) to satisfy even degree condition.

If we can prove that adding X to G does not decrease Flow augmentation cost, thenC(Opt) C(D) + C(U) + C(X)

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Lower bound proof

Input G New input HINOUT solution INOUT solution

Can CIO(H) < CIO(G)?

No. In fact, adding any edge in U does not decrease INOUT augmentation cost.

Alternating cycle

x

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Our Mixed algorithm (part 2)

Input G

Flow Double

C(Opt) C(D) + C(U)

C(S2) C(D) + 2C(U) 2C(Opt) - C(D)

Solution S2

D

U

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Input G

Our Mixed Algorithm (part 1)

Match Flow

H

Evenparity

Solution S1

C(S1) C(U) + C(X) + 2C(D)

C(Opt) + C(D) Balanced node

Unbalanced node

Solution S = min(S1, S2)

C(S) 3/2 C(Opt)

X

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Performance of Algorithms

C(D)

Cost of solution

C(Opt)

2C(Opt) C(new S1)

C(S2)

C(S1)

C(Opt)

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Concluding Remarks

• Worst case performance ratio for our algorithm is 1.5 (Analysis is tight).

• Windy Postman Problem: Asymmetric version of MPP

• LP relaxation of WPP is half-integral - gives 2 approximation algorithm.

• Recently we have improved ratio of WPP to 1.5

• Open: Improvements? Planar Graphs?

U VC(U,V) C(V,U)

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Further Research

• Explore MPP with additional constraints:– duplicate only undirected edges

– duplicate only directed edges

• Explore shortest path problem in mixed graphs when negative costs are allowed.

• Implement a few heuristic algorithms and analyze the performance.

Euler tours, postman tours and mixed graphs

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