Linear Programming Optimization

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5.4 T-Joins and Postman Problems

• Postman Problems
• Assume c ? 0 and G is connected.
• Thm 5.23 A connected graph G has an Euler tour
  if and only if every node of G has even degree.
• Let xe be the number of extra traversals of edge
  e in a postman tour. Construct the graph Gx by
  making 1 xe copies of e for each e.

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• Postman Problem is equivalent to the problem
• Minimize ?(cexe e ?E) (5.30)
• subject to x(?(v)) ? ?(v) (mod 2),
  for all v ?V
• xe ? 0, for all e ?E.
• xe integer, for all e ?E.
• There is an optimal solution for which x is 0,
  1-valued. (since c ? 0)
• We call a set J ? E a postman set of G if, for
  every v ?V, v is incident with an odd number of
  edges from J iff v has odd degree in G.

Postman Problem Given A graph G (V, E) and
c ?RE such that c ? 0. Objective To find a
postman set J such that c(J) is minimum.
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(T-Joins)

• Let G (V, E) be a graph, and let T ?V such that
  T is even.
• A T-join of G is a set J of edges such that
• J??(v) ? T?v (mod 2), for all v
  ?V
• J is a T-join iff the odd-degree nodes of the
  subgraph (V, J) are exactly the elements of T.

Optimal T-Join Problem Given A graph G (V,
E), a set T ?V such that T is even, and a cost
vector c ?RE . Objective Find a T-join J of G
such that c(J) is minimum.
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• Examples
• Postman sets Let T v ?V ?(v) is odd (
  need c ? 0)
• Even set
• Even set is a set A ?E such that every node of
  (V, A) has even degree. Let T ?.
• 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 G has no negative-cost
  circuit.
• (can find a negative-cost circuit of determine
  that none exists.)
• (r, s)-paths
• Let r, s ?V, and T r, s.
• Every T-join J contains the edge-set of an (r,
  s)-path (otherwise, the component of the
  subgraph (V, J) containing r has only one node of
  odd degree, which is impossible.). So minimal
  T-joins are edge-sets of simple (r, s)-paths.

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• Deleting a simple (r, s)-path from a ( not
  minimal) T-join J, we obtain even sets. Hence if
  the graph contains no negative-cost circuit,
  optimal T-join (and minimal) is minimum cost
  simple (r, s)-path.
• (can solve the shortest simple (r, s)-path
  problem for undirected graphs when negative edge
  costs are allowed, but negative cost circuits are
  not.)
• Prop 5.24 Let J' be a T'-join of G. Then J is
  a T-join of G if and only if J?J' is a
  (T?T')-join of G.
• (Pf) ( gt) Suppose J is a T-join and J' is a
  T'-join. Let v ?V.
• Then (J?J') ??(v) is even ltgt J??(v) ?
  J'??(v) (mod 2)
• ltgt v is an element of neither or both of T
  and T'.
• ltgt v? T?T'.
• (lt) apply "only if" part with J replaced by
  J?J' and T replaced by T?T'. ?

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Solving the Optimal T-Join Problem

• Assume c ? 0. Then there is always an optimal
  T-join that is minimal. Problems with negative
  costs can be transformed into nonnegative costs
  case.
• Prop 5.25 Every minimal T-join is the union of
  the edge-sets of T/2 edge-disjoint simple
  paths, which join the nodes in T in pairs.
• (Pf) ?Pi ? edge-sets of min T-joins
• To show reverse inclusion, show any T-join
  contains such a set of edge-disjoint paths.
• Let u ? T, and let H be the component of (V, J)
  that contains u. Then there is a node v ?u in T
  ? H. (otherwise u is the only node in H of odd
  degree)
• So there is a simple (u, v) path P such that
  E(P) ? J. Now J\E(P) is a T'-join, where T'
  T\u, v by Proposition 5.24. Repeat the
  argument. ?

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• Suppose an optimal T-join is expressed as union
  of edge-sets of paths and P is one of these
  paths, with P joining u, v ?T. Then P is a
  minimum cost (u, v)-path.
• Suppose there is a (u, v)-path P' in G that has
  smaller cost than P. By Proposition 5.24, J
  ?E(P) ?E(P') is a T-join. Since E(P) ?J, its
  cost is
• c(J\E(P)) c(E(P') - 2c( (J\E(P)) ? E(P') )
• ? c(J) - c(E(P)) c(E(P')) lt c(J),
• a contradiction. Hence
• Prop 5.26 Suppose that c ? 0. Then there is an
  optimal T-join that is the union of T/2
  edge-disjoint shortest paths joining the nodes of
  T in pairs.

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• For any pair u, v of nodes in T, let d(u, v) be
  the cost of a least cost (u, v)-path in G. Let
  T 2k. The minimum cost T-join is (assuming
  c ? 0)
• minimize ?( d(ui, vi) i 1, , k)
• s.t. u1v1, , ukvk is a pairing of the
  elements of T.
• Form a complete graph G (T, E), give edge uv
  weight d(u, v), and find a minimum-weight perfect
  matching of G. Join the selected pairs in G
  using shortest paths. If some edges overlap
  (since ce 0 allowed), take symmetric
  difference.

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Negative Costs

• Given c ?RE, let N e ?E ce lt 0. Let T' be
  the set of nodes of G that have odd degree in the
  subgraph (V, N). Then N is a T'-join.
• By proposition 5.24, J is a T-join ltgt J ?N
  is a (T ?T')-join.
• c(J) c(J\N) c(J ?N)
• c(J\N) - c(N\J) c(N\J) c(J ?N)
• c(J ?N) c(N).
• ( c is the vector defined by ce ce.)
• c(N) is a constant that does not depend on J.
  Hence
• J is an optimal T-join w.r.t. cost vector c
• ltgt J ?N is an optimal (T ?T')-join w.r.t.
  cost vector c.

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Optimal T-Join Algorithm Step 1. Identify the
set N of edges having negative cost, and the set
T' of nodes incident with an odd number of edges
from N. Replace c by c and T by T ?T'. Step 2.
Find a least-cost (u, v)-path Puv w.r.t. cost
vector c for each pair u, v of nodes from T. Let
d(u, v) be the cost of Puv. Step 3. Form a
complete graph G (T, E) with uv having weight
d(u, v) for each uv ?E. Find a minimum-weight
perfect matching M in G. Step 4. Let J be the
symmetric difference of the edge-sets of paths
Puv for uv ?M. Step 5. Replace J by J ?N.
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c-optimal T-join