ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Section number 5643
• Lecture 6

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Last class

• Bipartite (maximum cardinality) matching
• can be solved efficiently by augmenting
• more efficiently, by max flow solver
• Bipartite weighted matching
• Assignment problem
• primal-dual method

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Augmenting path and matching

• Given a matching M and an augmenting path P
  w.r.t. M, M M ? P (M?P) \ (M?P) is a
  matching w/ M gt M
• A matching M is optimal iff there is no
  augmenting path w.r.t. M

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Bipartite matching

• Given a bipartite graph B (V, U, E), find a
  maximum size matching in B (maximum cardinality
  matching)
• Start from a matching M (possibly empty),
• if an augmenting path P is found, augment M by
  replacing P?M by P\M
• if no augmenting path is found, M is optimal
• Augmenting path is of odd length and the graph is
  bipartite, one exposed node is in V and the other
  in U

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Idea of algorithm

• Idea
• systematically search for an augmenting path by
  enumerating all possible alternating paths from
  an exposed node
• breadth-first search simultaneous search

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Example
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Auxiliary digraph
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• Algorithm in Fig 10-3, PS p.224
• O(min(V, U)E) for B (V, U, E)

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Nonbipartite matching

• Can we use bipartite matching solver for
  nonbipartite matching?
• an augmenting path corresponds to a path in
  auxiliary digraph from an exposed node to a
  target node
• But the converse does not hold!

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Odd cycles in nonbipartite graphs

• the only difference between bipartite and
  nonbipartite graphs
• folding
• when? maximum matched edges... (blossom)

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Blossom

• cycle w/ 2k1 nodes and k matched edges
• 2 paths from v
• basis latest common matched node the only
  node not matched with another node in the
  blossom
• how to detect?
• mate of current node is already matched (matched
  edge (5, 6) is used twice)

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Shrinking a blossom

• replacing a blossom b with a single node
• G (V, E) ? G/b (V/b, E/b)
• shrinking does not add / omit augmenting paths!

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Shrinking and alternating path

• If while searching for an augmenting path from an
  exposed node u in G w.r.t. M, we discovered a
  blossom b, then there is an alternating path from
  u to any node of b ending with a matched edge.

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Shrinking and augmenting path

• If while searching for an augmenting path from an
  exposed node u in G w.r.t. M, we discovered a
  blossom b, then there is an augmenting path from
  u in G w.r.t. M iff there is one from u in G/b
  w.r.t. M/b.
• ?

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Shrinking and augmenting path

• ?

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Algorithm

• Start from an empty matching and repeatedly
  choose an exposed vertex u to find an augmenting
  path from it
• If a blossom is found, shrink it and continue
  search
• If no augmenting path is found, u will never be
  chosen again, choose a different exposed node
• If augmenting path is found, reconstruct blossom
  and backtrack the path and augment the matching

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Example
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Example
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Analysis of algorithm

• In G, there is no augmenting path starting from
  an exposed node u w.r.t. M. Let p be an
  augmenting path w/ endpoints two other exposed
  nodes v and w. Then there is no augmenting path
  from u w.r.t. M ? P either.

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Analysis of algorithm

• If at some state there is no augmenting path from
  a node u, then there will never be augmenting
  path from u.
• Fig 10-13, PS p. 237
• O(V4) time complexity

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Dynamic programming

• Basic idea
• Principle of optimality piece of optimal
  solutions are themselves optimal
• To find an optimal solution, first find optimal
  subsolution using recursion (define recurrence)
• Example Shortest path problem
• If the shortest path from s to t passes p,
  subpaths (s, p) and (p, t) are also shortest
  paths.
• d(v) length of shortest path from s to v,
  thend(v) mini ?V-(v) d(i) civ

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Example

• Shortest path problem
• recurrence
• Dk(i) length of a shortest path from s to i
  containing at most k arcs
• Dk(i) min Dk-1(i), mini ?V-(v) d(i) civ
• Increasing k from 1 to n-1, using the above
  recurrence, find optimal subsolutions
• takes O(mn) time and d(j) Dn-1(j)
• In order to backtrack the path, we need to keep a
  matrix containing the chosen subsolution

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Example
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Uncapacitated lot-sizing

• nonnegative demands dtnt1, costs of production
  ptnt1, storage htnt1, fixed set-up cost
  ftnt1
• Find a minimum cost production plan
• min ?nt1 ptxt ?nt1 htst ?nt1 ftyt
  st-1 xt dt st for 1 ? t ? n
  xt ? Myt for 1 ? t ? n s0 sn 0, s ?
  Rn1, x ? Rn, y ? Bn

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ULS

• two important structural properties
• There exists an optimal solution s.t. st-1xt 0
  for all t.
• There exists an optimal solution s.t. xt gt 0,xt
  ?tkit di for some k ? 0.
• dit ?tji dj
• st ?ti1 xi d1t?nt1 ptxt ?nt1 htst
  ?nt1 ptxt ?nt1 ht(?ti1 xi d1t)
  ?nt1 ctxt ?nt1 htd1t
  
  ct pt ?nit hiobjective function ?nt1
  ctxt ?nt1 ftyt ?nt1 htd1t
• recurrence
• H(k) minimum cost of a solution for periods 1,
  ..., k
• H(k) min1 ? t ? k H(t-1) ft ctdtk, H(0)
  0

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Example

• n 4, d (2, 4, 5, 1), p (3, 3, 3, 3),h
  (1, 2, 1, 1), f (12, 20, 16, 8)
• c (8, 7, 5, 4), (d11, d12, d13, d14) (2, 6,
  11, 12)?4t1htd1t 37
• H(1) f1 c1d11 28
• H(2) min H(0) f1 c1d12, H(1) f2
  c2d22 min H(1) c1d2, H(1) f2
  c2d2 min 60, 76 60
• H(3) min H(0) f1 c1d13, H(1) f2
  c2d23, H(2) f3 c3d33
  min 60 c1d3, 76 c2d3, H(2) f3
  c3d3 min 100, 111, 101 100
• H(4) min H(0) f1 c1d14, H(1) f2
  c2d24, H(2) f3 c3d34,
  H(3) f4 c4d4 min 100 c1d4, 111
  c2d4, 101 c3d4, H(3)f4c4d4 min
  108, 118, 106, 112 106
• H(4) 106 H(2) f3 c3d34. y3 1, x3 6,
  y4 x4 0H(2) f1 c1d12, y1 1, x1 6, y2
  x2 0
• x (6, 0, 6, 0), y (1, 0, 1, 0), s (4, 0, 1,
  0)original cost 106 37 69 6p1 f1 6p3
  f3 4h1 h3 69

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Example

• As a shortest path problem...
• for (i, j), the cost fi1 ci1di1, j

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Today

• Nonbipartite matching
• blossom makes it difficult
• detect / shrink blossom and find augmenting path
• Dynamic programming
• principle of optimality
• examples shortest path problem, ULS