Finding Minimum-Cost Circulations by Canceling Negative Cycles

1
Finding Minimum-Cost Circulations by Canceling
Negative Cycles
ANDREW V. GOLDBERG ROBERT E. TARJAN

• Advanced Algorithms Seminar
• Instructor Prof. Haim Kaplan
• Presentation Moshik Guttmann

2
Definitions

• A circulation network is a symmetric graph
  G(V,E) with a real value capacity u(v,w)
  function and a cost c(v,w) function for each arc
  s.t.
• c(v,w) -c(w,v)
• (cost anti-symmetry constraint)
• A circulation is a function s.t.
  for each
• f(w,v) u(w,v) (capacity constraint)
• f(w,v) f(v,w) (flow anti-symmetry constraint)
• (conservation
  constraints).
• The cost of a circulation is
• The minimum-cost circulation problem is that of
  finding a circulation of minimum cost

3
Definitions

• For a circulation f and an arc (v, w) the
  residual capacity of (v, w) is
• uf(v,w) u(v, w)-f(v, w).
• An arc (v, w) is a residual arc if
• uf(v,w) gt 0.
• We denote by Ef the set of residual arcs.
• A residual cycle is a simple cycle of residual
  arcs. The capacity of a residual cycle is the
  minimum of the residual capacities of its arcs.
• For a price function the reduced cost
  of (v,w)
• cp(v, w) c(v, w) p(v) - p(w)

4
Algorithm outline(origin)

• Basic negative cycle canceling algorithm
• KLEIN, 67 A primal method for minimal cost
  flows with applications to the assignment and
  transportation problems
• Use the Ford-Fulkerson maximal flow routine (or
  any other) to find a flow f of value v.
• Form the residual network Gf
• Test for the existence of a negative directed
  cycle C in Gf
• While Gf contains a negative cost cycle C do
• push along C.

5
Algorithm outline

• The cycle-canceling algorithm can run for an
  exponential number of iterations even if the
  capacities and costs are integers, and it need
  not even terminate if the capacities are
  irrational.
• FORD, JR., L. R., AND FULKERSOND, . R. Flows in
  Networks..

6
Algorithm outline

• A natural question
• Is there a rule for selecting cycles to cancel
  that results in a number of iterations bounded by
  a polynomial in n and m?
• Answer
• Selection rule -
• always cancel a residual cycle whose average arc
  cost is as small as possible (minimum-mean cycle)
• A minimum mean cycle can be found in O(nm) time
  using an algorithm of Karp 19 or in 0(vnm
  log(nC)) time using an algorithm of Orlin and
  Ahuja 26. C max abs cost

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Algorithms review
Strongly Polynomial Bound Weakly Polynomial Bound Reference Algorithmic Description
O(n2m3 log n) O(n2m2 log(nC) Goldberg and Tarjan 1988 Augment flow along a negative cycle with a minimum mean cost
O(nm2 log2n) O(nm log n log(nC)) Goldberg and Tarjan 1988 Augment flow along a cycle composed entirely of negative reduced cost arcs
Not available O(mn3log(mCU)) Barahona and Tardos 1989 Augment flows in negative cycles with maximum improvement
Not available O(nm2 log(nC)) Wallacher and Zimmerman 1991 Augment flow along a negative cycle with minimum ratio.
O(m2log n(mn log n)) O(mlogU (m nlog n)) Sokkalingam, Ahuja and Orlin 1996 Augment flow in a negative cycle with sufficiently large residual capacity
We call a network algorithm strongly polynomial
if it satisfies the following two conditions 1)
The number of arithmetic operations is
polynomially bounded in the number of nodes n,
and the number of arcs m. 2) The only arithmetic
operations in the algorithm are comparisons,
additions and subtraction.
8
Algorithm theory

• Theorem 2.1 Busacker and Saaty
• A circulation is minimum-cost if and only if
  there are no negative residual cycles.
• Proof (gt)
• If there are negative cycles in Gf we can obtain
  a cheaper circulation by increasing flow on
  them.
• Proof (lt)
• If the circulation f is not optimal, we show
  there exists a negative cycle in Gf
• Let f be an optimal circulation.We will look at
  the decomposition of the circulation f-f into
  cycles.
• Since it is of negative cost and feasible in Gf,
  it contains a negative cycle!

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Theorem 2.1 - example
10
Algorithm theory

• Observe that the cost of a residual cycle is the
  same whether the original arc costs or the
  reduced arc costs with respect to some price
  function are used. Furthermore, the flow
  conservation constraints (3) imply that the cost
  of any circulation is unaffected by replacing
  original costs by reduced costs.
• Theorem 3.1 10.
• A circulation f is minimum-cost iff there is a
  price function p s.t.

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Algorithm theory

• A circulation f is e-optimal with respect to a
  price function p if (we relax the optimality
  constraint)
• Observe that a circulation is e-optimal w.r.t. p
  iff for every residual arc.

12
Algorithm theory

• Theorem 3.2
• If all costs are integers and then an
  
• e-optimal solution is optimal 5
• Proof
• Consider a simple cycle in Gf.
• The e-optimality of f implies that the reduced
  cost of the cycle is at least .
• The reduced cost of the cycle equals the original
  cost which is non-negative (all costs are
  integers).
• Since there are no negative cost cycles, f is
  minimum cost circulation.

13
Algorithm definitions

• The connection between e-optimality and minimum
  cycle means
• For a circulation f
• e(f) the minimum e such that f is e-optimal.
• µ(f) the mean cost of a minimum-mean residual
  cycle

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Algorithm theory

• Theorem 3.3 16.
• Suppose f is a nonoptimal circulation. Then e(f)
  -µ(f).
• Proof (gt)
• Consider any cycle G in Gf Let the length of G
  be l.
• For any e, define c(e) by c(e)(v,w)c(v, w) e
  for (v, w) ? Ef
• Since f is e(f)-optimal, we have
• 0 c(e)(G)c(G) le , i.e. c(G) / l -e(f).
• Since this is true for any cycle G, µ(f) -e(f),
  i.e., e(f) -µ(f).

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Algorithm Theorem 3.3 16. Cont

• Proof (lt)
• Conversely, let G be the minimum-mean cost
  residual cycle then the mean cost of G is equal
  to µ(f).
• Since f is not optimal, there is a negative-cost
  residual cycle, and therefore µ(f) lt 0.
• Fix a price function p and an e such that e gt
  -µ(f).
• Since the cost of G is equal to the sum of the
  reduced costs of the arcs on G, for the minimum
  reduced cost arc (v, w) on G we have cp(v,w)
  µ(f) (the minimum is at most the average).
• Therefore cp(v,w) -e, so f is not e -optimal
  with respect to p.
• Thus e(f) -µ(f).

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Algorithm definitions

• The admissible graph G(f,p) is

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Algorithm Lemma 3.4

• Lemma 3.4.
• Suppose a circulation f is e-optimal with respect
  to a price function p and G(f, p) is acyclic.
  Then f is (1-1/n)e-optimal.
• Proof.
• Let G be a simple cycle in Gf, and let l be the
  length of G. By the e-optimality of f the cost
  of every arc on G is at least -e. Since the
  admissible graph is acyclic, at least one arc on
  G has a nonnegative cost. Therefore, the mean
  cost of G is at least
• Theorem 3.3 implies that f is (1-1/n)e-optimal.

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Example
Flow
Admissible graph
Residual graph
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Algorithm theory

• Lemma 3.5.
• Canceling a minimum-mean cycle cannot increase
  e(f).
• Proof
• Let G be the minimum-mean cycle that is canceled.
  Before G is canceled, every residual arc
  satisfies cp(v,w)-e by e-optimality.
• By Theorem 3.3, the choices of e and G imply that
  every arc (v,w) on G satisfies cp(v,w)-e before
  canceling.
• By antisymmetry, every new residual arc created
  by canceling G has cost e. (Every such arc is the
  reversal of an arc on G.) It follows that after
  cancellation of G, every residual arc still
  satisfies cp(v,w)-e.
• Thus after the cancellation e(f)-e

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Algorithm theory

• Lemma 3.6.
• A sequence of m minimum-mean cycle cancellations
  reduces e(f) to at most (1 - 1/n)e, that is, to
  at most (1- 1/n) times its original value.
• Proof
• Consider a sequence of m minimum-mean cycle
  cancellations. As f changes, the admissible graph
  G(f,p) changes as well. Initially every arc (v,
  w) ? E(f,p) satisfies cp(v,w)-e . Canceling a
  cycle all of whose arcs are in E(f,p) adds only
  arcs of positive reduced cost to Ef and deletes
  at least one arc from E(f, p).
• Consider the two cases

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Algorithm theory - lemma 3.6 cont

• Case 1.
• None of the cycles canceled contains an arc of
  nonnegative reduced cost.
• Then each cancellation reduces the size of E(f,
  p), and after m cancellations E(f, p) is empty,
  which implies that f is e-optimal.
• that is e(f) 0. Thus the lemma is true in this
  case.

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Example
Flow
Admissible graph
Residual graph
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Algorithm theory - lemma 3.6 cont

• Case 2.
• Some cycle canceled contains an arc of
  nonnegative reduced cost.
• Let G be the first such cycle canceled. Every arc
  of G has a reduced cost of at least -e, one arc
  of G has a nonnegative reduced cost, and the
  number of arcs in G is at most n.
• Therefore, the mean cost of G is at least
• -(n-1)e/n -(1-1/n)e.
• Thus, just before the cancellation of G,
• e(f) (1-1/n)e by Theorem 3.3.
• Since, by Lemma 3.5, e(f) never increases, the
  lemma is true in this case also.

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Algorithm Observation

• Lemmas 3.5 and 3.6 are enough to derive a
  polynomial bound on the number of iterations,
  assuming that all arc costs are integers.

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Algorithm theory

• Theorem 3.7.
• If all arc costs are integers, then the
  minimum-mean cycle-canceling algorithm terminates
  after
• O(nm log(nC)) iterations.
• Proof.
• Let f be the circulation maintained by the
  algorithm. Initially e(f) C. If e(f) lt 1/n,
  then e(f) 0 by Theorem 3.1.
• Lemmas 3.5 and 3.6 imply that if i is the total
  number of iterations, (1 - 1/n)??(i-1/m)?
  1/(nC).
• That is, ??(i-1/m)? -ln(nC)/ln(1-1/n) n
  ln(nC)
• since ln(1-1/n) -l/n for n gt 1.
• It follows that i O(nm log (nC)).

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Algorithm theory

• Theorem 3.8 16.
• Let e gt 0, suppose a circulation f is e-optimal
  with respect to a price function p, and suppose
  that for some arc (v, w), cp(v,w)l 2ne.
• Then (v,w) is e-fixed.
• Proof.
• By antisymmetry, it is enough to prove the
  theorem for the case cp(v,w) 2ne. Let f be a
  circulation such that f(v, w) ? f(v, w). Since
  cp(v,w) gt e, the flow through the arc (v, w) must
  be as small as the capacity constraints allow,
  namely -u(w, v), and therefore f(v, w) ? f(v, w)
  implies f(v,w)gtf(v,w). We show that f is not
  e-optimal, from which the theorem follows.

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Algorithm definitions

• We say that an arc is e-fixed if and only if the
  flow through this arc is the same for all
  e-optimal circulations.

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Algorithm theory Theorem 3.8 cont

• Consider
• Note that Ggt is a subgraph of Gf and (v, w) is
  an arc of Ggt.
• Since f and f are circulations, Ggt must contain
  a simple cycle G that passes through (v,w).
• Let l be the length of G. Since all arcs of G are
  residual arcs, the cost of G is at least
• cp(v,w) (l-1)e 2ne (n-1)e gt ne

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Algorithm theory Theorem 3.8 cont

• Now consider a cycle G obtained by reversing the
  arcs on G. Note that G is a cycle in
• and therefore a cycle in Gf
• by antisymmetry, the cost of G is less than -ne
  and thus the mean cost of G is less than -e.
• Theorem 3.3 implies that f is not e-optimal.

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Algorithm running time analysis

• Consider an execution of the cycle-canceling
  algorithm.
• Suppose an edge (v, w) becomes fixed at some
  point in the execution.
• Since by Lemma 3.5 the error parameter e(f) never
  increases, the flow through (v,w) henceforth
  remains the same.
• When all edges are fixed, the current circulation
  is optimal.
• To see this, observe that an optimal circulation
  is e-optimal for any e 0, and therefore it must
  agree with the current circulation on all (fixed)
  arcs.

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Algorithm theory

• Theorem 3.9.
• For arbitrary real-valued arc costs, the
  minimum-mean cycle-canceling algorithm terminates
  after O(nm2 log n) iterations.
• Proof.
• Let k m(n?ln n 1?). Divide the iterations
  into groups of k consecutive iterations. We
  claim that each group of iterations fixes the
  flow on a distinct arc (v, w), that is,
  iterations after those in the group do not change
  f(v, w). The theorem is immediate from the claim.
  
• To prove the claim, consider any group of
  iterations.
• Let f be the flow before the first iteration of
  the group, f the flow after the last iteration
  of the group, e e(f), e e(f).

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Algorithm Theorem 3.9 cont

• Let p be a price function for which f satisfies
  the e-optimality constraints.
• Let G be the cycle canceled in the first
  iteration of the group.
• The choice of k implies by Lemmas 3.5 and 3.6
  that
• e e(1 - l/n)?n(ln n l) e /(2n).
• Since the mean cost of G is - e, some arc on G,
• say (v, w), must have cp(v,w) -e -2ne.
• By Lemma 3.5 and Theorem 3.8, the flow on (v, w)
  will not be changed by iterations after those in
  the group. But f(v, w) is changed by the first
  iteration in the group, which cancels G. Thus
  each group fixes the flow on a distinct arc.

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Algorithm theory

• Theorem 3.10.
• The minimum-mean cycle-canceling algorithm runs
  in O(n2m3 log n) time on networks with arbitrary
  real-valued arc costs, and on networks with
  integer arc costs in
• O(n3/2m2 minlog2 (nC), vn log (nC), vn m log n)
  
• Proof.
• Immediate from Theorems 3.7 and 3.9.

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Algorithm improvement

• In this section we describe a variant of the
  minimum-mean cycle-canceling algorithm for which
  the time per cycle cancellation is O(log n)
  instead of O(nm).
• This improvement is based on a more flexible
  selection of cycles for canceling, explicit
  maintenance of a price function to help identify
  cycles for canceling, and a sophisticated data
  structure to help keep track of arc flows.

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Algorithm cancel-and-tighten - outline

• The cancel-and-tighten algorithm, maintains a
  circulation f and a price function p.
• Denote by e(f, p) the minimum e such that f
  satisfies the e-optimality constraints for p,
• e(f,p) max0, -mincp(v, w) uf(v, w) gt 0.
• (v, w) is an admissible arc if cp(v,w)lt0
• A residual cycle is an admissible cycle if all
  its arcs are admissible arcs.
• Initially f is any circulation and p is the
  identically zero price function.
• (Thus, e(f, p) C initially.)

36
Algorithm cancel-and-tighten - outline

• The algorithm consists of repeating the following
  two steps until the circulation f is optimal
• Step 1 cancel cycles.
• Repeatedly find and cancel admissible cycles
  until the admissible graph is acyclic.
• Step 2 tighten prices.
• Modify p so that e(f, p) decreases to at most
• (1-1/n) times its former value.
• Note that, by Lemma 3.4, a suitable price
  function can always be found in Step 2

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Algorithm cancel-and-tighten time analysis

• Theorem 4.1.
• Each iteration of Step 1 results in the canceling
  of at most m cycles. If all arc costs are
  integers, there are
• O(n log(nC)) iterations of Steps 1 and 2
• Proof.
• Let E(f,p) be the set of admissible arcs.
• Canceling an admissible cycle reduces the size of
  E(f,p) by at least one and cannot increase
  e(f,p), since all newly created residual arcs
  have positive cost.
• Since E(f,p) has maximum size m and minimum size
  0, after at most m cycle cancellations Step 1
  terminates.
• Lemma 3.4 implies that after Step 1, the mean
  cost of a residual cycle is at least
  -(1-1/n)e(f,p), which implies that p can be
  modified to satisfy the requirement in Step 2.
• The third part of the theorem follows as in the
  proof of Theorem 3.7.

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Algorithm theory

• Theorem 3.7.
• If all arc costs are integers, then the
  minimum-mean cycle-canceling algorithm terminates
  after
• O(nm log(nC)) iterations.
• Proof.
• Let f be the circulation maintained by the
  algorithm. Initially e(f) C. If e(f) lt 1/n,
  then e(f) 0 by Theorem 3.1.
• Lemmas 3.5 and 3.6 imply that if i is the total
  number of iterations, (1 - l/n)??(i-1/m)?
  1/(nC).
• That is, ??(i-1/m)? -ln(nC)/ln(1-1/n) n
  ln(nC)
• since ln(1-1/n) -1/n for n gt 1.
• It follows that i O(nm log (nC)).

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Algorithm cancel-and-tighten implementation

• Step 1 is the dominant part of the computation.
• A simple implementation runs in O(nm) time (O(n)
  per cycle canceled).
• A more complicated implementation runs in O(m log
  n) per cycle canceled).

40
Algorithm cancel-and-tighten implementation

• Performing Step 1 is essentially the same as
  converting an arbitrary flow into an acyclic flow
  by eliminating cycles of flow.

41
Algorithm cancel-and-tighten implementation

• depth-first search to find admissible cycles.
• Each search advances only along admissible arcs.
• Whenever a search retreats from a vertex v, this
  vertex is marked as being on no admissible cycles
• A search is allowed to visit only unmarked
  vertices.
• Whenever a search advances to a vertex it has
  already visited, an admissible cycle has been
  found.
• The cycle is canceled and a new search begun.
• A straightforward implementation of this
  algorithm has a running time of O(m) iterations
  with O(n) per cycle canceled, for a total of
  O(nm) time.

42
Algorithm cancel-and-tighten implementation

• Algorithms for the latter purpose, such as the
  O(m log n)-time algorithm of Sleator and Tarjan
  30, can be adapted to the former purpose.
• We shall describe the appropriately modified
  version of this algorithm.

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Algorithm cancel-and-tighten implementation

• Dynamic tree operations
• make-tree(v) Make vertex v into a one-vertex
  dynamic tree. Vertex v must be in no other tree.
• find-root(v) Find and return the root of the
  tree containing vertex v.
• find-value(v) Find and return the value of the
  tree arc connecting v to its parent. If v is a
  tree root, return infinity.
• find-min(v) Find and return the ancestor w of v
  such that the tree arc connecting IY to its
  parent has minimum value along the path from v to
  find-root(v). In case of a tie, choose the vertex
  w closest to the tree root. If v is a tree root,
  return v.
• change-value(v, x) Add real number x to the
  value of every arc along the path from v to
  find-root(v).
• link(v, W, x) Combine the trees containing v and
  w by making w the parent of v and giving the new
  tree arc joining v and w the value X. This
  operation does nothing if v and w are in the same
  tree or if v is not a tree root.
• cut(v) Break the tree containing v into two
  trees by deleting the arc from v to its parent.
  This operation does nothing if v is a tree root.

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Algorithm cancel-and-tighten implementation
step 1

• Step 1a initialize.
• For each vertex v,
• unmark v and perform make-tree(v).
• Step 1b finding starting vertex a search.
• If all vertices are marked,
• go to Step 1g.
• Otherwise,
• select an unmarked vertex v and go to Step 1c.
• Step 1c find end of path.
• r find-root(v).
• If there is no admissible arc (r, w) with w
  unmarked,
• go to Step 1f.
• Otherwise,
• let (v, w) be such an arc and go to Step 1d.
• Step 1d extend path.
• If find-root(w) ? r,
• link(r, w, uf(r,w)) and go to Step 1c.
• Otherwise,
• go to Step 1e.

45
Algorithm implementation step 1 cont

• Step 1e cancel cycle.
• Let d minuf(r,w)), find-value(find-min(w)).
• Perform f(r, w) f(r, w) d . If uf(r,w) 0,
  mark (r, w) inadmissible.
• Perform change-value(w, - d).
• While find-value(find-min(w)) 0, do the
  following
• z find-min(w)
• f(z, parent(z)) u(z, parent(z))
• cut(z).
• Go to Step 1b.
• Step 1f retract path.
• Mark r.
• For each vertex z such that r parent(z), do the
  following
• f(z, r) u(z, r) - find-value(z) cut(z).
• Go to Step 1b.
• Step 1g extract flow values of tree arcs.
• For each vertex v,
• if find-root(v) ? v,
• f(v, parent(v)) u(v, parent(v)) -
  find-value(v).
• stop.

46
Algorithm cancel-and-tighten implementation
step 2

• Step 2a compute levels.
• For each vertex v,
• compute a level L(v), defined recursively
• If v has no incoming admissible arcs,
• L(v) 0.
• Otherwise,
• L(v) maxL(u) 1 (u, v) is an admissible
  arc.
• Step 2b compute new prices.
• For all v ? V,
• p(v) p(v) - (e/n)L(v).

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The end

• Questions?