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CSC 373H / L0501        Lecture Summary for Week  9              Fall 2006
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Network Flow Algorithms
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Definition: a "network" is a directed graph N=(V,E) with
  - a "source" s in V with no incoming edge,
  - a "target" t with no outgoing edge (sometimes called "sink"),
  - a nonnegative integer weight (the "capacity") for each edge.

  - Example picture.  Networks can be used to represent, e.g., computer
    networks (capacity = bandwidth), electrical networks, etc.

Network flow problem: assign flow f(e) for each edge e such that we have
maximum flow in the network, subject to:
  - capacity constraint: 0 <= f(e) <= c(e) (flow does not exceed capacity);
  - conservation constraint: for each vertex v != s,t,
        flow into v = flow out of v
    (flow into v = sum_{e in E-(v)} f(e); flow out of v = ...E+..., where
    E-(v) = in-edges, E+(v) = out-edges).

  - Flow for network N = flow out of s = flow into t (by conservation).

Augmenting paths:

  - First idea: path P = s -> ... -> t where f(e) < c(e) for each e.
    Define "residual capacity" delta_f(e) = c(e) - f(e), and residual
    capacity delta_f(P) = MIN (delta_f(e) for e in P).
    Augment path by adding delta_f(P) to all edge flows.

  - Problem: notion too narrow, can get stuck with sub-optimal solution.
    (Example.)

  - Second idea: allow "backward" edges on path and re-define residual
    capacity of e is c(e) - f(e) if e is a forward edge on the path; it's
    f(e) if e is a backward edge.

  - Augmenting path = s-t path where each edge has positive residual
    capacity (i.e., c(e)-f(e) > 0 for forward edges e, f(e) > 0 for
    backward edges e).

    (A backward edge with positive flow represents extra flow that can be
    reassigned to forward edges.)

  - Augmentation: add delta_f(P) (defined as before) to forward edges,
    subtract it from backward edges.

  - Example.

Ford-Fulkerson algorithm:
    start with any flow f (e.g., f(e) = 0 for all e in E)
    while there is an augmenting path P
        augment f using P
    output f

Cuts, getting maximum flow:

  - A "cut" is a partition of V into V_s, V_t (i.e., V = V_s U V_t and no
    vertex belongs to both V_s and V_t) such that s in V_s, t in V_t;
      . an edge (u,v) with u in V_s, v in V_t is a "forward" edge;
      . an edge (u,v) with u in V_t, v in V_s is a "backward" edge.

  - Example.

  - Lemma:  For any cut X, flow for network |f| = flow across cut f(X),
    where f(X) = sum_{e forward} f(e) - sum_{e backward} f(e).

  - Lemma:  For any cut X, f(X) <= capacity of cut c(X)
    (c(X) = sum of capacities of forward edges).

  - Corollary:  For any flow f, any cut X, |f| <= c(X).

Theorem (Ford-Fulkerson): |f| is maximum (and equal to c(X) for some cut)
iff there is no augmenting path.

Proof: (=>) augment
(<=) Construct a cut X as follows:
  - s in V_s;
  - if (u,v) in E with u in V_s, v not in V_s, f(u,v) < c(u,v), then
    add v to V_s;
  - if (u,v) in E with v in V_s, u not in V_s, f(u,v) > 0, then
    add u to V_s.
Let V_t = V \ V_s.  Since there is no augmenting path, t in V_t.  By
definition of X, every edge crossing X has the property that f(e) = c(e)
for forward edges and f(e) = 0 for backward edges.  Hence, f(X) = |f| =
c(X).

Corollary (max-flow/min-cut theorem):  For any network, the maximum flow
equals the minimum cut capacity.