Chapter 1 Introduction

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Chapter 1Introduction
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1.1 Introduction

• Networks mathematical models of real systems
  like electrical and power/ telecom/ logistics/
  highway/ rail/ airline/ water, etc.
• Also network models frequently show up for
  problems that look irrelevant to physical
  networks. Important modeling/algorithmic tool.
• Graph, Network G (N, A) ( G (V, E) )
• N set of nodes, usually N n
• A set of arcs, A m
• (i, j) ?A, i, j? N (ordered/unordered pair
  of distinct nodes) (directed network/undirected
  network, graph)
• V vertex set
• E edge set
• parallel arcs, loops may be considered, but no
  hyper graph

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• Two major categories flow networks, topological
  graph theory
• Many flow network problems can be modeled as LP.
• Interpret the behavior of LP algorithms on
  network problems.
• Emphasis on specialized algorithms for network
  problems.
• Special attention on running times of
  algorithms(polynomial solvability, efficient
  algorithm)
• Implementation issues, data structures
• Basic problems to consider
• Shortest path problem
• Maximum flow problem
• Minimum cost flow problem

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1.2 Network Flow Problems

• Minimum cost flow problem
• Given directed G (N, A)
• For each arc (i, j)?A, cost cij , capacity uij,
  lower bound lij (usually 0)
• For each node i?N, supply demand b(i)
• b(i) gt 0 supply node
• b(i) lt 0 demand node
• b(i) 0 transshipment node
• decision variables xij ? amount of flow on arc
  (i, j)?A
• Want minimum cost flows that satisfy
  supply/demand at every node and arc capacity
  constraints on the arcs.
• Quite general problem. Has many problems as
  special cases

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• Formulation
• Minimize ?(i, j)?A cij xij
• subject to ?j (i, j)?A xij - ?j (j, i)?A
  xji b(i) for all i ? N, (1.1b)
• lij ? xij ? uij for all (i, j) ? A, (1.1c)
• Matrix notation
• Minimize cx
• subject to Nx b, (1.2b)
• l ? x ? u, (1.2c)
• N n?m matrix, called node-arc incidence matrix
• the column Nij corresponds to the
  variable xij
• column Nij has a 1 in the i-th row, a -1 in
  the j-th row others are 0.
• ? necessary condition for feasibility is ?i1n
  b(i) 0 (add the left hand
• side and right hand side of (1.1b)
  respectively, have 0x ?b(i) )

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• Minimize ?(i, j)?A cij xij
• subject to ?j (i, j)?A xij - ?j (j, i)?A
  xji b(i) for all i ? N, (1.1b)
• lij ? xij ? uij for all (i, j) ? A, (1.1c)
• (1.1b) called mass balance constraints or flow
  conservation constraints
• ( outflow inflow net flow at node i)
• (1.1c) called flow bound constraints
• Assume all data are integral (integrality
  assumption) Otherwise, multiply suitably large
  integer to rational numbers to convert them to
  integers.
• If all data integral, optimal solution is
  integer valued (later).

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• Shortest path problem
• Given directed G (N,A), arc cost (length) cij
  for (i, j)?A.
• Find a minimum cost (or length) path from a
  specified source node s to another specified
  sink node t.
• Transformation to MCF Let b(s) 1, b(t) -1,
  others 0, arc capacities 1 (or ?1).
• Variations
• All pairs shortest path
• k-shortest path (enumeration)
• Longest path (quite different, NPC)
• Be careful about the conditions on data
• cij ? 0 (easy)
• cij lt 0 allowed, but no negative cycle (takes
  more time, but still polynomial time algorithms
  exist)
• cij lt 0 allowed, negative cycle exists
  (difficult, NPC)

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• Maximum flow problem
• Given G (N, A), arc capacities uij ? 0, (i,
  j)?A, specified source node s, sink node t
• Find maximum flow that can be sent from source
  node s to sink node t.
• Transformation to MCF
• Let b(i) 0, for all i?N, cij 0, for all
  (i, j)?A
• Add arc from t to s, with capacity ? and cost
  -1
• Maximum flow problem is related to minimum cut
  problem.

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• Assignment problem
• G (N, A), N N1 ? N2, N1? N2 ?, N1 N2
• A ? N1 ? N2 , cost cij for each (i,
  j)?A
• Find assignments which pair each node in N1 to a
  node in N2 with minimum cost.
• Transformation to MCF
• Let b(i) 1, for all i?N1
• -1, for all i?N2
• uij 1, for all (i, j)?A

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• Transportation problem
• Special case of MCF
• N N1 ? N2, N1? N2 ?,
• N1 supply nodes, N2 demand nodes
• for each (i, j) ?A, i? N1, j? N2
• Ex) minimum cost distribution of goods from
  warehouses (N1) to customers (N2). Cost cij is
  the cost of a distribution channel from warehouse
  i to customer j (may involve many consecutive
  transportation means).

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• Circulation problem
• Minimum cost flow problem with only
  transshipment nodes, i.e. b(i) 0 for all i?N.
• lower bound (lij, may not be 0) and upper bound
  (uij) imposed on the flows.
• Find a feasible flow with minimum cost, or
  verify if a feasible flow exists (cij 0) for
  the network.

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• Convex cost flow problem
• Cost function for flow not linear function, but
  convex function.
• (cost increases more than linear as there are
  more flows congestion on arcs, ..)
• How about concave cost function? (ex fixed
  charge network flow problem)
• Generalized flow problem
• xij units of flow enter arc (i, j) ? ?ijxij
  units arrive at node j.
• 0 lt ?ij lt 1 lossy arc
• 1 lt ?ij lt ? gainy arc
• ex)
• Power transmission through through electric lines
• Flow of water through pipelines or canals
• Transportation of perishable commodity
• Cash management

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• Multicommodity flow problem
• Multiple commodities share the common network .
• Mass balance equations for each commodity.
• Each commodity has origins/destinations.
  Commodities can use the same arc together, but
  should observe the capacity of the arc.
• Multiple origin/destination for each commodity
• One origin, one destination for each commodity
• Transmission network, freight train

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• Other models
• Minimum spanning tree problem
• G (N, A) undirected, arc weight cij for (i,
  j)?A
• Spanning tree tree (connected acyclic graph)
  that spans all nodes of an undirected network
• Find minimum cost spanning tree.
• Simplest form of connectedness
• Variations Arborescence (directed tree),
  Steiner tree, capacitated tree, etc.

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• Matching problem
• G (N, A), undirected
• Matching set of arcs (edges) of G with the
  property that every node is incident to at most
  one arc in the set
• Find a matching that optimizes some criteria (min
  cost perfect matching, maximum cardinality
  matching, maximum weight matching, b-matching, )
• Assignment problem ? bipartite matching problem
  (N1, N2 may not be equal)
• Ex) matching roommates, matching pilots to
  compatible airplanes, scheduling airline crews
  for available flight legs, assigning duties to
  bus drivers, plotter scheduling
• Related problems Chinese postman problem,
  T-joins, edge coloring,
• Similar looking but quite different problem
  stable set (node packing) problem.
• stable set set of nodes such that no two of
  them are joined by an edge.
• maximum cardinality (or weighted) stable set
  problem is NP-hard
• traveling salesman problem (compare to Chinese
  postman problem) is NP-hard

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1.3 Applications

• Application 1.1 Reallocation of Housing
• House categories i 1, , n
• Tenants wants to move to the house of different
  categories.
• cyclic change is desirable.

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• Application 1.2 Assortment of Structural Steel
  Beams
• Steel beems of varying lengths, i 1, , n
• Di gt 0 demand of steel beam of length Li,
  L1 lt L2 lt lt Ln
• Can cut longer length beam and use it for
  shorted length (scraps result)
• Let Ki cost of inventory facility for beams of
  length Li
• Ci cost of a beam of Length Li
• Want to find inventory set up plan to minimize
  the total cost (facility cost scrap loss)
• Model nodes 0, 1, , n
• arc (i, j) represent we maintain inventory
  of length Lj and use them
• for the demand of beams of length Li1, Li2,
  , Lj
• cij Kj Cj?ki1j Dk.
• Find shortest path from 0 to n.

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• Application 1.4 Leveling mountainous terrain
• Building road networks through hilly or
  mountainous terrain
• Distribution of earth from high points to low
  points to produce a leveled roadbed.

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• Application 1.6 Pairing stereo speakers
• Must pair individual speakers to sell them as a
  set
• Measure the responses of the speakers at 20
  discrete frequencies
• matching coefficients for a pair calculated as
  the sum of absolute differences of responses at
  each frequency.
• Objectives
• Find as many pairs as possible whose matching
  coefficients do not exceed a specification limit
• Pairing speakers within specification limits to
  minimize the total sum of the matching
  coefficients.

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• Application 1.10 Racial balancing of schools
• In 1968, nondiscrimination rule for school
  system. Need to balance the ratios between races.
• S schools with capacity uj for school j.
• School j should have ljk, ujk student from the
  k-th ethnic group.
• L population centers.
• Sik denote the number of students of the k-th
  ethnic group at the i-th population center.
• fij distance between population center i and
  school j.
• Find assignment of students to schools so that
  the ethnic requirement for each school is
  satisfied and minimize the total distance
  traveled by the students.
• Multicommodity flow problem.

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(cost, lower bound, upper bound)
fij, 0, ?
0, 0, uj
b1
c1
d1
0, 0, Sik
0, ljk, ujk
a1
e1
?i13Si1
-?i13Si1
b2
c2
d2
a2
e2
?i13Si2
-?i13Si2
b3
c3
d3
Schools (input)
Schools (output)
Ethnic groups (sinks)
Ethnic groups (sources)
Population centers