chapter 1 introduction 1. 1.1 introduction networks: mathematical models of real systems like...

Chapter 1Introduction

1

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| = nA: set of arcs, |A| = m

• (i, j) A, {i}, {j} N (ordered/unordered pair of distinct nodes) (directed network/undirected network, graph)

V: vertex setE: edge setparallel arcs, loops may be considered, but no hyper graph

Network Theory and Applications 2010 2

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 considerShortest path problemMaximum flow problemMinimum cost flow problem


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 iN, supply demand b(i)b(i) > 0: supply nodeb(i) < 0: demand nodeb(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


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: nm 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 i=1n b(i) = 0 (add the left

hand

side and right hand side of (1.1b) respectively, have 0x = b(i) ) Network Theory and Applications 2010 5

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).


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).

VariationsAll pairs shortest pathk-shortest path (enumeration)Longest path (quite different, NPC)

Be careful about the conditions on datacij 0 (easy)

cij < 0 allowed, but no negative cycle (takes more time, but still polynomial time

algorithms exist)

cij < 0 allowed, negative cycle exists (difficult, NPC)


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 iN, 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.


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 iN1

= -1, for all iN2

uij = 1, for all (i, j)A


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).


Circulation problem:

Minimum cost flow problem with only transshipment nodes, i.e. b(i) = 0 for all iN.

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.


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 < ij < 1: lossy arc

1 < ij < : gainy arc

ex) Power transmission through through electric linesFlow of water through pipelines or canalsTransportation of perishable commodityCash management Network Theory and Applications 2010 12

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 commodityOne origin, one destination for each commodity

• Transmission network, freight train


Other modelsMinimum 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.


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


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.


Application 1.2 Assortment of Structural Steel Beams

Steel beems of varying lengths, i = 1, … , n

Di > 0: demand of steel beam of length Li, L1 < L2 < … < 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 Li+1, Li+2, … , Lj

cij = Kj + Cjk=i+1j Dk.

Find shortest path from 0 to n. Network Theory and Applications 2010 17

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.

Objectives1. Find as many pairs as possible whose matching coefficients do not exceed a

specification limit

2. Pairing speakers within specification limits to minimize the total sum of the matching coefficients.


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.



a1

a2

b1 c1 d1

e1

e2

b2

b3

c2

c3

d2

d3

Ethnicgroups(sources)

Populationcenters

Schools(input)

Schools(output)

Ethnicgroups(sinks)

i=13Si1

i=13Si2 -i=1

3Si2

-i=13Si1

0, [0, Sik]

fij, [0, ] 0, [0, uj]

0, [ljk, ujk]

(cost, [lower bound, upper bound])

chapter 1 introduction 1. 1.1 introduction networks: mathematical models of real systems like...

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