1. Chapter 1 Introduction 1

2. 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 Network Theory and Applications 2010 2

3.  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 Network Theory and Applications 2010 3

4. 1.2 Network Flow Problems  Minimum cost flow problem Given directed G = (N, A) For each arc (i, j)  A, cost c ij, capacity u ij, lower bound l ij (usually 0) For each node i  N, supply demand b(i)  b(i) > 0: supply node  b(i) < 0: demand node  b(i) = 0: transshipment node decision variables: x ij  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 Network Theory and Applications 2010 4

5.  Formulation Minimize  (i, j)  A c ij x ij subject to  {j: (i, j)  A} x ij -  {j: (j, i)  A} x ji = b(i) for all i  N,(1.1b) l ij  x ij  u ij for all (i, j)  A,(1.1c)  Matrix notation Minimize cx subject toNx = b,(1.2b) l  x  u,(1.2c) N: n  m matrix, called node-arc incidence matrix the column N ij corresponds to the variable x ij column N ij has a +1 in the i-th row, a -1 in the j-th row; others are 0.  necessary condition for feasibility is  i=1 n 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

6. Minimize  (i, j)  A c ij x ij subject to  {j: (i, j)  A} x ij -  {j: (j, i)  A} x ji = b(i) for all i  N,(1.1b) l ij  x ij  u ij 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). Network Theory and Applications 2010 6

7.  Shortest path problem Given directed G = (N,A), arc cost (length) c ij 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  c ij  0 (easy)  c ij < 0 allowed, but no negative cycle (takes more time, but still polynomial time algorithms exist)  c ij < 0 allowed, negative cycle exists (difficult, NPC) Network Theory and Applications 2010 7

8.  Maximum flow problem Given: G = (N, A), arc capacities u ij  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, c ij = 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. Network Theory and Applications 2010 8

9.  Assignment problem: G = (N, A), N = N 1  N 2, N 1  N 2 = , |N 1 | = |N 2 | A  N 1  N 2, cost c ij for each (i, j)  A Find assignments which pair each node in N 1 to a node in N 2 with minimum cost.  Transformation to MCF: Let b(i) = 1, for all i  N 1 = -1, for all i  N 2 u ij = 1, for all (i, j)  A Network Theory and Applications 2010 9

10.  Transportation problem: Special case of MCF N = N 1  N 2, N 1  N 2 = , N 1 : supply nodes, N 2 : demand nodes for each (i, j)  A, i  N 1, j  N 2  Ex) minimum cost distribution of goods from warehouses (N 1 ) to customers (N 2 ). Cost c ij is the cost of a distribution channel from warehouse i to customer j (may involve many consecutive transportation means). Network Theory and Applications 2010 10

11.  Circulation problem: Minimum cost flow problem with only transshipment nodes, i.e. b(i) = 0 for all i  N. lower bound (l ij, may not be 0) and upper bound (u ij ) imposed on the flows. Find a feasible flow with minimum cost, or verify if a feasible flow exists (c ij = 0) for the network. Network Theory and Applications 2010 11

12.  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: x ij units of flow enter arc (i, j)   ij x ij units arrive at node j. 0 <  ij < 1: lossy arc 1 <  ij <  : gainy arc ex)  Power transmission through through electric lines  Flow of water through pipelines or canals  Transportation of perishable commodity  Cash management Network Theory and Applications 2010 12

13.  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 Network Theory and Applications 2010 13

14.  Other models  Minimum spanning tree problem: G = (N, A) undirected, arc weight c ij 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. Network Theory and Applications 2010 14

15.  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 (|N 1 |, |N 2 | 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 Network Theory and Applications 2010 15

16. 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. Network Theory and Applications 2010 16

17.  Application 1.2 Assortment of Structural Steel Beams Steel beems of varying lengths, i = 1, …, n D i > 0: demand of steel beam of length L i, L 1 < L 2 < … < L n Can cut longer length beam and use it for shorted length (scraps result) Let K i : cost of inventory facility for beams of length L i C i : cost of a beam of Length L i 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 L j and use them for the demand of beams of length L i+1, L i+2, …, L j c ij = K j + C j  k=i+1 j D k. Find shortest path from 0 to n. Network Theory and Applications 2010 17

18.  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. Network Theory and Applications 2010 18 1 2 5 3 4 6 -7 3 6 5 10 15 4 5

19.  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 1.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. Network Theory and Applications 2010 19

20.  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 u j for school j. School j should have [l jk, u jk ] student from the k-th ethnic group. L population centers. S ik denote the number of students of the k-th ethnic group at the i-th population center. f ij : 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. Network Theory and Applications 2010 20

21. Network Theory and Applications 2010 21 a1a1 a2a2 b1b1 c1c1 d1d1 e1e1 e2e2 b2b2 b3b3 c2c2 c3c3 d2d2 d3d3 Ethnic groups (sources) Population centers Schools (input) Schools (output) Ethnic groups (sinks)  i=1 3 S i1  i=1 3 S i2 -  i=1 3 S i2 -  i=1 3 S i1 0, [0, S ik ] f ij, [0,  ] 0, [0, u j ] 0, [l jk, u jk ] (cost, [lower bound, upper bound])