2 GRAPH AND NETWORK OPTIMIZATION. E. Amaldi Introduction to Operations Research Politecnico Milano 1

Transcription

1 2 GRAPH AND NETWORK OPTIMIZATION E. Amaldi Introduction to Operations Research Politecnico Milano 1

2 A variety of decision-making problems can be formulated in terms of graphs and networks Examples: - transportation and distribution problems (freight and people), - network design (communication, electrical and others), - location problems (services and facilities), - project planning, resource management, - financial planning, timetable scheduling, - production planning,... E. Amaldi Introduction to Operations Research Politecnico Milano 2

3 2.1 Graphs and algorithms E. Amaldi Introduction to Operations Research Politecnico Milano 3

4 2.1.1 Graphs Example Road network which connects n cities n= city node connection edge 4 Model: A graph G = (N, E) which consists of a set N={1,2,3,4,5} of nodes (vertices) and a set E={[1,2],[1,3],[1,4],[1,5],[2,3],[2,5],[3,4],[3,5],[4,5]} N N of edges connecting them. [, ] indicate an unordered pair of nodes E. Amaldi Introduction to Operations Research Politecnico Milano 4

5 Two nodes are adjacent if they are connected by an edge. An edge e is incident in a node v if v is an endpoint of e. G nodes 1 and 2 are adjacent edge [1,5] is incident in nodes 1 and 5 4 The degree of a node is the number of incident edges. Example: node 1 has degree 4 and node 4 has degree 3 E. Amaldi Introduction to Operations Research Politecnico Milano 5

6 Given a graph G=(N, E) with n = N e m = E n=4 m=4 v 1 v 2 v 3 N = { v 1, v 2, v 3, v 4 } E = { [v 1, v 2 ], [v 2, v 3 ], [v 2, v 4 ], [v 3, v 4 ] } v 4 A sequence of consecutive edges [v 1, v 2 ], [v 2, v 3 ],, [v k-1, v k ] is a path which connects nodes v 1 and v k v 1 v 2 v 3 v k E. Amaldi Introduction to Operations Research Politecnico Milano 6

7 v i, v j N are connected if there exists a path connecting them v j v i G = (N, E) is connected if v i, v j are connected v i, v j N connected 4 not connected E. Amaldi Introduction to Operations Research Politecnico Milano 7

8 If some connections can be travelled only in one direction: directed graph G = (N, A), where A is a set of ordered pairs of nodes (v i, v j ) called arcs v i v j (v i, v j ) A 1 2 G E. Amaldi Introduction to Operations Research Politecnico Milano 8

9 A sequence of consecutive arcs (v 1, v 2 ), (v 2, v 3 ),, (v k-1, v k ) is a directed path from v 1 to v k v 1 v 2 v k directed path from 1 to 2 4 E. Amaldi Introduction to Operations Research Politecnico Milano 9

10 A cycle ( circuit ) is a ( directed ) path with v k = v 1 v 2 v 1 = v k v 3 v k cycle C 4 E. Amaldi Introduction to Operations Research Politecnico Milano 10

11 Compatibility and incompatibility relations Esempio 3 tasks and 2 engineers i j N 1 N 2 N 2 = N \ N 1 edge [i, j] indicates that task i can be executed by engineer j G is bipartite if there exists a partition (N 1, N 2 ) of N s.t. no edge connects nodes in the same N i (i = 1, 2 ) E. Amaldi Introduction to Operations Research Politecnico Milano 11

12 G is complete if E = { [v i, v j ] : v i, v j N, i j } n=4 m=6 Property For any undirected graph with n nodes, the number of edges m is such that n( n 1) m 2 For directed graphs we have m n(n-1). In both case we have equality for complete graphs. E. Amaldi Introduction to Operations Research Politecnico Milano 12

13 Precedence constraints between entities A project is composed of n activities {a i } 1 i n with m precedence relations between pairs of activities a i a j (a j cannot start before a i is completed) Model: directed graph G=(N, A) with n= N and m= A s.t. activity node i j a i a j E. Amaldi Introduction to Operations Research Politecnico Milano 13

14 2.1.2 Graph reachability Problem Given a directed graph G = (N, A) and a node s, determine all the nodes that are reachable from s. s Successor lists : S(1) = { 2, 4 } S(2) = { 5 } S(3) = { 5, 6 } S(4) = { 2 } S(5) = { 4 } S(6) = { 2 } E. Amaldi Introduction to Operations Research Politecnico Milano 14

15 Efficient algorithm which allows to find all nodes reachable from s? input G = (N, A) with n = N and m = A, described by the successor lists, and a node s output Subset M N of nodes of G reachable from s Q = queue containing the nodes reachable from s and not yet processed (First-In First-Out policy) E. Amaldi Introduction to Operations Research Politecnico Milano 15

16 Example Q = { 1 } e M = Ø Q = { 1 } e M = {1} Q = { 2, 4 } e M = M {2} Q = { 4, 5 } e M = M {4} M Q = { 5 } Q = e M = M {5} Subset M = { 1, 2, 4, 5 } of nodes that have been labeled is the subset of nodes reachable from s = 1. NB: There are no arcs exiting M and entering in N \ M! E. Amaldi Introduction to Operations Research Politecnico Milano 16

17 Graph reachability algorithm output Labeled nodes (in M) are reachable from s BEGIN Q := {s}; M := ; WHILE Q DO /* process a node h Q */ Select a node h Q e set Q := Q \ {h}; M := M {h}; /* label h */ FOR EACH j S(h) DO IF j M AND j Q THEN Q := Q {j} END-IF END-FOR END-WHILE END FIFO queue Q breadth-first search node exploration E. Amaldi Introduction to Operations Research Politecnico Milano 17

18 Given a set of nodes S N, the cut induced by S, δ(s) = { (v,w) A : v S, w N \ S o w S, v N \ S }, is the subset of edges with an endpoint in S and the other one in N \ S. Example N \ S S δ + (S) = { (v, w) A : v S, w N \ S } outgoing cut δ - (S) = { (v, w) A : w S, v N \ S} entering cut E. Amaldi Introduction to Operations Research Politecnico Milano 18

19 Example N \ M M The algorithm (exploration) stops because δ + (M)=Ø δ - (M) is the set of arcs with head in M and tail not in M E. Amaldi Introduction to Operations Research Politecnico Milano 19

20 2.1.3 Complexity of algorithms Algorithm: sequence of instructions that allow to solve the instances of a problem Since the computing time is computer dependent, we consider the elementary operations (arithmetic operations, comparisons,...) we assume they all have the same cost Examples: 1) Dot product of a, b R n requires n multiplications and n-1 additions 2n-1 elementary operations 2) Given two nxn matices A and B, the product AB requires (2n-1)n 2 elementary operations E. Amaldi Introduction to Operations Research Politecnico Milano 20

21 In general it is hard to determine the exact number of elementary operations (e.o.) as a function of the size of the instance (here n o m) We consider the speed of growth in the worst case: We look for a function f(n) which is an upper bound on the number of e.o. needed to solve every instance of size n number of e.o. for every instance of size n f (n) E. Amaldi Introduction to Operations Research Politecnico Milano 21

22 A function f (n) is order of g(n) and we write f (n) = O( g (n) ) if c > 0 such that f (n) c g(n), for n sufficiently large asymptotically c g f n 0 Examples 3n 3 + n = O(n 3 ) 6n = O(n 2 ) E. Amaldi Introduction to Operations Research Politecnico Milano 22

23 We distinguish between algorithms whose order of complexity (in the worst case) is polynomial: O( n d ) for a given constant d The algorithms with a higher order polynomial complexity (such as O(n 10 ) ) are not efficient in practice! exponential: O ( 2 n ) E. Amaldi Introduction to Operations Research Politecnico Milano 23

24 Polynomial versus exponential growth E. Amaldi Introduction to Operations Research Politecnico Milano 24

25 E. Amaldi Introduction to Operations Research Politecnico Milano 25

26 Example: complexity of exploration algorithm At each iteration of the cyclewhile: select one node h Q non yet processed, extract it from Q and label it by inserting it in M insert in Q all unlabeled nodes j that are directly reachable from h Since each node h is insereted in Q at most once, each arc (h, j) is considered at most once Overall complexity O(n + m), where n = N e m = A NB: for dense graphs m = O(n 2 ) E. Amaldi Introduction to Operations Research Politecnico Milano 26

27 2.1.4 Subgraph Example Design a communication network that connects n cities ( offices ) Model: Undirected graph G = (N, E) with n = N, m = E 1 2 n=5 5 3 The edges represent the possible links 4 E. Amaldi Introduction to Operations Research Politecnico Milano 27

28 G = (N, E ) is a subgraph of G = (N, E) if N N and E E only contains edges with both endpoints in N. G=(N, E) G =(N, E ) N = {1,2,3,5} N E = {[1,2],[1,5],[2,3],[2,5],[3,5]} E E. Amaldi Introduction to Operations Research Politecnico Milano 28

29 Desired properties of a communication network: 1) every pair of cities must be connected connected subgraph containing all the nodes of G 2) do not waste resources acyclic subgraph (without cycles) A tree G T = (N, T) of G is a subgraph of G that is both connected and acyclic. 1 2 G T 5 3 T={[1,5],[2,5],[3,5]} E. Amaldi Introduction to Operations Research Politecnico Milano 29

30 G T = (N, T) is a spanning tree of G = (N, E) if it contains all the nodes of G ( namely N = N ) 1 2 G T The leaves of a tree are the nodes of degree 1. E. Amaldi Introduction to Operations Research Politecnico Milano 30

31 Trees properties 1 ) Every tree T with n 2 nodes has at least 2 leaves. Proof By contradiction: if T has 0 or 1 leaves, it would be possible to travel along its edges starting from the leaf (if there is one) or from any node, using each edge at most once. Since a tree has no cycles, the nodes cannot be visited twice. If there is no (other) leaf, we can leave each node along an unused incident edge. an infinite path in a finite graph! E. Amaldi Introduction to Operations Research Politecnico Milano 31

32 2 ) Every tree with n nodes has n 1 edges. Dim. By induction: Inductive base : true for n = 1 (1 node and 0 edges) Inductive step : if it true for the trees with n nodes, it is also true for those with n + 1 nodes. Consider a tree T 1 with n + 1 nodes. By deleting one leaf and its incident edge, we obtain a tree T 2 with n nodes. Since by assumption Property 2) holds for T 2, T 2 has n 1 edges. T 1, which has one more edge than T 2, has n edges. E. Amaldi Introduction to Operations Research Politecnico Milano 32

33 3 ) Any pair of nodes is connected via a unique path otherwise there would be a cycle! 4 ) By adding to a tree any edge that it does not contain, we create a unique cycle consisting of the path of Property 3) and the new edge. E. Amaldi Introduction to Operations Research Politecnico Milano 33

34 5 ) Exchange property Let G T = (N, T ) be a spanning tree of G = (N, E) Consider an edge e T and the unique cycle C of T {e} (Property 4). For each edge f C \ {e}, the subgraph T {e} \ {f } is also a spanning tree of G. G T f e 3 C T {e} \ {f } E. Amaldi Introduction to Operations Research Politecnico Milano 34