The Multiple Traveling Salesman Problem with Time Windows: Bounds for the Minimum Number of Vehicles

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1 The Multiple Traveling Salesman Problem with Time Windows: Bounds for the Minimum Number of Vehicles Snežana Mitrović-Minić Ramesh Krishnamurti School of Computing Science, Simon Fraser University, Burnaby, British Columbia, Canada V5A S6 SFU CMPT TR 00- ftp://fas.sfu.ca/pub/cs/techreports/00 School of Computing Science, Simon Fraser University November 00 Abstract This paper deals with finding a lower and an upper bound for the minimum number of vehicles needed to serve all locations of the multiple traveling salesman problem with time windows. The two types of precedence graphs are introduced the start-time precedence graph and the end-time precedence graph. The bounds are generated by covering the precedence graph with minimum number of paths. Instances for which bounds are tight are presented, as well as instances for which bounds can be arbitrary bad. The closeness of such instances is discussed. Keywords: The multiple traveling salesman problem with time windows, bounds for the minimum number of vehicles. Introduction The Multiple Traveling Salesman Problem with Time Windows (m-tsptw) deals with finding a set of optimal routes for a fleet of vehicles in order to serve a set of locations, each one within a specified time window. This article describes a way of finding a lower and an upper bound for the minimum number of vehicles needed to serve all the locations. The m-tsptw is a generalization of the Traveling Salesman Problem (TSP), a well-known NP-hard problem. Furthermore, checking feasibility of the m-tsptw when the number of vehicles is fixed is an NP-complete problem (Savelsbergh, 985). Therefore, the problem of finding the minimum number of vehicles v min needed to serve the set of locations of m-tsptw is NP-hard.

2 The m-tsptw belongs to the class of time-constrained vehicle routing problems, which has been extensively studied in the last few decades. For extensive surveys and bibliography see the state-of-the-art books by Ball, Magnanti, Monma and Nemhauser (995) and Toth and Vigo (00).. Problem formulation Let G = (V, E) be a complete (undirected) graph, where V = {,,,..., n} is a set of locations and E is a set of edges. A travel time t i,j = t j,i > 0 is associated with every edge (i, j) E, i j, such that the triangle inequality is satisfied. Each location i V has a time window I i = [a i, b i ] within which i has to be served, where a i is the release time and b i is the deadline. If a vehicle arrives too early at a given location, it is allowed to wait. The Multiple Traveling Salesman Problem with Time Windows (m-tsptw) consists of determining a set of optimal routes for a fleet of vehicles in order to serve each location of set V exactly once satisfying time windows constraints. A route is a sequence of locations assigned to one vehicle. A route schedule is a sequence of pairs (A i, D i ) where A i is the scheduled arrival time at location i and D i is the scheduled departure time from location i. This article presents two types of precedence graphs that we have used for finding a lower bound v and an upper bound v for the minimum number of vehicles v needed to serve all the locations of the m-tsptw. The reminder of the article is organized as follows. Section introduces the two precedence graphs and describes an algorithm for finding bounds. Section describes the precedence graphs and the bounds for some variations of m-tsptw. Section 4 describes classes of m-tsptw instances for which the bounds are tight. Instances with bad bounds are discussed in Section 5. Conclusion follows in Section 6. Precedence graphs and bounds on vehicles. Precedence graphs A precedence graph is a directed graph which mirrors precedence relation between pairs of locations in V, imposed by the travel times and the time window constraints. We present two types of precedence graphs, the start-time precedence graph and the end-time precedence graph. Definition The start-time precedence graph, G a = (V, E a ), is a directed graph (digraph) of the precedence relation among locations in V with regard to their release times: E a = {(i, j) (V V ) a i + t i,j b j }.

3 Edges from E a have been known in the literature of time-constrained vehicle routing problems as admissible edges. Definition The end-time precedence graph, G b = (V, E b ), is a digraph of the precedence relation among locations in V with regard to their deadlines: E b = {(i, j) (V V ) b i + t i,j b j }. It is clear that E b E a, implying that graph G b is a subgraph of graph G a. Figure shows an m-tsptw instance whose precedence graphs are given in Figure. G 4 time I I 5 I 0 I I locations Figure : An m-tsptw instance with five locations. There are five locations in the instance, and their time windows are I = [a, b ] = [0, 0], I = [a, b ] = [, ], I = [a, b ] = [, 4], I 4 = [a 4, b 4 ] = [0, 4], I 5 = [a 5, b 5 ] = [, 4]. The travel times are: t, =, t, =, t,4 =, t,5 = t, =, t,4 =, t,5 = 4 t,4 = 4, t,5 = t 4,5 = Ga 4 Gb Figure : The start-time precedence graph G a, and the end-time precedence graph G b.

4 . Number of vehicles needed to serve locations of a path of precedence graph Any two locations i, j V can be served by one vehicle if and only if (i, j) is in E a, due to the definition of the start-time precedence graph. Lemma For serving locations of a path p of graph G a more than one vehicle may be needed. Proof: Each location in p can surely be reached from its predecessor (on p) if the predecessor is left at its release time. Therefore, if each location in p may be left at its release time, all locations in p would be served by one vehicle. But it is possible that more than one vehicle is needed to serve the locations in p. Consider three consecutive locations (i, j, l) p, and assume that a i + t i,j > a j, If, in addition, inequality a i + t i,j + t j,l > b l holds, it implies that for serving locations (i, j, l) at least two vehicles are needed, e.g., one for serving (i, j) and the other for serving l. Lemma All locations on a path p of graph G b may be served by one vehicle. Proof: Each location in p may be reached on time from its predecessor if the predecessor was left at its deadline. Since all locations have to be served within their time windows, all locations along any path in G b may be served by one vehicle.. A lower and an upper bound The two precedence graphs of an m-tsptw instance may be used for finding a lower and an upper bound for the minimum number of vehicles needed. Theorem The minimum number of directed paths that cover all the nodes of the start-time precedence graph G a is a lower bound v on the minimum number of vehicles v needed to serve all locations in G. Proof: It is a corollary of Lemma. Theorem The minimum number of directed paths that cover all the nodes of the end-time precedence graph G b is an upper bound v on the minimum number of vehicles v needed to serve all locations in G. Proof: It is a corollary of Lemma. An acyclic, transitive directed graph uniquely defines a partial order relation among its nodes. Problem of finding the minimum cover of nodes by directed paths over such a graph is equivalent to the decomposition by chains, a problem from the theory of partially ordered sets. Dilworth s theorem (Dilworth, 950) 4

5 states that the minimum number of chains (totally ordered sets) which partitions a partially ordered set is equal to the maximum size of an antichain (set of incomparable elements). The decomposition by chains is solvable in polynomial time (Ford and Fulkerson, 96) by transformation to the maximum bipartite matching. The end-time precedence graph G b is transitive and acyclic. Suppose that (i, j) E b and (j, k) E b, i.e., b i + t i,j b j and b j + t j,k b k. Since the triangle inequality is satisfied the following holds: b i +t i,k b i +t i,j +t j,k b j +t j,k b k implying that (i, k) E b. Therefore, G b is transitive. Assume that there exist a cycle C = (i, i,..., i k ) in G b. All travel times being positive implies b i < b i <... < b ik < b i which is a contradiction. Thus, G b is acyclic. time G a 0 locations Figure : A TSPTW problem with collinear locations whose start-time precedence graph has a cycle. The start-time precedence graph G a may have a cycle. An example is given in Figure where three locations are collinear, the time windows are I = [0, ], I = [0, ], I = [0, ], and the travel times are t, =, t, =, t, =. When G a has a cycle we will use its condensation. The condensation of a digraph D = (V, E) is a digraph D c = (V c, E c ) where V c = {C,..., C c } is a set of strongly connected components of D and E c = {(C u, C v ), u v i C u and j C v such that (i, j) E}. (Condensation of any digraph is an acyclic digraph.) The condensation of a graph may be done in Θ(V +E) time. Given that the start-time precedence graph G a is used for finding a lower bound on the number of vehicles, the condensation G c a of G a may be used, too. The new lower bound may be worse because it assumes that each component of G a is served by one vehicle, even though this might not be feasible (Lemma ). Intuitively, the lower bound will be better if the average size of components is small, i.e., if the number of components is close to the original number of locations. The average size of components of G a depends on the number of overlapping time windows, the travel times between the locations, and the size of time windows. (Note that if all the time windows are mutually disjoint, then the corresponding G a is acyclic.) The time window of a component C is set to [a C, b C ] = [min {a i}, max {b i}]. i C i C The start-time precedence graph G a is not necessarily transitive. An example is given in Figure 4 where the time windows are I = [0, 0], I = [0, ], I = [0, ], and the travel times are t, =, t, =, 5

6 time G a 0 locations Figure 4: A TSPTW problem with collinear locations whose start-time precedence graph is not transitive. t, =. When G a is not transitive we will use its transitive closure. The transitive closure of a graph increases the number of edges, thus enlarging the set of feasible solutions of the MCC problem. But, since graph G a is used for finding a lower bound, the transitive closure of G a may be used as well. Solution of the decomposition by chains is a set of disjoint chains. It seems that the number of directed paths in the cover of nodes may be smaller than the number of chains in the decomposition, because the paths do not have to be disjoint. However, in a transitive digraph each set of directed paths may be directly transformed into a set of mutually disjoint chains. Algorithm for finding a lower bound on the minimum number of vehicles needed to serve locations of an m-tsptw may be written as follows: Input: Output: An m-tsptw instance. The number of chains is a lower bound on the minimum number of vehicles needed to serve all locations of m-tsptw. begin create precedence graph G a G a condensation of G a G a transitive closure of G a solve the decomposition by chains problem over G a by solving the corresponding bipartite matching problem end Algorithm for finding an upper bound on the number of vehicles needed to serve locations of an m- TSPTW is similar, except that condensation and transitive closure are not needed. Complexity of the algorithm for finding a lower bound is O( V ) due to the time needed for generating the transitive closure. Complexity of finding an upper bound is O( V E ) due to the complexity of the maximum bipartite matching. 6

7 Variations of m-tsptw and their precedence graphs. The m-tsptw with a depot When one location of the m-tsptw is designated as a depot, the problem of finding the vehicle bounds can be transformed into a no-depot problem by setting the release time of location i to a i max (a i, a 0 + t 0,i ), i, where 0 is the depot and a 0 is the earliest departure time of a vehicle from the depot. Now, the precedence graphs may be built over the set of locations excluding the depot and the algorithms described in Subsection. may be used for finding the bounds.. The m-tsptw with many depots When there are several depots, first find the closest depot to each of the locations, and then set the release time of location i to a i max (a i, a d(i) + t d(i),i ), i, where d(i) is the depot closest to i and a d(i) is the earliest departure time of a vehicle from depot d(i). In other words, a i max (a i, min (a d + t d,i )), where d minimum is found over all depots. Now, the precedence graphs may be built over the set of locations excluding the depots, and the algorithms described in Subsection. may be used for finding the bounds. Note that the algorithms are correct only if each depot has sufficient number of vehicles. The number of sufficient vehicles in depot d is equal to the number of chains (in the decomposition by chains solution) whose first location is closest to depot d. Unfortunately, it can happen that the chain decomposition generates solution with k chains all starting with locations closest to depot d and depot d has less than k vehicles available. In this situation only the lower bound remains valid. The upper bound may be found by decomposition in chains of the graph that includes the depots nodes. Additional constraints are that each chain has to start with a depot node. Also, each depot may be a starting node for just a certain number of chains. The corresponding matching problem is the degree-constrained matching problem, where the matching degree of a each depot is bounded above by the number of vehicles available in the depot. Another way of finding the upper bound is by transforming the problem into the m-tsptw version in which each vehicle has its own starting position. This particular problem is discussed in the next subsection.. The m-tsptw with different vehicle starting positions A variation of m-tsptw is the problem where each vehicle j (j =, m) has its own starting position s j. In order to find vehicle bounds, first include the starting vehicle positions S = {s, s,... s m } in graph 7

8 G. Build the precedence graphs and solve the chain decomposition problem with additional constraints: each chain has to start with s j for some j, each chain has to start with a distinct node, and not all s j have to be covered. The first constraint may be made redundant by setting the time window of s j to [a sj, a sj ], where a sj is such that a i + t i,sj > a sj holds for any location i. In other words, a starting vehicle position cannot be served after any of the locations. The second constraint is redundant because a solution to chain decomposition contains disjoint chains. The third constraint makes the problem more complex. The constrained covering of the greaph nodes by paths: Having a directed graph G = (V S, A), cover all nodes in V by a minimum number of directed paths such that each path starts at a node from S. Not all nodes from S have to be covered. The problem may be solved by the following algorithm. Input: Output: An m-tsptw instance with n locations and m vehicles. Starting position of vehicle v j is s j. A lower bound for the minimum number of vehicles needed to serve all the locations. begin for each vehicle j {,..., m} create the start-time precedence graph G a (s j ) as if s j is the only depot of the given m-tsptw instance with n locations find the minimum chain decomposition over G a (s j ) denote this partition of locations by P j end denote by P = {P,..., P m } family of all m partitions of set V each partition corresponding to one of m vehicles find minimum cover of set V by subsets from P j, j =, m such that at most one subset is used from each partition end The minimum set cover the last step of the algorithm may be written as a linear program. Before presenting the formulation, a simple example will be discussed. Consider an m-tsptw instance having three locations and two vehicles. Suppose that the chain decomposition of the start-time precedence graph G a (s ) generates partition P = {L, L } = {{}, {, }}, while the other start-time precedence graph G a (s ) generates partition P = {L, L 4, L 5 } = 8

9 {{}, {}, {}}. We have that P = m =, L = {L, L, L, L 4, L 5 } = 5, and V = n =. Generate (0, )-valued matrix A whose each column corresponds to one set in L and each row to one location. The matrix entry A[i, j] is if location i is in set L j. Generate, also, (0, )-valued matrix B whose each column corresponds to one set in L and each row to one partition P j. The matrix entry B[i, j] has value if set L j is in partition P i. In our example we have A = B = Let generate column vector x = [x, x, x, x 4, x 5 ] of variables such that x j = corresponds to set L j being part of the solution. The described generalized minimum set cover problem may be written as following linear program: min such that xi Ax e Bx e x {0, }, where e is vector of s..4 When vehicles ending positions are given When the vehicles have to return to depot(s) at the end of the service period, or the ending vehicle positions are specified, together with the latest arrival times, the end-time precedence graph may be changed similarly as the start-precedence graph is changed due to the existence of depot(s). The formula b i min (b i, min (b d t i,d )) may be used or similar one if each vehicle has to return to its own depot. d 4 When the bounds are tight? This section presents classes of m-tsptw instances for which one or both of the bounds are tight. First, the instances in which G a = G b are considered. Since G b G a it is enough to find the instances in which G a G b ((i, j) G a (i, j) G b ). In other words, we are seeking an instance whose each two 9

10 locations i, j satisfies a i + t i,j b j b i + t i,j b j. The second subsection gives some characteristics of instances whose lower bound v is equal to the minimum number of vehicles v. 4. When the two precedence graphs are equal? Lemma Consider a m-tsptw instance whose time windows are of a constant width, i.e., b i a i = c, for a given constant c. Then, G a = G b if (i, j)(a i + t i,j a j ) (a i + t i,j > b j ). Proof: The first case a i + t i,j a j implies (i, j) G a. Since b i + t i,j = a i + c + t i,j a j + c = b j the edge (i, j) belongs to G b, too. The second case imposes that (i, j) does not belong to G b, and we have to prove that it does not belong to G a either. Precisely, the following holds (i, j) G b b i + t i,j > b j a i + t i,j > b j c = a j. This implies that a i + t i,j > b j because the pairs of the locations satisfy either a i + t i,j a j or a i + t i,j > b j. Further, a i + t i,j > b j (i, j) / G a. Label by T the class of instances described in this lemma. Instances with a constant time window widths not belonging to T are those for which (i, j) such that a j < a i +t i,j b j, i.e., for which a vehicle leaving location i at a i reaches location j within its time window, I j. These instances are characterized by G a G b, and may have v v. Lemma 4 Consider a m-tsptw instance in which the travel time between any pair of two locations is constant, t i,j = c, the size of each time window is equal to the constant, b i a i = c, each time window starts at a multiple of the constant, a i mod c = 0, and a i a j, i j. Then, the two precedence graphs are identical. Proof: We prove that (i, j) G a (i, j) G b, i.e., that a i + t i,j b j implies b i + t i,j b j. Inequality a i + t i,j b j means that a i + c = b i b j a i a j a i < a j. Therefore a j = a i + cp for some integer p > 0. Then we have b i + t i,j = a i + c + t i,j = a i + c + c a i + cp + c = a j + c = b j. Label by T the class of instances described in this lemma. Notice that T T holds. Lemma 5 Consider a m-tsptw instance in which travel time between any pair of two locations is a constant, t i,j = c, the size of each time window is equal to a constant, b i a i = c ε > 0, and each time window starts at a multiple of c, a i mod c = 0. Then, the two precedence graphs are identical. 0

11 Proof: We prove that (i, j) G a (i, j) G b, i.e., that a i + t i,j b j implies b i + t i,j b j. The left inequality of the implication is a i + t i,j = q i c + c b j = a j + c ε = q j c + c ε, for some integers q i, q j > 0. This implies (q i + )c (q j + )c ε, which implies q i < q j because q i, q j are positive integers and ε < c. On the other side we have b i + t i,j = a i + c ε + c = q i c + c ε and b j = a j + c ε = q j c + c ε. Therefore, b i + t i,j b j because q i < q j (q i + )c q j c. Label by T the class of instances described in this lemma. Notice that the following T T holds. Notice also that the constant ε appears in the assumptions of this theorem in order to remove the restrictions of Lemma 4 of having two equal time windows, and to avoid the situation [a i, b i ] = [a j, b j ] t i,j = c implying (i, j) G a (i, j) / G b. Lemma 6 Consider a m-tsptw instance in which the travel time between any pair of two locations is a multiple of a constant, t i,j = p i,j c, the size of each time window is equal to a constant, b i a i = c ε, and each time window starts at a multiple of c, a i mod c = 0. Then, the two precedence graphs are identical. Proof: Let us prove that a i + t i,j b j implies that b i + t i,j b j. a i = q i c b i = q i c + c ε a j = q j c b j = q j c + c ε a i + t i,j b j q i c + p i,j c q j c + c ε (q i + p i,j )c (q j + )c ε q i + p i,j < q j + q i + p i,j + q j +. The last equivalence relation is true because p i,j, q i are integers. On the other, starting from b i + t i,j b j we have: b i + t i,j b j q i c + c ε + p i,j c q j c + c ε (q i + + p i,j )c ε (q j + )c ε q i + + p i,j q j +,

12 which is true when a i + t i,j b j. Label by T 4 the class of instances described in this lemma. It is obvious that T T 4 T. Relation between time window widths and travel times between locations makes the difference. As long as the time windows are of equal widths and small enough such that the width may be taken as a time unit for the travel times, a situation may arise in which the two precedence graphs are equal. The bounds coincide and they are equal to the minimum number of vehicles needed to serve all the locations. Thus, if all the time windows are of width c = 5 = 4 minutes, and travel times are multiples of 5, i.e., t i,j {5, 0, 45,...} the lower and the upper bounds coincide. Instances with these characteristics may be good enough for modelling the Dial-a-Ride Problem (DARP) except that in DARP usually has additional constraint: waiting is not allowed when a vehicle is carrying a passenger. (With this additional DARP constraint, the lower bound would still be valid, but there would be no indication how good it is.) 4. When the lower bound is equal to the optimum? Lemma 7 If locations are collinear and if G a is a transitive graph, the lower bound coincides with the minimum number of vehicles, i.e., v = v. Proof: When a path of the start-time precedence graph G a contains only collinear locations and when it is transitive, all locations on the path may be served by one vehicle. The last path locations is the last location in the sequence of locations. The last path location may be served by a vehicle starting at the release time of the first path locations of the path. Thus, all other path locations may be served on the way because the path contains only collinear locations. Even waiting at an intermediate location i (if i is reached before a i ) will not impact feasibility of serving the rest of the route: location i is the first location a subpath of the original path, and it is a transitive path whose all locations can be served by one vehicle under the condition that i is left at its release time, which is satisfied. Each path in G a is transitive and contains only collinear locations. Thus, the lower bound coincides with the minimum number of vehicles needed.

13 5 Arbitrary bad bounds 5. Arbitrary bad lower bound The m-tsptw, as many time-constrained vehicle routing problems, is a problem for which arbitrary bad instances can be built. For reference see Solomon (986) where it has been proven for a variety of widely used heuristics that the worst-case ratio performance is Ω(n). The examined objectives were the minimization of the number of vehicles, the minimization of the total distance travelled and the minimization of the total schedule time. time... 4 n locations Ga... 4 n Gb 4... n Figure 5: The m-tsptw instance containing n locations whose v = << v = n/ < v = n. Travel time between each two neighboring locations is. The width of each time window is. Figure 5 shows an instance whose lower bound determined by the precedence graphs and the chain decomposition is arbitrary bad, i.e., v v = Ω(n), while Figure 6 depicts an instance whose upper bound is arbitrary bad, i.e., v v = Ω(n) Therefore, even if time windows are small and of equal width bad instances can be made. Similar situation is in the problems with depot(s). 5. Two similar instances: one with tight bounds and the other with loose bounds Only slight change in an instance with tight bounds v = v can create an instance with arbitrary loose bounds v v = n = Ω(n). Consider an instance from class T whose locations are collinear, the travel time between each two adjacent locations is t i,i+ = c, and the time windows are of the same width b i a i = c ε. If the time windows are such that a i+ = a i +c the lower bound will be equal to the upper

14 time n... 4 n locations G a... 4 n Gb 4... n Figure 6: The m-tsptw instance containing n locations whose v = = v = << v = n. Travel time between each two neighboring locations is. The width of each time window is n. bound (Lemma 5). Consider a similar instance that differs only in the time windows being slightly moved down : set a i+ = a i +c δ. Then it is valid a j = a i +(j i)(c δ) and a n = a +(n )c (n )δ. Fixing constant δ = c ε n implies that for ( i, j)i < j (i, j) G a. It is so because a i + t i,j = a i + c b j, i < j, which is a consequence of the following a i + t i,j = a i + c(j i) a j + c ε a i + c(j i) a i + (j i)(c δ) + c ε 0 c ε δ(j i) ( ) c ε 0 c ε (j i) n 0 (c ε)( (j i)) n ( ) (n ) (j i) 0 (c ε) n which is true for n > because c ε > 0 due to the definition of ε, and j i n. Consequently, the lower bound v =. 4

15 On the other hand, the upper bound is v = n, because for every pair (i, j) of locations the inequality b i + t i,j > b j is valid. Assume the opposite: b i + t i,j b j a i + c ε + c(j i) a j + c ε = a i + (j i)(c δ) + k ε 0 δ(j i) which is not true because δ > 0 and j > i. Consequently, i < j, (i, j) / G b. Since i < j, (j, i) / G b, the upper bound is v = n. 6 Conclusion This paper introduces two precedence graphs that we have used for finding a lower bound and an upper bound for the minimum number of vehicles needed to serve all locations of the m-tsptw problem. The paper presents an algorithm for finding the bounds based on the decomposition by chains problem from the theory of partially ordered sets. We present certain classes of m-tsptw instances that guarantees tight bounds. Also the closeness between instances with tight and loose bounds is discussed. We can conclude that based on the size of time windows alone no conclusion may be made of how good the bounds are. The paper indicates that important factor is mutual relation of the time windows together with travel times between corresponding locations. References Ball, M.O., T.L. Magnanti, C.L. Monma and G.L. Nemhauser (eds.) (995) Network Routing, Handbooks in Operations Research and Management Science, Volume 8, North-Holland, Amsterdam. Cormen, T.H., C.E. Leiserson and R.L. Rivest (99) Introduction to Algorithms, McGraw-Hill, New York. Dilworth, R. (950) A decomposition theorem for partially ordered sets, Annals of Mathematics 5, Ford, L.R. and D.R. Fulkerson (96) Flows in Networks, Princeton University Press, Princeton, NJ. Garey, M.R. and D.S. Johnson (979) Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York. Mitrović-Minić, S. (00) The dynamic pickup and delivery problem with time windows, Ph.D. Dissertation, School of Computing Science, Simon Fraser University, Burnaby, Canada. Robinson, D.F. and L.R. Foulds (980) Digraphs: Theory and Techniques, Gordon and Breach Science Publishers, New York. Savelsbergh, M.W.P. (985) Local search in routing problems with time windows, Annals of Operations Research 4,

16 Solomon, M.M. (986) On the worst-case performance of some heuristics for the vehicle routing and scheduling problem with time window constraints, Networks 6, Toth, P. and D. Vigo (00) The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia. Trotter W.T. (99) Combinatorics and Partially Ordered Sets, The Johns Hopkins University Press, Baltimore. 6