NOT FOR QUOTATON WTHOUT PERMSSON OF THE AUTHOR THE HRSCH CONJECTURE FOR DUAL TRANSPORTATON POLYHEDPA M.L. Balinski FeSruary 983 CP- 8 3-9 CoZZaborative Papers report work which has not been performed solely at the nternational nstitute for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the nstitute, its National Member Organizations, or other organizations supporting the work. NTERNATONAL NSTTUTE FOR APPLED SYSTEMS ANALYSS A-236 Laxenburg, Austria
ABSTRACT An algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most (m- ) (n- ) extreme edges.
THE HRSCH CONJECTURE FOR DUAL TRANSPORTATON POLYHEDRA M.L. Balinski Laboratoire d'~conom6trie de 'Ecole Polytechnique, Paris, France The distance between a pair of extreme points of a convex polyhedron P is the number of extreme edges in the shortest path that joins them. The diameter of P is the greatest distance between any pair of extreme points of P. The Hirsch conjecture (see [5], pp. 60 and 68, [7]) is that the diameter of a convex polyhedron defined by q halfspaces in p-dimensional space is at most q-p. n linear programming jargon it is that given r linearly independent equations in nonnegative variables it is possible to go from one feasible basis to any other in at most r pivots all the while staying feasible. For unbounded polyhedra in dimension 4 or more the Hirsch conjecture is false [7], so it is false in general. However, it has been proven to be true for certain special cases: the polytopes arising from the shortest path problem [8], Leontief substitution systems [6], the assignment problem [4], and certain classes of transportation problems []. will show it is true for the unbounded polyhedra arising from dual transportation problems. The approach will also establish a combinatorial characterization of extreme points that has proven to be very useful. is The dual transportation polyhedron for the m by n matrix c w
where R ( = m and c ~ = n. Setting ul = 0 is an arbitrary choice that rules out lines of solutions (ui+6) (vj-6). t is natural to study the extreme points of D in terms of a bipartite graph model. Let the set of m nodes R stand for the rows of c and the set of n nodes C for the columns. The equations N u =o,ui+v - for (i,j)et, ier, j EC, form a maximal j 'ij linearly independent set if and only if T is a spanning tree.." - only if ui + vj 2 cij for i, j The unique solution u, v to T is an extreme point of D if and 4 T : will say T has the extreme point u, v and for clarity will sometimes write T(u,v). Given a U N U U spanning tree T the unique solution to U u3 U4 R: Row Nodes v2 v3 v4 v5 V6 C: Column Nodes Figure. Spanning tree. Signature (3,2,,3) its equations is immediate. that have extreme points u, v are considered: corresponds exactly one extreme point. - -4 n the sequel only spanning trees to any T there To one extreme point, however, there can correspond many trees T: this happens when ui + v - called "degeneracy". for (i,j) j - 4 T and is 'ij n this case any spanning tree chosen from {(i,j);ui+vj =cij} has the same extreme point. The (row) signature of a tree T is the uniquely defined vector of the degrees of its row nodes a = (al,..., a lai = m+n-, -4 m ai = >. Lemma. Two different trees T, T with the same signatures have one and the same extreme point.
Proof. Let the extreme points of T and T' be u,v - and u,y. U 4.l will show that u = u, v = v. T # T means there is some node il ER (il,jl) ET but (iltjl)$t (see Figure 2 for what follows). n T let (i2,jl) with i2 # il be the edge on the unique path that joins jl to il, and consider node i2..v Figure 2. Solid lines in T; dashed lines in T. t must have degree at least 2 in T and so in T. Therefore there exists an edge (i2, j2) ET, j2 # jl. NOW, let (ijtj2) be the edge on the unique path that joins j2 to il in T. Continue to build this path until a node already on it is encountered again,forming a cycle: (ih,jh), (ih+ltjh)..-t(irjr)' (ihtjr)- Call the edges of type (ik,jk) of the cycle odd, and of type (ik+ lk) and (ih,jr) even. Then for (i,j) odd, ui j + = 'ij and Ui < c. for (i,j) odd, + Vj = lj ui + vj = < c. lj for (i,j) even ui + vj = cij for (i,j) even. Summing, implying equality holds throughout and so
u + v = = U + V' i j 'ij i j for all (it j) in the cycle- Transform T' by taking from it all even edges and putting in it all odd edges. The new T' has the same signature and the same u,vl but more edges in common with T. Repeat until T = T,..) U 4 showing u = u, v = v...).y.y U Given T(u,v), let (kt&) be one of its edges with k and R U U both of degree at least 2. A pivot on (ktk) obtains T (ul,vl) N U as follows (see Figure 3) : drop (k,r) from T to obtain two connected components, T k containing k and T' containing 2. Let k E =min {cij-ui-v ~ET',~ET 2 0 and (g,h) be some j ' - k R edge at which this minimum is obtained. Set T = T UT u (g,h) ( (g, h) is the "incoming" edge). f row node E T~ define - ui - ui + E, iet, ui = ui otherwise, v = v - E j ET, v = v otherwise ; j j j j and if row node E T' define ui = ui - E, k iet, ui = ui otherwise, k v = v + E, jet, v = v otherwise. j j j j E 2-0 because u,v satisfies all inequalities. The choice of E - guarantees that u,! satisfies them all as well and that it " belongs to T. T': a'= (, 3,, 2, 3) C Figure 3. Pivot from T to T.
f E: = 0 then (u,v) = u, ) and we have two different Y -4-4 trees having the same extreme point (degeneracy). f E > 0 then u,v. and u,vl are neighbors, connected by an extreme edge of D. Y 4 - n either case, if a is the signature of T then the signature a' - -4 of T' is the same except that a: = ak - and a = a +. g g - Theorem. The diameter of D m,n (c) is at most (m-)(n-). This bound is the besz possible, Proof. give a method that constructs a path of at most (m-l)(n-) extreme edges between any pair of extreme points. The idea is to begin at one extreme point (the "initial" one) and to pivot in order to obtain a tree T whose signature is equal to that of the other extreme point (the "destination"): for then, by the lemma, T has as its extreme point the desired one. Let a be the signature of the current tree T (e.g., the U J initial one), and at the signature of the destination extreme *- point. f ai < ail i is a deficit node. f there are d deficit * nodes, m-d is the number of nondeficit nodes. f a, > a,, i is * * ~ a surplus node. The net deficit is {E(ai- ai) ;ai> ail. The method has the property that the number of surplus nodes never increases and within at most m-d pivots the net deficit must decrease by. Choose some surplus node and designate it the source s and some deficit node and designate it the target t. Pivot on the edge (s,r) incident to the source s that is on the unique path joining s to the target t (see Figure 4). Call Q the set of row nodes of the component of T- (s,r) that contains t. S~Q. The degree of some geq increases by : either (i) it was not a deficit node of T or (ii) it was. Fiaure 4.
(i) f not, name it the new source s and repeat: pivot on (s, l? ) the edge on the path joining s' to t in T. The set of row nodes Q of the component T. (s, l? ) containing t belongs to Q but must be smaller: s Q. Each time a nondef icit node s degree goes up it is immediately brought down and cannot again increase unless the target node is changed. most m-d pivots a case (ii) must occur. (ii) The net deficit decreases by. zero, the desired tree is found. target nodes and continue. Therefore, in at f the net deficit is Otherwise, name new source and The net deficit can be at most n-; the number of nondeficit nodes at most m-: number of pivots and so on the distance. this gives the upper bound (m-) (n-) on the The bound is best possible. Consider the polyhedron D m,n (c).y with cij = m i ) ( - Suppose il < i2 and jl < j2: it is impossible to have both (il, j2) and (i2, j ) in a tree T (u,v). This.y a, implies that the trees T of this D (c) are characterized as all m,n those that have "no crossings" (see Figure C) -- it being under- stood that the row and column nodes are drawn in their natural orders. n particular, this polyhedron admits no degeneracy. n pivoting from one tree to a neighbor if node its degree de- creases by then the degree of either node i+l or node i- must increase by. Therefore, to decrease the degree of node by and increase that of node m by it takes m- steps. This shows that to go from the extreme point with signature (n,l,..., ) to that with (,...,l,n) it takes (m- ) (n-) steps. Figure 5. A tree with "no crossings".
An immediate result of the foregoing is: Theorem 2. To every integer vector a, ai 2, lai = m+n- L. there corresponds an extreme point u,v. Proof. * - * Given any such vector a the method of the above proof finds an extreme point having a as its signature. L. So for nondegenerate polyhedra D (c) there is a one-to-one m,n. correspondence between extreme points and signatures. This char- acterization enables one to describe and count all faces of D (c) [3]. t has also motivated a new algorithm for the asm,n - signment problem that is guided entirely by the signatures and terminates in at most (n-)(n-2)/2 pivots [2.. L. Acknowledament t is a pleasure to acknowledge the fact that part of this work was done during my visit to..a.s.a. during the summer of 982.
References [] M.L. Balinski, "On two special classes of transportation polytopes," in M.L. Balinski (ed.), Pivoting and Extensions, Mathematical Programming Study, North-Holland Publishing Co., Amsterdam, ( 974), 43-58. [2] M.L. Balinski, "Signature methods for the assiqnment problem," research report, Laboratoire d~conom6trie de 'Ecole Polytechnique, (November, 982). [3] M.L. Balinski and A. Russakoff, "Faces of dual transportation polyhedra," research report, Laboratoire deconom6trie de llecole Polytechnique, (November, 982). [4] M.L. Balinski and A. Russakoff, "On the assignment polytope," SAM Review, 6 (974), 56-525. [5] G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, N.J. (963). [6 R.C. Grinold, "The Hirsch conjecture in Leontief substitution systems," working paper, Center for Research in Management Science, University of California at Berkeley, (March, 970). [7] V. Klee and D.W. Walkup, "The d-step conjecture for polyhedra of dimension d < 6," Acta Mathematics, 7(967), 53-78. [8] R. Saigal, "A proof of the Hirsch conjecture on the polyhedra of the shortest route problem," SAM JournaZ of Applied Mathematics 7 (969, 232-238.