Minimax
Trees,
Paths,
and
Cut
Sets
H. w. Corley
Department of Industrial Engineering. The University of Texas at Arlington.
Arlington, Texas 76019
H. Golnabi
Rockwell International. Collins Telecommunications SystemsDivision. P.O. Box
10462, Dallas. Texas 75207
The notions of minimax paths,trees, and cut setsare definedfor undirectedgraphs;and
relationshipsbetweentheseentitiesare established.Simple solutionproceduresbasedon
theserelationshipsare presented.
Consider the undirected, weighted, and connected graph G = (V ,A) with set of
vertices V and set of arcs A, where undefined terms will be taken as in [1]. Let djj
denote the weight assigned to the arc a = (Vj,Vj) EA with terminal vertices Vi and
Vj. A minimax spanning tree of G = (V,A) is a spanning tree (V,T*) which solves
the problem minTE:Im~V;.VpETdjj, where any T E '3 is a set of arcs in A determining
a spanning tree of G. Similarly, a minimax path between vertices r, s E V is a solution
p* E g>,the set of all paths between r and s, to minpE!Pmax(V;.VpEP
dij, and a maximin
cut set between vertices r and s is a cut set C E e, the set of all cut sets relative to r
and s, solving maxcEemax(V;.VpEC
d;j. The shortest of such entities involve in an obvious
way the minimization of the sum of the appropriate dijs as a secondary criterion. The
purpose of this note is to relate the above problems and to provide simple solution
procedures based on the relationships. The modification to directed graphs is evident.
RESULT I: A tree is a shortest minimax spanning tree of G if and only if it is a
shortest spanning tree. Furthermore, if p is the path between vertices r and s in any
shortest spanning tree, then p is a minimax path in G between r and s .
Result 1, stated withqut proof, is a refinement of known facts (see [1, p. 139,2,
3]): for example, a shortest spanning tree is a minimax spanning tree. However, a
minimax spanning tree is not necessarily a shortest spanning tree unless stated as
above. Similarly, any path from r to s in simply a minimax spanning tree is not
necessarily a minimax path.
Result 1 provides the basis for finding all (shortest) minimax trees and paths between
r and s in G = (V,A). For (shortest) minimax trees, let a be the maximum d;j in any
shortest spanning tree obtained using some standard algorithm, eliminate all arcs from
A with dij > a, and find all (shortest) spanning trees for the remaining graph using
established procedures. For (shortest) minimax paths, let r3 be the maximum dij in the
path between r and s in any shortest spanning tree, eliminate from A all arcs with
djj > r3, and find all (shortest) paths in the remaining graph using existing algorithms.
Naval ResearchLogistics Quarterly, Vol. 30, pp. 665-666 (1983)
Copyright @ 1983by John Wiley & Sons,Inc.
CCC 0028-1441/83/040665-02$01.20
An analogous method for obtaining a maximin cut set is based on the following
r~sult, where I]' is the collection of all paths in G between vertices r and s and e the
collection of all cut sets relative to r and s. The proof is similar to the proof of a result
established in [ 1, p. 175] relating maximin paths and minimax cut sets.
RESULT 2:
min max dji = max max dij.
PE!P(vj.v;JEP
cEe (vj.v;JEC
In the following new algorithm based on Results 1 and 2 for determining a maximin
cut set between vertices r and s, a minimax path between r and s is first determined
to give the maximin cut value 'Y = maxCEemin(vi.vjJEc
dij. The set of (Vi' Vi) with dki ~ 'Y
is then a maximin cut for G, though not necessarily proper. A proper maximin cut set
of G is obtained below.
STEP I. Find a shortest spanning tree of G = (V,A) and the maximum arc length
'Y in the path between r and s. Set k = 1.
STEP 2. Set Ek = {(Vi'VJ: dij ~ 'Y} and C = 0. Eliminate from A all arcs (Vj,vJ
in Ek to give a graph Gk = (V,Ak).
STEP 3.
Step 4.
Stop if Ek = 0; C is then a proper maximin cut set. If Ek ~ 0, go to
STEP 4. Select an
Ek+1 = Ek\{(Vj,VJ}.
arc
(vivi) E Ek such that
dij = min(v...voJEEt
dk/.
Set
STEP 5. Add (Vj,vJ to Ak for Gk to give a graph of Gk. If there is a path from r
to s in Gk, redefine C to be {(Vj,vJ} U c. If not, set Gk+l = Gk, k = k + 1, and
go to Step 3.
Choosing the minimum weight in Step 4 expedites the process only in that an arc
of weight 'Ymust be included in the maximin cut set. This minimization is not required.
The above algorithm does not necessarily yield a shortest maximin cut set. To obtain
such a cut set, one approach is to form the graph GM with the same vertices and arcs
as G but with any weight dij < 'Y replaced by a very large positive number M (much
as in the big M method from linear programming). Any shortest, or minimum, cut set
of GM is a shortest maximin cut set of G.
REFERENCES
[1] Christofides,N., Graph Theory:An Algorithmic Approach, Academic, New York, 1975.
[2] Fulkerson,D. R. , , 'Flow NetworksandCombinatorialOperationsResearch,' , TheAmerican
MathematicalMonthly, 73, 115-138 (1966).
[3] Hu, T. C., "The Maximum CapacityRoute Problem," OperationsResearch,9, 898-900
(1961).