On minimal cost-reliability ratio spanning trees and related problems

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E L S E V I E R Operations Research Letters 19 (1996) 65-69

On minimal cost-reliability ratio spanning trees and related

problems l

Y u n g - C h e n g C h a n g a, L i h - H s i n g H s u b,,

a Institute of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC b Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC

Received 1 July 1994; revised 1 October 1995

Abstract

The minimal cost-reliability ratio spanning tree problem is to find a spanning tree such that the cost-reliability ratio is minimized. This problem can also be treated as a specific version of a more generalized problem discussed by Hassin and Tamir. By Hassin and Tamir's approach, the minimal cost-reliability ratio spanning tree problem can be solved in O(q 4) where q is the number of edges in the graph. In this paper, we reduce the complexity of the algorithm proposed by Hassin and Tamir to O(q 3 ). Furthermore using our approach, related algorithms proposed by Hassin and Tamir can also be improved by a factor of O(q).

Keywords: Combinatorial algorithms; Complexity; Spanning trees

I. Introduction and notation

The minimal cost-reliability ratio spanning tree problem, or M C R R S T problem for abbreviation, was first discussed by Chandrasekaran et al. [2, 3]. A non-polynomial algorithm for the M C R R S T problem was proposed in [3]. Then a polynomial but not strongly polynomial algorithm was introduced by Chandrasekaran and Tamir in [4]. Their algorithm is based on the fact presented in [4] that a query o f the form "Is a b >>, ca? '' can be solved in time which is polynomial in the binary encoding o f the numbers

* Corresponding author. Fax: 886-35-721490. E-mail: lhhsu@cc. nctu.edu.tw.

1This work was supported in part by the National Science Council of the Republic of China under contract NSC83-0208- M009-034.

a, b, c, and d. Later, Hassin and Tamir [5] developed a different, unified approach that yields a strongly polynomial algorithm for classes o f optimal spanning tree problems which include the M C R R S T problem. By Hassin and T a m i r ' s approach, the M C R R S T problem can be solved in O(q 4) where q is the number o f edges in the graph. In this paper, we reduce the complexity o f the algorithm proposed by Hassin and Tamir to O(q 3 ).

N o w we formally introduce the M C R R S T problem. Most o f the graph definitions used in this paper are standard (see, e.g., [1]). Let G = ( V , E ) be a graph. We associate with each edge ei E E an ordered pair o f rational numbers (ai, hi), namely a non-negative cost ai and a positive probability bi. For a spanning tree T, the cost o f T , c ( T ) , is defined as ~ e i e r ai and the reliability o f T, r ( T ) , is defined as I-Iei~T bi. Natu- rally the cost-reliability ratio o f T, w ( T ) , is defined 0167-6377/96/$15.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved

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66 Y.-C. Chano, L.-H. Hsu/Operations Research Letters 19 (1996) 65-69 as c(T)/r(T). The MCRRST problem is to find a

spanning tree T in G such that w ( T ) <<. w ( T ' ) for every spanning tree T t in G. The MCRRST problem is a specific version of the following generalized problem discussed in Hassin and Tamir [5]. Let G -- (V,E) be a graph. In each edge e~ E E is associated with an ordered pair of rational numbers (ai, bi). For a subset E ~ of E, we define A ( U ) = ~ , ~ , ai and B ( U ) = ~e,~e' hi. Let g be a real-valued function defined in E2. The problem is to find a spanning tree T which maximizes g ( A ( T ) , B ( T ) ) over all spanning trees T of G. In [5], various g(A(T),B(T)), such a s l"IeiET bi/~-'~e~CT ai, (~--~,cr ai)2 -1- ( ~ e , CT bi)2'

EeiEl" ai-~t- I-[eicT bi or I-Ie, E T ai'q- ~XeiET bi are stud- ied. In the MCRRST problem, we need to find a spanning tree T that minimizes ~ r ai/I'IeiET bi. This is equivalent to finding a spanning tree T that maximizes I-Ie, cv b i / ~ , ~ r ai. Since the log function is strictly increasing, it is also equivalent to finding a T that m a x i m i z e s ~eiC T log bi - log ( ~ e , ~ r ai ). Thus, the MCRRST problem can be modeled in terms of maximizing g(A (T), B'(T)) = B' (T) - log A (T) with B~(T) = ~-]~r log bi and can be solved in O(q 4) using Hassin and Tamir's algorithm. We propose a modification of their approach which can reduce the time complexity to solve the MCRRST problem to O(q 3) and moreover, improve related algorithms reported in [5] by a factor of O(q).

2. Previous work

To make this paper self-contained, we first outline the basic strategy of Hassin and Tamir's approach. Let g be a real-valued strictly convex function defined in E2. Without loss of generality, we assume that (ai, bi) ~ (aj, bj) if ei ~ ej. Let the value of a spanning T, g(T), be defined as g(A(T),B(T)). A spanning tree T* in G is called an optimum spanning tree (with respect to g) if g(T*) >~ g(T') for all spanning trees T ~ in G. We call a spanning tree T a local optimal spanning tree if there is no pair of elements el, ej E E, such that eiET, ej~: T, and T' = T - {el} + {ej} is a spanning tree which yields a larger value than T does. Hassin and Tamir divided the (A,B) plane into a number of cells and showed that each cell produces at most one local optimal spanning tree. The optimum spanning tree T* is one of these local optimal span-

ning trees. T* will be contributed by the unique cell containing (A(T* ), B(T* )).

More precisely, let (A,B) be a point in E2. Define a directed graph DA,8(G) with the vertex set being the edge set E of G. Let e~, ej be distinct elements in E. [ei, ej] is an arc in DA,B if and only ifg(A - ai -[-

aj,B - bi + bj) > g(A,B). An equivalence relation in E2 can be defined by (A,B) ~ (C,D) if and only if D,~,B(G) -- DC, D(G). We use W to denote the set of equivalence classes induced by " ~ " . For any c E W, we use Dc to denote the directed graph DA,8(G) with (A, B) ~ c.

Let E(D~) be the arc set of D~. Let Tl and T2 be two distinct spanning trees of G. We say that/'2 is a D~-improvement of T1 if there exist ei E TI and ej f Ti such that [ei, ej] EE(Dc) and T2 = TI - {ei} + {ej}, i.e., /'2 is obtained from T1 by a single edge swap. A spanning tree T of G is Dc-optimal if there exists no spanning tree T ~ of G which is a D~-improvement of T. In [5], the following theorem is presented. Theorem 1. There is at most one Dc-optimal span- ning tree o f G for every (A,B) in ~2.

We use Tc to denote the Dc-optimal spanning tree if it exists. Let F ( Dc )( e i ) be the set { e j I [ e i, e j ] E E ( Dc )} for ei E E. Also, letXc be the set {ei[ the two endpoints of e~ in G are on different connected components of the graph H = ( V, F(D~)(ei)) }. The following theorem is also from [5].

Theorem 2. I f the De-optimal spanning tree Te exists, then the edge set o f Tc is exactly Xe.

This theorem states that a necessary condition for the existence of the Dc-optimal spanning tree is that Xc forms a spanning tree of G. IfXc forms a spanning tree, it is a candidate solution. It is suggested in [5] that we do not have to verify that the candidate solution is Dc- optimal. To reduce the computational complexity, it will suffice simply to find the candidate solution. The optimum spanning tree is a candidate solution that has maximum value. The following algorithm proposed in [5], Algorithm 1, finds Xc and then tests whether it forms a candidate solution in a Dc.

Algorithm 1

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Y.-C Chano, L.-H. Hsul Operations Research Letters 19 (1996) 65-69 67

Step 2. Set Xc = 0.

Step 3. For each ex E E, do the following:

If the two endpoints o f ex are disconnected in H =

(V,I'(D~)(e~)), setXc = Xc t2 {ex}.

Step 4. If Xc does not form a spanning tree, stop and conclude that the D~ has no solution. Otherwise, Xc forms the candidate solution.

Obviously, Step 1 in Algorithm 1 takes O(q 2) time. Step 3 needs q tests to see if the endpoints of ex are not connected in H = (V, F(Dc)(ex)). Each test takes O(q) time. Hence, Step 3 takes O(q 2) time. Step 4 is completed in O ( p ) time. Hence, the complexity of the Algorithm 1 is O(q2).

We can also describe the set W as follows: Let ei, e j E E , ei 7 A ej. Define the function gij(A,B) by 9ij(A, B) = 9(,4 - ai + a j, B - bi + by) - 9(A, B ). Since 9 is strictly convex, every topological component in- duced on ~2 by the set of 9ij(A,B) is an equivalent class induced by ,-~. Define Rij = {(A,B) [9(A - ai -+- a j , B - b i + b j ) > 9 ( A , B ) } . L e t f : I~ 2 ~ I~k b e a m a p - ping of ~2 into R k. Further, let Tij, ei, ej E E, ei # ej be a collection o f subsets in ~k such that ( A , B ) E Rij if and only if f ( A , B ) E Tij for el, ej E E , i # j. Sup- pose there exists a polynomial hij(xl . . . xk) such that Tij = {(Xl,... ,xk) I hij(Xl . . . Xk) > 0}. Then the num- ber of elements in W is bounded by the number of topological components induced on ~k by the set of polynomials hij. If d, the maximum degree ofhij, and k, the dimension of ~k, are constant and independent o f q, then it can be proved that the number of equivalence classes will be a polynomial in q.

Hassin and Tamir suggest that we can pick any point for every topological component induced by the set of Rij (or corresponding Tij ) and apply Algorithm 1 to obtain a spanning tree as a candidate solution. Then the optimum spanning tree is the candidate solution that has the maximum value.

For the MCRRST problem, 9 ( T ) = g(A(T), B ' ( T ) ) = B ' ( T ) - log A ( T ) , with A(T) = ZeiET ai, B ' ( T ) = ~-~e, ET log bi. Rij =- { ( A , B t ) [ ( B t - log bi -}- log bj) - log(A - ai + a j ) > B ' - l o g A } = {(A,B')] logA + log bj > l o g bi + log (A - a i + a j ) } = {(A,B')[Abj > bi(A - a i + aj)}. Set f ( A , B ' ) = A and

r , j = z l x > bj - bi J"

Hence, (A,B') ERij if and only if f ( A , B ' ) E Tij for every pair of distinct edges ei, ej E E. Let

bi(aj - ai)

dij - , ei, ej EE, ei ~ ej. b~ - bi

Let W ---- {c [ c is a positive interval induced by the set of 0 and dij, or a set containing a positive dij}. Then, any c E W is either the set of a positive dij for some i and j or the open (positive) interval defined by two consecutive points in the set of 0 and the sorted se- quence of positive {dij}. Assume there are s elements in W. For each c E W, we pick any point r(c) E c as the representative of c. Let S = {r(cl ), r(cz) . . . r(cs)}, with r(ci) < r(cj) if i < j. The following algorithm proposed in [5], Algorithm 2, solves the MCRRST problem.

Algorithm 2

Step 1. Compute and sort the positive num- bers {dij} and obtain the sorted sequence of S, {r(cl ), r(c2) . . . r(cs)}.

Step 2. Construct Dc~ for each ck E W as follows: Add a r c [ei, ej] if and only if one of the following conditions is satisfied.

(a) bj = bi and aj < ai. (b) bj > bi and r(ck) > dij. (c) bj < bi and r(ck) < dij.

Step 3. Use Algorithm 1 to find the candidate solution Xck for each Dc k. Then compute the value for each candidate solution.

Step 4. Find an optimal solution for the objective. Step 1 takes O(q 2 log q). Step 2 needs O(q 4) time since the number of arcs in a Dc~ is O(q 2) and there are O(q 2) elements in S. Since Algorithm 1 takes O(q 2) time, Step 3 takes O(q 4) time. Obviously, Step 4 takes O(q 2) time. Hence Algorithm 2 takes O(q 4) time.

3. Our algorithm

Observe that Steps 2 and 3 of Algorithm 2 are repeated several times. To avoid repeated execution of these steps, we should extract and reuse information from what we have solved. Thus, we need the following observation.

Without loss of generality, we assume that bi # bj ifei y~ ej and each dij is different. There are s elements

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68 E-C Chang, L.-H. Hsul Operations Research Letters 19 (1996) 65-69 in S where s <~ 2q(q - 1 ) + 1. Moreover, {r(ck) I k is

even} is the set of positive dij's. Following the rule that constructs De k's in Step 2 of Algorithm 2, we have the following theorem, Theorem 3.

Theorem 3

(1) Assume that r(ck) = dij for some i and j. Then, if bj < hi, E(D~k ) = E(Dck_~ ) - {[ei, ej]}. Otherwise, E(Dc~ ) = E(D~k_, ).

(2) Assume that r ( c k - l ) = dij for some i and j. Then, if bj > b~, E(Dc, ) = E(Dck_, ) + {[e~,ej]}. Otherwise, E(D~ ) = E(D~_, ).

The following corollary, Corollary 1, follows from Theorem 3.

Corollary

1. Assume that r(ck ) = d i j for some ei and

ej. F ( D c ~ )(ex) = F(Dck )(e~ ) = F(D~,_, )(ex)for every ex such that e~ :fi ei and ex ~ ej.

From Corollary 1 and Step 3 in Algorithm l, which constructs X~ for a De, we have the following corollary, Corollary 2.

Corollary

2, Assume that r(ck )=dij Jbr some ei and

ej. We have

(1) Xc~ = X,~_, - {ei, ej} t._) { e x l e ~ E { e i , ej}, the two endpoints o f ex are disconnected in H =

(v, r(D~,

)(ex))}.

(2) X¢~+, = X,., - {ei, ei} tA {ex[ex<{ei, ej}, the two endpoints of ex are disconnected in H =

(v, r(Dc,_,

)(ex ))}.

We now propose the following algorithm, Algorithm 3, for the MCRRST problem.

Algorithm 3

Step

1. Compute and sort the positive num-

bers {dij} and obtain the sorted sequence of S,

{r(cl

),

r(c2)

. . . r(Cs)}.

Step

2. Set k = 1. Construct De, as follows:

Add arc [ei, ej] if and only if one of the following conditions is satisfied.

(a) bj = bi and aj < ai. (b) b / > bi and r(ck) > d~/. (c) bJ < bi and r(ek) < dij. Compute F(D< )(ex) for all ex C E.

Step

3. Set X~., = qS.

For each ex E E, do the following:

If the two endpoints of ex are disconnected in H = (V,F(Dc.)(ex)), set X~, =Xc, tO {ex}.

Step

4. If k = s, go to Step 8.

Step

5. Set k = k + 1. Construct Dck with the rules

in Theorem 3.

For ei and ej where r(ck) = dij or r ( c k - l ) = dij, compute F(Dc~ )(ei) and F(Dck )(e j).

Step

6. Set Xck =X~_, - {ei, e j } .

For e~ = ei and ex = e j, do the following:

If the two endpoints of ex are disconnected in H = (tl, r(Dc, )(ex)), set X~, = Xc~ u {ex}.

Step

7. Go to Step 4.

Step

8. For 1 <~ k ~< s, find the X~ k which forms the

candidate solution with the optimal objective value then stop.

Step 1 takes O(q 2 log q). Step 2 needs O(q 2) time. Step 3 computes Xc, and takes O ( q 2) time. Steps 5 and 6 can be finished in O(q) time. Since Steps 5 and 6 are executed O(q 2 ) times, the complexity is O(q 3). Step 8 finds the Xc~ that yields the optimal objective value in O(q 3 ) time. Hence, the complexity of Algorithm 3 is O(q3).

Algorithm 3 reuses the Dc, F(D~)(ex) for all ex E E, and X~ generated from a previously computed adjacent equivalence class. Steps 1-3 actually do the same thing Algorithm 2 does to Dc,. Steps 5 and 6 apply Theorem 3 and Corollaries 1 and 2 derived in this sec- tion to compute Dc,+,, F( Dc,+, )( ex) for all ex E E, Xck~, from D~k, F(D~ k )(ex) for all ex E E, and Xck, respec- tively. Compared with using Algorithm 1 for every ek, Algorithm 3 reduces the time complexity by O(q) for every ck where k > 1. If there are {ei, ej,em, en} C_E, where dij and dmn coincide, Steps 5 and 6 are still ex- ecuted O(q 2) times if we use perturbation on dij. Step 4 verifies that all ck are computed and finds the op- tirnal candidate solution. It follows that Algorithm 3 correctly solves the MCRRST problem, just as Algo- rithm 2 does, but with a complexity of O(q 3).

The methodology of our improved algorithm for the MCRRST problem can be used to improve the other algorithms that apply the unified approach proposed in [5]. Applying Theorem 3 and the corollaries, we can obtain X~ for any equivalence class e from Arc, in O(q) time, where c' is an adjacent equivalence class of c. In [5], since the set of" equivalence classes is induced

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Y.-C. Chang, L.-H. Hsu/ Operations Research Letters 19 (1996) 65-69 69 class has adjacent classes. Hence, the related optimal

s p a n n i n g tree algorithms p r o p o s e d in [5], w h i c h m a x -

i m i z e g ( A ( T ) , B ( T ) ) = ( ~ e i E T ai) 2 + ( ~ e ~ r bi) 2,

E e , ET ai q- 1-Ie, ET bi or HeiET cti q- I"IeiET hi, c a n also be i m p r o v e d b y a factor o f O ( q ) w h e n we use the approach in [5] a n d that p r o p o s e d in this paper.

References

[1] J.A. Bondy and U.S.R. Murty, Graph Theory with

Applications, Elsevier, New York, 1976.

[2] R. Chandrasekaran, Minimal ratio spanning trees, Networks

7, 335-342 (1977).

[3] R. Chandrasekaran, Y.P. Aneja and K.P.K. Nair, Minimal cost

reliability ratio spanning tree, Ann. Discrete Math. 11, 53-60

(1981).

[4] R. Chandrasekaran and A. Tamir, Polynomial testing of the

query "is ab>~ ca? '' with application on finding a minimal

cost reliability ratio spanning tree, Discrete Appl. Math. 9,

117-123 (1984).

[5] R. Hassin and A. Tamir, Maximizing classes of two-parameter

objectives over matroids, Math. Oper. Res. 14, 362-375