On some algorithmic problems of multicriterion optimization on graphs

117 Ten trials were run. We see that stratified sampling with better estimate of the maximum.

m-N

on average gives a

REFERENCES 1. ERMAKOV S.M., On the admissibility of Monte-Carlo procedures, Dokl. Akad. Nauk SSSR, 172, 7, 262-263, 1967. 2. ZHIGLYAVSKII A.A., Mathematical Theory of Global Random Search, Izd. LGU, Leningrad, 1985. 3. GALAMBOS J., The asymptotic theory of Extreme Order Statistics, Wiley, New York, 1978. 4. SOBOL I.M., MultidimensionalQuadrature Formulas and Haar Functions, Nauka, Moscow, 1969. 5. VINOGRADOV I.M., Fundamentals of Number Theory, Nauka, Moscow, 1981.

Translated by Z.L.

U.S.S.R. Comput.Maths.Math.Phys.,Vo1.29,No.l,pp.117-125,1989 Printed in Great Britain

0041-5553/89 $lO.OO+O.OO 01990 Pergamon Press plc

ON SOME ALGORITHMIC PROWLERS OF M~LTICRITER~O~ OPTIMIZATION ON ~RAP~S~~

V.A. EMELICHEV and V.A. PEREPELITSA

The problem of finding the set of alternatives for multicriterion problems of matchings, paths, and chains on a graph is considered. A probabilistic analysis of the computational complexity of the problem is performed and the effectiveness of algorithms based on linear scalarization of the criteria is considered. 1. Statementof the problem,definitConsand Zem~wzs. Any real optimal design problem (of circuits, constructions, networks, etc.) involves choice and decision making by multiple performance criteria 11-31, and we accordingly need to find some set of alternatives. These multiple criteria are generally conflicting and heterogeneous in the sense that the performance of the alternatives compared cannot be adequately expressed by a single composite or integral criterion representing some weighted sum of the initial partial criteria. In other words, prior procedures of multicriterion optimization /4/ do not apply. Choice and decision making in this case must rely on adaptive procedures and a posteriori type procedures /4/, which have to be applied to some set of alternatives. We will consider multicriterion problems on m-weighted graphs, m>l. following the terminology and the notation of /5, 6/ and qualifying whenever the graphs are digraphs. The graph (digraphl G=(V, E) is called m-weighted if each edge (arc) e& is assigned the weights u.,(e)E(1, 2,...,r), k=J, Z,..., m; @(m,n.r) (or @'(m,n,r)) is the set of all n-vertex m-weighted graphs (or digraphsf. For any problem on the graph G, we denote its feasible solution by 5, implying that z=( if,. E,) is a subgraph of the graph G with a set of vertices V&V and a set of edges which satisfies certain conditions; K={x} is the set of all feasible solutions E,GE of the problem. On the set X we define a vector objective function (vof) F(r)=(F,(~),....F"(r).....FN(I)).

(1.1)

which is formed from the partial criteria

‘) F,(r)-+min(max). y=f .-,...,,V.

(1.7)

then the vof (l.l), (1.2) defines a Pareto-optimal set /l-4/, which we denote If x+0, by x. For N=l, i.e., in the l-criterion case, .y is the set of all optima of the given problem on the graph G. In terms of the theory of Pareto-optimalchoice and decision making /l/, the Paretooptimal set -%'may be regarded as the set of alternatives required. However, a more interesting mathematical object is a set of alternatives of a particular kind, the so-called complete

118

set of alternatives. The complete set of alternatives (csa) is defined as the subset X’~X of minimum cardinality such that F(X')-F(X), where F(X')=(F(r):s=X') Vx'sX. The csa is a generalization of the classical optimum: to find the csa for the case N=l is to find an optimum of the l-criterion problem (for N=l, the cardinality of the csa is /X0/=1 for any problem with a non-empty X). The main problem of discrete multicriterion optimization is to find the csa. More precisely: the main issue is to construct a sufficiently effective algorithm that will find the csa for the given problem. We know /7/ that for some problems on graphs the cardinality of the Pareto-optimal set and the cardinality of the csa iX'I for IV>' are of the order of O(ezp(n)). IXI Exponential lower bounds on cardinalities of these sets of alternatives also hold for the problem of paths (chains between a pair of vertices, for the perfect matching problem, and for other problems considered in this paper. In other words, the problem of finding the Pareto-optimal set and the complete set of alternatives is no longer NP-hard: it is intractable 18, 9/. In this context, it is relevant to estimate the fraction of graphs G~@(rn. n, r), for which (XI and 1X0/ are not exponential and also to estimate the effectiveness of the most common algorithms based on linear scalarization of the criteria /l-3/. We will especially focus on the conditions when it is possible to construct and justify statistically effective methods of solving these problems. Particular results relating to these issues are derived for the problems of paths, chains, and matchings. We start with two auxiliary propositions, which are true for any problem in which the vof (1.11, (1.2) is defined by a mapping of the form F :X--i4.v. tT is the Euclidean space of finite dimension N. we have the equality Lenvna1. For any problem with vof (1.1) of the form F:X-R", of cardinalities I*Yl=iF(S)l. The notation @(tn.n.r) stands for the sequence of sets @(n, n, r),n=l, L..., where and r=r(n) are either constants or functions that increase without limit as m=n(n) n increases. We say that almost all graphs G~@(rn, n. r) have the property y if the fraction of these graphs goes to 1 as n increases, i.e., if lim ~W(l~B(m,n,r)

n-m

I-1,

where @iI is the set of all graphs GE@{?& ?Z, r), with the property r. Consider the problem of finding a certain set of alternatives (sa) for some problem on graphs (in particular, this sa may consist of a single element). Suppose that some algorithm a is used to find this sa. We say that a is a statistically effective algorithm (sea) if it finds the required sa for almost all graphs G&(m,n,r) in a time having an upper limit set by a polynomial in n. In what follows, we use the notation

assuming, to fix our ideas, that all the partial criteria (1.2) are also to be minimized. As it applies to the problem of finding the complete set of alternatives in discrete multicriterion problems, the proof of statistical effectiveness of algorithms relies on the following lemma. Lenmia2. For any problem with vof (1.1) of the form are equivalent:

h,w,

IF(X)I=i,

F:X+lRN

the following statements

(XOI=i.

When the values of all the parameters of the vof (l.l), (l-2) and the set X on which the vof is defined are fixed, we obtain a so-called individual problem /a, 9/. If the entire domain of definition is specified for some of these parameters or for the set X, then we have a mass problem /8, 9/ or, alternatively,a problem which differs from other problems by its proper name. For any problem with the vof (l.l), (1.2), linear scalarization of the criteria is the expression

where

119 3. errthe effectiveness of using Zinear scalarisation of criteria to find the Set of alternat ‘ves. Eonsider the problem of finding sets of alternatives that are subsets of the Pareto-optimal set. Among the methods of finding the Pareto-optimal decisions (i.e., the elements x=X), the most popular are the so-called linear scalarization algorithms (Isa) /l-3/. These algorithms are based on the following well-known fact: for a positive definite vof (1.11, (1.2), the element X:ES minimizing (maximizing)the linear scalarization F’^‘( rc) is Pareto-optimal. Consider some individual problem with N criteria to be minimized, and denote by X' the required set of alternatives (sa) of this problem. If for each 5.E.Y’ there is a vector X*=.1,. satisfying the equality

then we say that the problem of finding the sa x' is Isa-solvable. If Isa-solvability defined in this way holds for all individual problems of the given problem, then we say that the problem of finding the given sa is Isa-solvable for this problem. Finally, we say that the problem of finding the sa is unsolvable if for the given problem there is an individual problem with the set of alternatives x' containing an element x%5X on which the scalarization F'"'(2) VJ,E.iN, does not attain the required extremum, i.e., Pu'(r')>minF@ (5) %.X

V&AN.

There are very few descriptions in the literature of individual multicriterion problems of mathematical programming for which the problem of finding the Pareto-optimal set has been proved to be Isa-unsolvable /lo/. The unsolvability of the problem of finding, by a linear scalarization algorithm, the complete set of alternatives (csa) for the integer multicriterion transportation problem was proved in ill/. So far, it has been impossible to establish the Isa-unsolvability (or solvability)of the problem of finding the sa (the csa) for the 3-criterion problem of covering a l-weighted graph by chains /12/ in which the vof (1.1) consists of the following criteria: the weight of the covering F,(J), the number of chains in the covering Fz(x), and the number of different types of chains in the covering F,(J). In this paper, we establish the unsolvability of the problem of finding the sa (the csa) for three problems which are formulated in mathematical terms below. In order to give a mathematical formulation of any problem on m-weighted graphs, it suffices to define its feasible solution x and to provide the expressions of the criteria (1.2). We will use the following notation: the v-the weight of the subgraph 5=(V*.E,) is

wobtem zp (the perfect matching problem) is formulated on the graphs G=Q
(the perfect matching problem on bipartite graphs) is formulated on Probtem 2,” with parts of equal cardinality (/I',i=jllzj=n). GE@(rn. 92. bipartite graphs C=(V,, V2, E) P): %r=(V,,Y,,E,) is the perfect matching in the given graph G. For each of the problems 2,'.&", Zsm. defined on appropriate sets of feasible solutions S=(X), criteria, and the criteria (1.2) are defined by we have a vof (1.1) with .V=m the expressions * . , n. (1.1) F,(x)=r&(.r)-min. \-=I( -, Theorem 1. The problem of finding the sa K (the csa S") of the perfect is Isa-unsolvable. and even n>4 problem Z," for any ma2

matching

To this end, consider the Proof. We will first prove the theorem for n=4, rn=l. with the set of vertices V={l. 2. 3, 4) and the set of edges E=(e,,..., complete graph K, The elements e,EE are defined by the following list of vertex pairs (v', v"): 4. e,=(2.4). e,=(l,2), e,=(3,4),e,=(l, 4).e&=(2.3).e5=(l, 3), Let these edges be weighted as follows: u;,(e,)=l, s-t,2,

uil(e,)=l, s=3,4.

UJ,(e,)=4. s=3.r,.

u?z(P,)=4. s=i.2.

w,(e,)=w,(e,)=3, s=3,6.

120 The set X for the given graph K, consists of three perfect matchings: z,-(1'. E,),E,=fe,, (V. EA,

El-(es,

e,f,

w-(V,

Ed, E2={eJ, e,l,

a-

es).

Evaluating F(s,)-(2,8),F(z,)=(8,2),F(23)1~1(6,6), we see that for the given individual problem of 2,' we have the equality XO=X=X-(Zr,srrxs)~

('.“)

We also have min(F'"'(r,), F"'(z,))
F'"'(z,)=6V'h=&.

(1.3)

which is the unique pre-image of F(r,)=F(X) cannot Therefore, the element x,=X@ be found by a Isa. This combined with Lemma 1 and equality (2.2) implies that Theorem 1 is proved for m+?,n=4. and n(>4) is an arbitrary even number becomes The proof for the case when m=2 consisting of obvious if we consider the n-vertex graph G=(V, E). E={e,},s=l, 2.. . ..n/2+4 (n-2)/2 connected components, the first of which is the graph K, and each of the others are weighted as above, and the remaining edges is an edge of this graph. The edges of K, have the weights w,(e)=wl(e)=l. Then, as before, (2.2) and (2.3) hold, with the sole difference that (2.3) takes the form min(F"'(s,), PL'(x$))<&+.J.

F'"'(xy)=A "+6

V:'h=A,,

This proves Theorem 1 for m=2. where .I,,=(n--'lfi?. is obtained if the weights rc?,(e,), The final proof of Theorem 1 (m>?) %(e,) of all m we take w,(e,)=w,(e.),if v the edges e,erl? are left unchanged, and for v=3, 4,.... if v is odd. is even and Ic,(e,)=w,(e,), Theorem 2. The problem of finding the sa _r (the csa X”) is Isa-unsolvable. for any mb2. n>4 Proof.

. . . (n-i-l, 04,

of the problem of paths

Z1"'

where the set of arcs E=(e,}.s=l, 2. Consider the n-vertex digraph G=(V, E), is defined by the following list of ordered vertex pairs (0'. 0"): e,=(1.2), e?=(2.4),

e3=(l,4),

e,=(i,3),

e,=(3,4);

for the other s>6 and nbi we set e,=(s-2,s-i),s=6, i’,..., &-I. Let m==2 and let the arcs e,EE be weighted by the following equalities: s=6, 7,. . . , nfl, ~,(e,)=w,(e,)-w,(e.)=w2(e,)=l,

and v=n. Then the set X consists of three paths The distinguished vertices are V’Z each defined by its set of arcs E,: ZP(VI, E*),t=1, 2, 3, Et-{et,ei, es,et,. . . , e,+,f,

E&es,

es, e7, .

. . , es+ll, Es=

fe,, e,, e,, er,. . . , en++). Evaluating F(x,)=(nf2,nf4). F(s,)=fn+4,n-2), (2.2) holds. Also min(P(.r,), P'(.r,))
F(s,)=(ni-3,

d-3),

we see that equality

F'L'(x,)=n+3Vhe,\,.

Therefore, the element x,EX' which is the unique pre-imageof F(z~)EF(.~) cannot be found by a Isa. This combined with Lemma 1 and equality (2.21 implies that Theorem 2 is proved for m-2. The final proof of Theorem 2 (m&?) is obtained if the weights ate,), of all u’~(P.) the edges fZ,=E are left unchanged and for s=3. 4 . . . ..m we set w4e.)-wo(e,), if v is even and W,(e.)--w,(e.)if II is odd. Remarks. 1. A theorem similar to Theorem 2 holds for the problem of chains on a graph. Lsa-unsolvabilityof the problem of chains is proved on a graph which is obtained by replacing the arcs of the digraph from the proof of Theorem 2 with undirected edges. 2. For the perfect matching problem on bipartite graphs 2~. the problem of finding the sa (the csa) is also unsolvable for m22 n>& and This is proved as a corollary of Theorem 4 of 1111, which has established the Isa-unsolvabilityof the problem of finding the csa in the multicriterion integer assignment problem. This corollary follows from the well-known equivalence of the assignment problem and the perfect matching problem on a bipartite graph. Let us now consider the problem of finding at least one representative of the csa. We will use the following notation: 05 is the Lawler algorithm /a/ which finds the minimum-weightperfect matching on a n-vertex weighted graph, its time complexity is r(a,)=O(n'):

121 ap is the Dijkstra algorithm l%l which finds the shortest path (or the shortest chain) between a pair of vertices on a n-vertex weighted digraph (or graph), ~(a*)~o(n~): os is the Hungarian algorithm, or a method with special organization of storage and computations /6, 6/, which finds the optimal solution of the assignment problem (the minimumweight perfect matching on a weighted complete bipartite graph G=(V,, v:,E). IV,l=/V?I=n)* z(a,)=G(n*). Using Karlin's lemma /4/, we obtain the following proposition. Proposition.At leastone Pareto-optimal solution I of the problem Z," (or Z2_,Z,") with vof (1.11, (2.1) can be found by applying the algorithm a, (or a+a,) to the given with every edge e=E weighted by graph G=(V,E)

. w(e)=

the time to find % is

EWV(e);

(2.4)

O(mn2+nJf (or O(mnL), ~(~~~+~~)).

3. Statistically effective atgorithms. For Iv-m-l the csa is the optimum of the classical extremum problem. Therefore, for the single-criterionproblems Z,',k=l,2,3, there exist polynomial algorithms that find the csa /a/, which we respectively denote by a(Z,l). As we have noted in Sect.1, for the problems Z,".k=l, 2. 3, with the cardinality of the sa and the cardinality of the m,2, csa have exponential lower bounds, and therefore the problem of finding these sets of alternatives is of exponential complexity. However, even if the cardinalities /K/ and ix*/ are bounded by a constant, the question of sufficiently effective algorithms for rnb' remains open, because there exist individual problems 2,".k=l. 2, 3. for which, by Theorems 1 and 2 (and also Theorem 4 of /ll/), we cannot guarantee finding the sa or the csa by using linear scalarization of criteria and the algorithms a(Zr') described above. It is therefore relevant to determine the fraction of individual problems for which these sets of alternatives can be found in polynomial time. Below we derive the sufficient conditions for which the fraction of these problems goes asymptotically to 1 and also describe the corresponding statistically effective algorithms (sea) that find the complete set of alternatives. Let a be the algorithm that finds a certain sa of the given mass problems Z,X, the result produced by this algorithm when applied to some individual problem from Z,X' the required sa. The algorithm d is called approximate if Z contains an individual problem such that the difference F(X")\F(X,) is a non-empty set. The algorithm described below for finding the csa (the sa) in the problems Z,'" will be denoted by or"(%) respectively. Algorithm cz3" to solve the perfect matching problem on bipartite graphs Zarnconsists of two stages %' and ~1~'. Stage x:'. For every given m-weighted graph G=l [‘,. 1':. E) compute the sum of weights (2.4). For each vertex ~.ez(l*,l~ 1.:) identify the set of all its incident edges and denote by an asterisk the edges with minimum weight (c(e)in this set. Remove from G all the edges not labelled by any asterisk, the result is the bipartite graph G.=(V,, V,,E‘) whose edges are treated as unweighted. Applying the Hopcroft-Karp algorithm /8/ to the graph G', we construct the maximum matching I+ Then compute a,=minw,(e) and check the conditions rsE r=l!J , -,

Li', (lo) =na,.

, m.

(3.1)

If (3.1) holds, then the algorithm (LJI'ends, because in this case the individual perfect problem on the bipartite graph satisfies the conditions of Lemma 2, i.e., iX*I=i. and X3 is the required csa. If (3.1) does not hold, then go to stage r*,L. Stage as? is a Isa which in general is as follows. From the set .\, the decision maker chooses the vectors h'=(?.,', . . Amy, t-i, 2.. , Gil. Idhereco is a constant. This choice is made either at random or allowing for the relative importance of each of the criteria (2.1) for the given individual problem (see /2, 4, lo/). To the edges g of the graph G assign the scalarized weights

... wt(e) = t:

h,‘w, tel.

denoting the bipartite graphs obtained after this weighting by G,,t=1.2,....co. For each C,, apply the Hungarian algorithm /a/ or other methods with special organization of storage and computations 1131 to obtain the optimal perfect matching s,. Denote the set of matchings obtained in this way by X,={.r,). t=l.2,....c,. Choosing from X1 one pre-image for each image !:(ia), we obtain the subset X3&X", which is the output of stage orZ. Once X," has been constructed, the algorithm a,' ends. The computational complexity of any algorithm o is measured by the number of elementaryoperations (arithmetic,comparison, etc.) that determine its running time; this time complexity is denoted by (T)c(. Let us estimate the time complexity of the algorithm al":

122 O(mn'), the time complexity of the Hopcroft-Kraft the graph G is constructed in time algorithm is O(#*) (see /8/I, hence ~(a~'~=O(~~*~~~a):the time to compute the weights of the edges of the graph Gi is O(mn?, the time to find the solution z1 is O(n? Thus r(a,“)=O(mn2in3). (see /8, 13/I, hence ?(a,")== O(mnZ+nJ). r+=(p(n), ~$,==lp~(n) be two arbitrarily slowly increasing functions of n, limcp(n)-limgr(n)-m, \p,(n)-O(cp(n)), z--1,2, n-._ n-e is the set of all m-weighted bipartite graphs GE @(m, 2n, r) with parts of 01(m, 2n, r) equal cardinality; G(r,n) (or F(r, 2n, n)) is the probabilistic n-vertex graph /14/ (or Zn-vertex bipartite graph with parts of equal cardinality, IVll=IVzl=n) defined by the following condition. For each pair of vertices v',v”(v’~V~. v”EV?) we may have r"+i U") exists and this edge exists weighted by one of equiprobable events: no edge e= (I,,', the rm vectors Let

Z....,r}, (IQ,(~),... ,~~(e)....+tu,(e)),20"(e)E{1, s=;i,Z,....m. including the existence of an edge weighted The probability by the vector (I.P'fi;!e~-"' "~~"e,",'a:"t"~~'?~'< For r-=1, we obtain a random graph /L5/. We also use the concept of probabilistic graph G,(n) (see /Pi/) (bipartitegraph defined as follows: for every pair of vertices v',V"E(l,2,...,n) G,("n),IV,I= IV?I=n), (U’EV,, U”EVI) v") occurs with probability p and does not occur with the edge e"=(u), belongs to the probability l-p. We assume that each realization of the graph G,(n) set @(I. n, I), i.e., every edge occurs with weight w,(e)-1. Let the property y indicate that the given graph includes the required subgraph X with every edge weighted by the vector (I..... 1)E R"'. Note that in the terminology of /15/, the property y is edge monotone. Alongside this monotonicity, we also use the fact that for the edges of the m-weighted graph have a single weight vector (i..... f)=W, r=l any m with i.e., the m-weighted graph for r=i can be treated as a l-weighted graph. Below we will is the set of all graphs GE need the following notation: Q%(m,n.r) ipr 41,(m. 2n. r)) GE@(m, 2n.r)) each with lEj=ls edges. (or bipartite graphs Bim, n, i-) Theorem 3. If PGn/(In ntcp), then for almost all graphs C&(m. 2n,r) the algorithm where aSo finds the csa of the perfect matching problem on bipartite graphs ZSn', /PI=1 and the time to find the csa is T(C&)=O(n."). Proof. By Theorem 1 of /16/, in almost all bipartite graphs G~g~(rn,Pn,1) a perfect matching exists if and only if k=k(n)=n(lnnf$,). $,-O(p). A relationship between the number k and the probability p ensuring that any edge monotone property,in particular property :, is satisfied for almost all graphs G=Q,(m, n. I) and almost always for the probabilistic was established in /l!i/. According to this relationship, graph G,(n) (or C(r, n), r-p-‘-l) Theorem 1 of /lb/ implies that the graph C,(2n) (or C(r,2n)) almost always has the property 1 if p=(r”+f)-‘>(lnn+lg,)ln,

$?=o(+d.

(3.2)

According to this relationship, 13.2) indicates that almost every bipartite graph r”=W G+&C{m.Zn.r) has the property :, i.e., contains the required perfect matching, if (in n+(F). Since the edges e' of this matching iwe denote it by r') have the minimum possible weight m

w(d)=

x 1

ru,(e*)-m,

v-i

then x' is contained in the graph G'=(I:,, F:,E') constructed by stage a,'. Then, by the definition of G', we have u,(e)-m for any e=E’. Hence it follows that stage '1,' almost always produces a matching S,? that satisfies the condition : and with it satisfies the conditions of Lemma 2, i.e., XV={20} and iS"/=l. Since the latter is equivalent to the easily checked condition (2.2), then almost T(bsO)=T(Ti,'). always The required equality Tj~r')-#(mn?+n-')=O(n;'~) follows from the fact that when the conditons of Theorem 3 are satisfied we necessarily have the inequality m
123

a,= minw,(e), v-i.2
w, (x0) =nad2,

,...,

m.

(3.3)

In order to prove the statistical effectiveness of the algorithm c&O, we repeat the proof of Theorem 3 with the sole dif_ferencethat instead of the probabilistic bipartite we consider respectively the probabilistic graphs G(r.?n). CP(2n) and the set @r,(m, 3% r) graphs C(r.n). G,(n) and the set @,(m.II. r). Instead of Theorem 1 of /16/ we use Theorem 4.18 of /15/, which implies that the graph Gp(n) (or G(P.n) ) almost always has the Hence almost every graph G&(m. n. r) has the property y if p=(~l’l+l)-‘~(lnn+~~):2n. property y if r"'<:!:(III ,~~q). We thus have r) with even Theorem 4. If r <_"n!(innrq), then for almost every graph GE @(m, II, number of vertices, the algorithm n," will find the csa X" of the perfect matching problem s(a,0)=0(&). Z,'". where ix"/=1 and Algorithms to findthe Pareto-optimal set of the problems Z”’ 2;. First let u3 define the following concept. Assume that the algorithm CL is used to find a certain sa of some problem Z," and X, is the output produced by this algorithm when applied to the individual problem from Z,"'defined by a particular graph Gd@(m, n, r); T(X*) is the time to find this particular set of alternatives X,. The specific computational complexity of the algorithm 3 (finding the particular sa of the given problem Z,,) is defined as ij(~)=n~~~.~ j.X,~l-'T(Xs). where the maximum is over all graphs G&(m. 12. ,'). Consider the graph G' produced by the first stage q,' (or rl,')and the set 9' of all perfect matchings of this graph. Under certain conditions, the determination of the Pareto-optimal set .y of the problem 2," (or Z,") almost always reduces to the construction of the set X'. and we thus have the following remark. Remark 3. If the conditions of Theorem 4 (Theorem 3) hold, then for almost every graph 0,. 5. I)) the Pareto-optimal set S of the problem Gr@im. ,!. r) (or bipartite graph j/-:6( Z,"‘ (or Z,") consists of perfect matchings whose edges are weighted by the vector (i.....l)-? and for these graphs we have the equality S=l' As an approximate algorithm to find the Pareto-optimal set .y of the problem Z;~'(or Z,"'). consider the algorithm +, (or al). which differs from the algorithm a," (or %a' ) only in that instead of running the procedure that finds one perfect matching in the graph G' when the Condition.(3.3)(or (3.11)is satisfied we apply the algorithm of/17/to find the set 9’. The specific computational complexity of this algorithm is 0(n31. Hence, using Remark 3, we obtain Theorem 5. If the conditions of Theorem 4 (or Theorem 3) are satisfied, then for almost (or bipartite graphs GE @(m, 2n.r)) all graphs GE@(~, 2n.T) the algorithm 3, (or
=

In n

Incl+lnlnn ’

cp,=O(lnlnn).

Consider the graph GE@(m. n. r), G=(V, E), with a distinguished pair of vertices is the set of all simple chains from L'#to uO in c; X” is the set of all L‘", uo;s chains x=.X that are optimal by the criterion I;,(.r)=rc;,(r): C is the graph corresponding to the given graph (J', E) with every edge weighted by the sum (2.4). r=E The algorithm ?ir.consists of the stages CL!'. t-l,?,% In stage 5' we first apply the Dijkstra algorithm/S, 191, say, to find one representative from each subset I_EX~. r=l. &....m, and also the shortest chain I from ciito u, on the graph G. Then we check the conditions ? r“,(I)=f,(s,). v=l ,,....,n. (:I. ;) If these conditions are satisfied, then go to stage azZ, otherwise go to stage a:'. and consists of the substages cl?," . s= I.:! . . . . . Stage a,% is applied to the graph G

1, where 2 is the least number such that

n zjS(.

As the substage

a;,' we take any of

the known algorithms that find the first K shortest chains between a distinguished pair of vertices, e.g., Yen's algorithm /20/ (or the generalized Dantzig algorithm, or the generalized Floyd algorithm /19/). For the substage q.l?" we have ?i=n”. The stage ?;' ends for s=l when the substage a$" starts producing chains that violate condition (3.4). These chains are eliminated, and the remaining chains by Lemma 2 constitute the required Pareto-optimal set 5' because the stage (z: operates when the conditions of Lemma 2 are satisfied, i.e., when .f-;x,P@. Y-L

124 The stage a%' is also the previous stage

al'

applied

and it comprises the substage a?

to the graph G

such that

s-o.

When

of

a:*@ ends, we delete the strictly

dominated elements of the set of chains produced by this substage /2/. The remaining set in general is an approximation of the required Pareto-optimal set Xand XilX(a,9)+@. X(azJ) According to /19, 20/, we have the following complexity bounds for the stages of the Therefore algorithm a,: r(a2')==O(rmn2), r(ar*)-O(n‘lXJ+mn'),?(r*1~)==o(n~+J+tnn*). z(&)= r(~,)=O(n'~X~+mn*)otherwise. Thus, for "large" ParetoO(n@+3+rnn?), if 1X[=ny-' opF;lal sets (/Xl>ma-') Ihear$ecific computational complexity is e(*,)
r"4ni~lnn, then almost all graphs

G&(,,

a,r)

have the property 7,.

Proof. By /15/, the property y, is edge monotone. Consider the probabilistic graph the diameter of G,(n) and apply Theorem 4.25 of /15/, which implies that for k>$,n Inn is d(G)<@. almost every graph G&&(1, n,1) A relationship between the probability p and the number of edges k ensuring that any the edge monotone property, and thus also property I,,is satisfied almost always for and for almost all graphs of the set probabilistic graphs Gp(n) (or _G(r, n), ~=p-‘-1) is established in 115/. Moreover, from Theorem 4.25 of /15/ it follows that %(1, m, 1) almost every graph G&J(m, n, r) has the property 7, if p& (~*ln~)!n. Since p=(p+l)-',. then clearly the last inequality is equivalent to r%((cplnn)/n. Lemma 3 is proved.

Proof of Theorem 6. Let the graph G have the property y,. Then there exists a chain that connects the distinguished pair of vertices uo,uo and consists of no more X0=X i.e., than o edges, each weighted by the vector (i,...,l)~[R'",

F”(ZO)~-o,

v=i, 2,...,m.

(3.5)

We therefore have p(Z)<0

V?EX,

where p(P) is the number of edges in the chain x. such that p(t')>o. then by (3.5) F,(x’)>o+l>F,(suj,

Indeed, if there exists a program v=l,

(3.6) X'EX

2 ,...,m.

But this implies that x%X. The algorithm ?ZZ produces the set of chains S(a2'). (or X(az")) generated by stage ai" ior a:'). We obviously have the inclusion .fs;X(a83), if the algorithm d2 stops in stage al'. Therefore, in order to complete the proof of Theorem 6 it suffices to show that we have the inclusion X~X(als). By definition of stage az4, if the condition y, holds, the set X(a,“j consists of all chains SEX satisfying (3.5) and (3.6). But in this case, all the Pareto-optimal chains also satisfy these xelationships. Thus, X~X(ccl’). This, combined with Lemma 3, completes the proof of Theorem 6. REFERENCES MOISEYEV N.N., Mathematical Problems of Systems Analysis, Nauka, Moscow, 1981. 2. MIKHALEVICH V.S. and VOLKOVICH V.L., Computational Methods of Analysing and Designing Complex Systems, Nauka, Moscow, 1982. 3. POSPELOV G.S., IRIKOV V.A. andKURILOV A.E.,Procedures and Algorithms for Constructing Complex Programs, Nauka, Moscow, 1985. 4. DUBOV YU.A., TRAVKIN S.I. and YAKIMETS V.N.,MulticriterionModels for Generating and Choosing System Alternatives, Nauka, Moscow, 1986. 5. HARARY F., Graph Theory, Addison-Wesley,Reading, Mass., 1969. 6. KOZYREV V.P. and YUSHMANOV S.V., Graph theory (algorithmic,algebraic, and metric problems), in: Itogi Nauki i Tekhniki, ser. Teoriya Veroyatnostei,MatematicheskayaStatistika, TeoreticheskayaKibernetika! 23, 68-117, VINITI, Moscow, 1985. 7. EMELICHEV V.A. and PEREPELITSA V-A., On algorithmic problems of vector optimization on graphs, in: Software Systems for Solving Optimal Scheduling Problems, 9th All-Union Symp., Minsk, 23, February - 3 March 1986, TsEMI Akad. Nauk SSSR, Moscow, 1986. 8. PAPADIMITRIOU C.H. and STEIGLITZ K., Combinatorial Optimization: Algorithms and Complexity, Prentice Hall, Englewood Cliffs NY, 1982. 9. GAREY M.R. and JOHNSON D.S., Computers and Intractability,Freeman, SF 1979. 1.

125

10. PODINOVSKII V.V. and NOGIN V.D. Pareto-optimal Solutions of Multicriterion Problems, Nauka, Moscow, 1982. transportation 11. EMELICHEV V.A. and PEREPELITSA V.A., Complexity bounds of multicriterion problems, Dokl. Akad. Nauk BSSR, 30, 7, 593-596, 1986. 12. PEREPELITSA V-A., On one class of multicriterion problems on graphs and hypergraphs, Kibernetika, 4, 62-67, 1984. 13. ADEL'SON-VEL'SKII G.M., DINITS E.A. and KARZANOV A.V., Flow Algorithms, Nauka, Moscow, 1975. 14. PEREPELITSA V.A., On two problems in graph theory, Dokl. Akad. Nauk SSSR, 194, 6, 12691272, 1970. 15. KORSHUNOV A.D., Fundamental properties of random graphs with many vertices and edges, Uspekhi Matem. Nauk, 40, 1(241), 107-173, 1985. 16. KORSHUNOV A.D., An algorithm for finding matching in finite graphs, Kibernetika, 1, 1-8, 1975. 17. KAKHICHKO A.A., The construction of perfect matchings of a graph, in: Methods of Solving Non-linear Problems and Data Processing, 41-44, Izd. DGU, Dnepropetrovsk, 1986. 18. CORLEY H.W. and MOON I.D., Shortest paths in networks with vector weights, J. Optimiz. Theory Appl. 46, 1, 79-86, 1985. 19. MINIEKA E., Optimization Algorithms for Networks and Graphs, Marcel Dekker, New York, 1978. 20. CHRISTOFIDES N., Graph Theory, An Algorithmic Approach, Mir, Moscow, 1978.

Translated

U.S.S.R. Comput.Matks.Matk.Pkys.,vol.29,No.l,pp.l25-132,1989 Printed in Great Britain

by Z.L.

0041-5553/89 $lO.OO+O.OO 01990 Pergamon Press plc

COMPUTATIONAL PROBLEMS FOR HYDRAULIC SYSTEMS3' K.R. AIDA-ZADE Numerical methods are given for solving the problems that arise in gas and fluid transport systems. Design expressions are obtained by analysing mathematical models of hydraulic systems. The static states of the gas motion are calculated hydraulically, and the resistance factors and the optimum source distributions over the specifications are determined. Computations of hydraulic systems have been made by various authors, see, in particular, /l-3/, but less attention has been paid to the numerical aspects of solving the relevant optimization problems. In the present paper we give expressions, and algorithms based on them, for making hydraulic calculations of the static modes of gas motion in the piping and for solving optimization problems such as the identification of the resistance factors in pieces of the system and the flow rate distribution over the sources. We will decompose the optimization problem and represent its solution as a two-level process; at the lower level we will solve the problem of hydraulic calculation, and at the upper level, the actual optmization problem; in this way we can obtain expressions in which the most laborious computations of the two levels are combined (such as inversion of the matrices, or evaluation of the function gradients). The hydraulic calculation of the stationary gas motion over the piping of a 1. composite ring system amounts to finding the gas flows over pieces of the system and the values of the pressure at its vertices. Let the system contain m pieces and n (nl>,L);k, vertices are gas inflows vertices (the set of vertices I,), and k, vertices are gas outflows (the set 12).k=k,+k,. I=t,UI,. !i is the total number of sources with positive and negative values of the gas flow rate Q, and with pressure values P,, i=l. Let j', be the pressure at the i-th vertex, i=l. 2,....n. and j=l.2.....m. We are given a priori the directions '/. the flow rate on the j-th piece, of the assumed gas motion over the pieces, i.e., if the flow rate over the j-th piece