Drago{ CVETKOVI] Mirjana ^ANGALOVI] 1. INTRODUCTION

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1 Yugoslav Journal of Operatons Research 12 (2002), Number 1, 1-10 FINDING MINIMAL BRANCHINGS WITH A GIVEN NUMBER OF ARCS Drago{ CVETKOVI] Faculty of Electrcal Engneerng Unversty of Belgrade, Belgrade, Yugoslava Mrjana ^ANGALOVI] Faculty of Organzatonal Scences Unversty of Belgrade, Belgrade, Yugoslava Abstract: We descrbe an algorthm for fndng a mnmal s -branchng (where s s a gven number of ts arcs) n a weghted dgraph wth an asymetrc weght matrx. The algorthm uses the basc prncples of the method (prevously developed by J. Edmonds) for determnng a mnmal branchng n the case when the number of ts arcs s not specfed n advance. Here we gve a proof of the correctness for the descrbed algorthm. Key words: Combnatoral optmzaton, weghted graphs, mnmal branchng. 1. INTRODUCTION We consder an arc weghted dgraph G wthout loops and wth an asymmetrc weght matrx. As usual, the weght dh ( ) of any subgraph H of G s defned to be the sum of weghts of arcs of H. We start wth some specfc defntons. Defnton 1. A tree n whch each edge s drected (so that t becomes an arc) s called a drected tree. Defnton 2. A drected tree s called an arborescence f the followng condtons are fulflled: 1. There s a node x wth no enterng arcs; 2. Exactly one arc enters each of the other nodes. Node x s called the root.

2 2 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs Defnton 3. A dgraph whose weakly connected components are arborescences s called a branchng. A branchng wth s arcs s called an s -branchng branchng. We consder the problem of fndng a mnmal s -branchng n G,.e. a branchng wth mnmal weght. An algorthm for ths problem has been developed n [6] (see also [8], pp ) and we descrbe t here. The algorthm uses the basc prncples of a method for determnng a mnmal branchng (wthout a specfcaton of the number of arcs n t) n a weghted dgraph usually accredted to Edmonds [10], [13], [15], [17] although t was prevously dscovered n [4] and later ndependently n [2]. (Some effcent mplementatons of ths method are gven n [3], [12], [16]). Edmonds' algorthm cannot of course be drectly appled to the problem of determnng a mnmal s -branchng, but t was a startng pont n developng our algorthm. In fact, our algorthm s dentcal to the one gven by Edmonds, except that t stops when the number of the chosen arcs s equal to s. Ths s true n any greedy algorthm when a lmt s put on the number of elements. However, Edmond's algorthm s not greedy snce t may change the arcs selected n prevous steps. Here we gve a proof of the correctness for our algorthm. The proof cannot be mmedately derved from the exstng correctness proofs of Edmond's algorthm (see [10], [13], [15]), as the fxed number of arcs requres some delcate addtonal consderatons. An ncomplete verson of our paper has been presented n [7]. The motvaton for consderng the problem of fndng a mnmal s -branchng s related to a knd of the travelng salesman problem (TSP). Consder the m travelng salesmen problem ( m -TSP) wth mk ctes and wth an asymmetrc weght matrx where each of the m salesmen should vst the gven number of k ctes, whle ther tours should be dsjont. The problem of fndng a mnmal mk ( 1) -branchng can be used as a relaxaton n a branch-and-bound procedure for solvng m -TSP [8]. Note that the above verson of the m -TSP cannot be solved by a standard transformaton [1] whch reduces the standard m -TSP to the ordnary TSP. Our m -TSP s smlar to the so-called clover leaf problem [9] whch has not been much studed n the lterature. 2. THE ALGORITHM The algorthm for fndng a mnmal s -branchng s based on a modfcaton M (defned n [10]) whch transforms the weghted dgraph G nto a weghted dgraph G n the followng way: Let C be a cycle n G. (By cycle we always mean a drected cycle). An arc whch does not belong to C, but enters (leaves) a node of C, s called the enterng (leavng) arc of C. The dgraph G s obtaned from G by contractng all nodes of C nto a node v, called a supernode. In G all arcs of G not ncdent to a node of C are kept, all arcs from C are removed, whle for all enterng and leavng arcs of C, ther endnodes belongng to C are replaced by v.

3 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs 3 For each arc ( xv, ) n G a new weght s determned as p, ( x, v) = d( x, y) + d( p, q) d( z, y ) (1) Cy where ( xy, ) s the correspondng enterng arc of C, ( pq, ) an arc n C havng the maxmal weght, ( zy, ) the arc of C whch enters the same node as ( xy, ), whle dxy (, ), dpq (, ), dzy (, ) are ther weghts n G, respectvely (see Fgure 1). Weghts of all the other arcs from G are the same as n G. It s obvous that G could have multple arcs,.e. t could be a mult-dgraph. Therefore, the expresson dgraph, used n all further consderatons, ncludes the possblty that multple arcs exst. Fgure 1. The algorthm uses a concept of mnmal arcs. For each nternal node v of G (.e. a node wth an enterng arc n G ) the mnmal arc of v s one of ts enterng arcs wth the mnmal weght. Remark 1. If v has more than one enterng arc wth the same mnmal weght, the mnmal arc of v can be any of them. Remark 2. As the mnmal arc s unque for each nternal node of G, then obvously an arbtrary set of mnmal arcs, whch do not form any cycle, nduces a branchng n G. Now a short descrpton of the algorthm for fndng a mnmal s -branchng s gven: The algorthm Intalzaton = 0, s0 = 0, A0 =, G0 = G Phase 1 (forward phase) Step 1. Determne the set R of all mnmal arcs from If R < s s, then STOP: there s no s -branchng n G. Step 2. Order all mnmal arcs from Step 3. Choose mnmal arcs from ether G whch do not belong to R accordng to ther nondecreasng weghts. A. R, one by one, respectng ther orderng, untl

4 4 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs the frst s s chosen arcs do not form, mutually or wth arcs from A, any cycle n G : These arcs and arcs from A determne a branchng T ; go to Phase 2. or the last chosen arc forms, wth the prevously chosen arcs from R or wth arcs of A, a cycle C n G. Step 4. Transform G nto a dgraph G +1 by modfcaton ( M ), contractng all nodes of C nto a supernode v. Determne A +1 as the set of all mnmal arcs, chosen n Step 3, whch belong to G +1. Update s +1 as j j= 0 s = A + ( m 1) where m j s the total number of nodes n the cycle C j. Set = +1 and go to Step 1. Phase 2 (backward phase). Step 5. If = 0, then STOP: T T 0 s a mnmal s -branchng n G. Step 6. A branchng T 1 n the dgraph G 1 s formed as follows: T 1 contans all arcs from T not ncdent to v 1 and also arcs correspondng to the arcs of T leavng v 1. If T ncludes the mnmal arc of v 1, then T 1 contans the correspondng enterng arc ( xy, ) of C 1 and all arcs of C 1, except ( zy, ) enterng the same node as ( xy, ). If T does not nclude the mnmal arc of v 1, then T 1 contans all arcs of C 1, except an arc of the maxmal weght. Set = 1 and go to Step 5. Remark 3. In the case when the weghts of some mnmal arcs from G are mutually equal, the set R and the orderng of ther elements (Steps 1 and 2 of Phase 1) need not be determned n a unque way. Remark 4. It can be easly estmated that the numercal complexty of the algorthm s 2 On ( ), where n s the number of nodes of the dgraph. In fact, the most tme consumng parts of the algorthm are the determnaton of all mnmal arcs (whch 2 obvously has complexty On ( )) and ther sortng (wth complexty no more than 2 On ( )). Remark 5. The problem of fndng a mnmal s -branchng becomes NP-hard when the resultng branchng s requred to be connected (.e. to consst of a smple arborescence spannng s + 1 out of n nodes n the dgraph). Ths follows from the NP-hardness of the k -cardnalty tree problem proved n [11].

5 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs 5 3. THE ALGORITHM CORRECTNESS PROOF It s well-known that the problem of fndng a mnmal branchng can be formulated as the weghted matrod ntersecton problem [14], [5]. Namely, branchngs are common ndependent sets for the forest matrod of G and for the matrod of subgraphs of G havng ndegrees at most 1. (However, branchngs for themselves do not consttute a matrod). The correctness of our algorthm follows from the correctness of the weghted matrod ntersecton algorthm, n partcular, from a theorem (Theorem 9.1 of [14],.e. Theorem 8.24 of [5]) justfyng a procedure for extendng a maxmum weght common ndependent set of cardnalty k to the one of cardnalty k + 1. However, our algorthm avods some steps present n the general 2 4 algorthm and has a lower complexty ( On ( ) nstead of On ( ) n general case). In addton we offer an elementary correctness proof usng graph theoretcal termnology, thus avodng more general structures of the matrod theory. Accordng to the algorthm defnton, t s obvous that the branchng T, f t s obtaned n Step 5 of Phase 2, has s arcs (f there s no s -branchng n G, the algorthm stops at Step 1 of Phase 1). We show there that T s a mnmal s -branchng n G. Our proof s an elaboraton of the one outlned n [6]. Frst, several necessary lemmas should be proved. Lemma 1. Let mnmal arcs of all nternal nodes n a weghted graph G be ordered accordng to nondecreasng weghts. If the frst k mnmal arcs do not form any cycle n G, then they nduce a mnmal k -branchng of G. The proof s straghtforward and thus omtted. Let G0 G, G1,..., Gq, q 0 be dgraphs, consdered n Phase 1, and T0, T1,..., T q be the correspondng branchngs, generated n Phase 2 of the algorthm. For q = 0 t follows, drectly from Lemma 1, that T T 0 s a mnmal s - branchng n G. Therefore we shall further suppose that q 1. Let H be a subgraph of G, { 01,,..., q 1}, nduced by all arcs enterng nodes of the cycle C (formed n Step 3 of Phase 1). For each ( xy, ) from H, whch s an enterng arc of C, p( x, y ) denotes a value, defned by (1) n the modfcaton ( M ),.e. p ( x, y) = d( x, y) + d( u, w) d( z, y ), where ( uw, ) s an arc of enters the same node as ( xy, ). Let C wth the maxmal weght and ( zy, ) belongs to G be a subgraph of whch do not belong to H. Now the followng lemmas can be proved: C and G, nduced by all arcs of G

6 6 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs Lemma 2. Each arc of that the weght of any arc n Proof: Each arc of G not enterng any nternal node of T, has weght not smaller C. G, not enterng any nternal node of T, enters ether a node T not belongng to C. It s suffcent to prove whch does not belong to T or a root of that for each such node y weght (, ) dxy of ts mnmal arc ( xy, ) s not smaller than weghts of arcs n C. For = q 1, weght dxy (, ) cannot obvously be smaller than weghts of arcs n C. (Otherwse, arc ( xy, ) would be chosen n Step 3 of Phase 1 before closng the cycle C and, as ( xy, ) would not be ncluded nto any supernode durng Phase 1, t would belong to T,.e. node y would represent an nternal node of T ). Let us consder the case when < q 1 for q > 1 and let us assume that dxy (, ) be smaller than a weght of at least one arc n C. Then arc ( xy, ) would be chosen n Step 3 of Phase 1 before closng cycle C. As ( xy, ) does not belong to T, ths arc would be ncluded to a cycle Cj, j { + 1,..., q 1} of a dgraph G j and then excluded from T j n Step 6 of Phase 2 such that ether (a) T j contans the enterng arc of C j correspondng to node y and all arcs from C j except ( xy, ) ; or (b) T j does not contan any enterng arc of cycle C j, but contans all ts arcs except arc ( xy, ) of the maxmal weght. Consequently, n G there would be a path P from node y to node x belongng to T and consstng of arcs whch ether belong to C j or they are contracted nto supernodes of C j. Now we shall derve the followng contradctons: If case (a) occurred, node y would obvously be an nternal node of T, whch leads to contradcton. The weght of the arc ( xy, ) n G j would be the same as n G and therefore n case (b) dxy (, ) would not be smaller than weghts of all the other arcs from Moreover, f C j contaned a supernode (formed after closng C j ), dxy (, ) would not be smaller than weghts of all arcs from C j. C and before closng G contracted nto ths supernode. (It follows from the fact that, always when a new supernode s obtaned by contractng a cycle, new weghts of ts enterng arcs are not smaller than weghts of the arcs n ths cycle). Consequently, weghts of arcs n path P would not be greater than dxy (, ). It means that there would be a cycle n G (composed of ( xy, ) and P ) whch would be closed n Step 3 of Phase 1 before C, whch leads to contradcton.

7 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs 7 Lemma 3. If T +1 s a mnmal branchng n G +1, then T s a mnmal branchng n G, { 01,,..., q 1}, where mnmal branchngs are consdered wth respect to the number of arcs n T +1 and T, respectvely. Proof: Let m be the number of nodes n the cycle C of G. If T +1 has r arcs, then T has m + r 1 arcs. Let us suppose that T s not a mnmal ( m + r 1) -branchng, but that there exsts a mnmal ( m + r 1) -branchng U n G such that du ( ) < dt ( ). We denote by T and U parts of branchngs T and U, respectvely, on the subgraph H, and wth T and U parts of these branchngs on the subgraph G of G. Two cases need to be consdered. The frst case. T has m arcs,.e. T has r 1 arcs. It s obvous that dt ( ) = dc ( ) + dxy (, ) dzy (, ), (2) where ( xy, ) s the enterng arc of C wth mnmal p ( x, y ) and ( zy, ) the correspondng arc n C. Accordng to the modfcaton ( M), G correspondng to a subgraph of G +1, nduced by all arcs not enterng the supernode formed by contractng C. Therefore, we have a) U has dt ( + 1) = dt ( ) + p( xy, ). m arcs,.e. U has r 1 arcs. It s obvous that the nternal nodes of U must belong to of the mnmal branchng U, and C contans only mnmal arcs, then m 1 arcs of C and an enterng arc ( x1, y 1) of C. Therefore C. As U s a part U conssts of du ( ) = dc ( ) + d( x, y ) d( z, y ), (3) where ( z1, y 1) s the correspondng arc n C whch does not belong to U. T s mnmal r -branchng n G +1, then As +1 dt ( + 1) = dt ( ) + p( xy, ) du ( ) + p( x1, y 1). (4) From (2)-(4) t follows that du ( ) dt ( ), whch leads to contradcton. b) U has m l arcs, 1 l m,.e. U has r 1 + It s obvous that l arcs.

8 8 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs l du ( ) = dc ( ) t, (5) = 1 where t1,..., t l, are the weghts of l arcs of C havng the greatest weghts. G contans r 1 nternal nodes of T. Therefore, accordng to Lemma 2, n U there exst at least l arcs wth weghts not smaller than weghts t1,..., t l. Among these arcs we choose arbtrary l 1 arcs and denote ther weghts wth t1, t2,..., t l 1. As Let U be a part of U not contanng arcs wth weghts t1, t2,..., t l 1,.e. l = + 1 = 1 du ( ) du ( ) t. (6) U has r arcs, then 1 = + du ( ) dt ( + ) dt ( ) p( xy, ). (7) Now, from (2), (5)-(7) t follows that du ( ) dt ( ), whch leads to contradcton. The second case. T has m 1 arcs,.e. T has r arcs. Accordng to Phase 2 of the algorthm, dt ( ) = dc ( ) duw (, ), (8) where ( uw, ) s an arc n C wth the maxmal weght, and dt ( ) = dt ( 1 ). a) U has m 1 arcs,.e. U has r arcs. The nternal nodes of U must belong to C. As U s a part of the mnmal branchng U and C conssts of mnmal arcs, then obvously U contans m 1 arcs of C. Therefore du ( ) dt ( ). As U has r arcs, then du ( ) dt ( + 1) = dt ( ). Consequently, du ( ) dt ( ), whch leads to contradcton. b) U has m l arcs, 1 < l m,.e. U has r 1 + l arcs. In the same way as for the frst case b), t can be proved that (5), (6) hold. Also, du ( ) dt ( + 1) = dt ( ). (9) From (5), (6), (8), (9) t follows that du ( ) dt ( ), whch leads to contradcton. c) U has m arcs,.e. U has r 1 arcs. Now, accordng to the consderatons n the frst case a), du ( ) = dc ( ) + d( x, y) d( z, y ), (10) where ( xy, ) s an enterng arc of have C and ( zy, ) the correspondng arc n + C. Also we

9 D. Cvetkov}, M. ^angalov} / Fndng Mnmal Branchngs wth a Gven Number of Arcs 9 du ( ) + p( x, y) dt ( ) = dt ( ). (11) + 1 From (8), (10), (11) t follows that du ( ) dt ( ), whch leads to contradcton. Snce n all consdered cases we come to a contradcton, T s a mnmal ( m + r 1) branchng n G. Lemma 4. The branchng T q n G q s a mnmal branchng wth respect to the number of arcs n T q. Proof: Let r be the total number of arcs n T q and T q be the set of all arcs from T q, chosen n prevous steps of Phase 1 (before formng G q ). If T q = t follows, drectly from Lemma 1, that T q s a mnmal r -branchng. If T q, we prove that the weght dxv (, ) of a mnmal arc ( xv, ) n G q such that ( xv, ) T q, s not smaller than weghts of arcs from T q. If v s not a supernode formed by contractng the cycle C q 1 n G q 1, then dxv (, ) cannot be smaller than weghts of arcs n T q (Otherwse, ( xv, ) would be chosen n one of the prevous steps of Phase 1,.e. ( xv, ) would belong to T q ). If v s a supernode formed by contractng the cycle C q 1 n G q 1, then ( xv, ) corresponds to an enterng arc ( xy, ) of C q 1. Therefore, dxv (, ) = p ( xy, ) duw, (, ) q 1 where ( uw, ) s an arc of G q 1. As ( uw, ) s a mnmal arc last chosen n C q 1 wth the maxmal weght, whle duw (, ) s ts weght n G q 1 (Step 3 of Phase 1), then duw (, ) T q and, consequently, the same holds for s not smaller than weghts of arcs from dxv (, ). Accordng to the above consderatons and Lemma 1, all arcs of Step 3 of Phase 1, together wth arcs from G q. G q chosen n T q, represent a mnmal r -branchng n Now the correctness of the algorthm can be proved n the followng way: Theorem. The branchng T T 0, generated by the algorthm, s a mnmal s - branchng n the dgraph G. Proof: Accordng to Lemma 4, T q s a mnmal branchng of G q wth respect to the correspondng number of arcs. Therefore, from Lemma 3 (teratvely appled to T for = q 1,..., 0 ) t follows that T 0 s a mnmal s -branchng n G0 G.

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