DEVELOPMENT OF A BRANCH AND PRICE APPROACH INVOLVING VERTEX CLONING TO SOLVE THE MAXIMUM WEIGHTED INDEPENDENT SET PROBLEM

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1 DEVELOPMENT OF A BRANCH AND PRICE APPROACH INVOLVING VERTEX CLONING TO SOLVE THE MAXIMUM WEIGHTED INDEPENDENT SET PROBLEM A Thesis by SANDEEP SACHDEVA Submitted to the Office of Graduate Studies of Texas A&M Uniersity in artial fulfillment of the requirements for the degree of MASTER OF SCIENCE December 2004 Major Subject: Industrial Engineering

2 DEVELOPMENT OF A BRANCH AND PRICE APPROACH INVOLVING VERTEX CLONING TO SOLVE THE MAXIMUM WEIGHTED INDEPENDENT SET PROBLEM A Thesis by SANDEEP SACHDEVA Submitted to Texas A&M Uniersity in artial fulfillment of the requirements for the degree of MASTER OF SCIENCE Aroed as to style and content by: Wilbert E. Wilhelm (Chair of Committee) Illya Hicks (Member) Arun Sen (Member) Brett A. Peters (Head of Deartment) December 2004 Major Subject: Industrial Engineering

3 iii ABSTRACT Deeloment of a Branch and Price Aroach Inoling Vertex Cloning to Sole the Maximum Weighted Indeendent Set Problem. (December 2004) Sandee Sachdea, B.Tech., Indian Institute of Technology, Delhi, India Chair of Adisory Committee: Dr. Wilbert E. Wilhelm We roose a noel branch-and-rice (B&P) aroach to sole the maximum weighted indeendent set roblem (MWI). Our aroach uses clones of ertices to create edge-disjoint artitions from ertex-disjoint artitions. We sole the MWI on sub-roblems based on these edge-disjoint artitions using a B&P framework, which coordinates sub-roblem solutions by inoling an equialence relationshi between a ertex and each of its clones. We resent test results for standard instances and randomly generated grahs for comarison. We show analytically and comutationally that our aroach gies tight bounds and it soles both dense and sarse grahs quite quickly.

4 i DEDICATION To my arents

5 ACKNOWLEDGEMENTS I would like to sincerely thank my adisor Dr. Wilbert E. Wilhelm for roiding motiation and guidance in ursuing this research. In addition, I would like to acknowledge the suort of Dr. Illya Hicks and other members of the NSF roject team for their aluable suggestions. I am also thankful to my friends for their suort throughout my stay at Texas A&M Uniersity. This material is based in art uon work suorted by the Texas Adanced Technology Program on Grant Number and by the National Science Foundation on Grant DMI

6 i TABLE OF CONTENTS Page ABSTRACT...iii DEDICATION... i ACKNOWLEDGEMENTS... TABLE OF CONTENTS... i LIST OF FIGURES...iii LIST OF TABLES... ix CHAPTER I INTRODUCTION Oeriew Motiation and Objecties Basic Notations Organization of Thesis... 3 II LITERATURE REVIEW... 5 III VERTEX CLONING APPROACH Concet Formulation IV ANALYSIS OF BOUNDS V IMPLEMENTATION ISSUES Selecting Vertices for Cloning Assigning Weights Soling the MWI Partial Cloning VI COMPUTATIONAL EVALUATION... 29

7 ii CHAPTER Page VII SUMMARY, CONCLUSIONS AND FUTURE RESEARCH REFERENCES VITA... 48

8 iii LIST OF FIGURES FIGURE Page 1 Examle grah G Vertex disjoint artitions of G Edge disjoint artitioning of G through ertex cloning Edge disjoint artitioning of G by cloning different ertices Z LP s P for johnson824com Z LP s P for manna9com Run time s P for Johnson 824 com Run time s P for manna9com... 34

9 ix LIST OF TABLES TABLE Page 1 Performance measures for DIMA instances Performance measures for -grahs Additional erformance measures for DIMA instances Additional erformance measures for random -grahs Performance measures for sarse random -grahs... 41

10 1 CHAPTER I INTRODUCTION 1.1 Oeriew Gien a grah, G = ( V, E) where V reresents the set of ertices; and E, the set of edges, a subset of ertices I V such that no two ertices in I are adjacent to each other constitutes an indeendent set (IS). The roblem of finding the indeendent set of largest cardinality in a grah is known as the maximum indeendent set roblem (MI). The cardinality of the maximum indeendent set is known as the indeendence number or the stability number of the grah. Extending the MI to ertex-weighted grahs, the MWI is to find the indeendent set of maximum weight. Letting w reresent the weight associated with ertex for such that I V, the MWI is to find the indeendent set I w is maximized. Both MI and MWI are known to be NP-Hard [12]. Een though the MWI can be soled in olynomial time on some secialized grah structures ([1], [11]); the roblem remains NP-Hard on arbitrary grahs. MI and MWI are among the most researched roblems in the field of grah theory. They hae large numbers of ractical alications in dierse fields, including rotein structure realignment [8], coding theory [7], comuter ision [2], exerimental design [2], signal transmission [2], and information retrieal [2]. This thesis follows the style and format of the Euroean Journal of Oerational Research.

11 2 1.2 Motiation and Objecties The aroach exlored in this study inoles soling the integer rogramming formulation of the MWI (in edge inequality form) which may be stated as Z MWI V { x B : x + x 1 ( u, E } = Max w x : x Q, where Q = + u ), (1) V where x = 1 if ertex is included in the indeendent set, and x = 0 otherwise. Warrier et al [25] deeloed a branch-and-rice (B&P) aroach to sole the MWI and showed that their aroach gies cometitie results for sarse grahs. Howeer, their aroach suffers from two major drawbacks: their restricted master roblem (RMP) gies bounds that are not tight and comrises a large number of constraints, requiring lengthy run times. This study contributes a new B&P aroach, which is directed towards oercoming these shortcomings. This new aroach, which we call Vertex Cloning, is designed to facilitate solution by yielding a RMP with fewer constraints. We also show that Vertex Cloning roides a tighter formulation, imroing bounds in the branch-and-bound (B&B) tree. The rimary objecties of this study are: (1) Formulation of the Vertex Cloning aroach, (2) Analysis showing that Vertex Cloning yields a tighter formulation, (3) Effectie methods to imlement Vertex Cloning, and (4) Analysis of the comutational efficacy of Vertex Cloning.

12 3 1.3 Basic Notations We consider only simle, undirected, and finite grahs. Most of the notation we use in this thesis is the same as that used by Warrier et al [25]. We reresent an edge as e E or, alternatiely, by denoting its end ertices as ( u, ) E where u, V, u. We use G = ( V, E ) to denote the comlement grah of G, where V = V and { u, ) E: u, V u } E = (,. We use N() to denote the set of s neighbors, { u ( u, ) E} :. We decomose grah G into P sets of ertex-induced artitions. We use G = ( V, E ) for P to denote the sub-grah (artition), where V and E denote the set of ertices and edges in artition, resectiely. Furthermore, we use Ê to reresent the set of edges that connect ertices in different sub-grahs, Eˆ = E \ U P E ; and similarly, Vˆ to denote the set of ertices at the ends of edges in Ê. For V, we use π to identify the artition into which is assigned. We use N () to denote the neighbors of in artition P. Vertex Cloning may dulicate certain ertices into artition P. We use an oer bar to denote the ertex and edge sets in artition after dulication (i.e., V and E for P ). 1.4 Organization of Thesis The remainder of this thesis is organized in six chaters. Chater II resents a reiew of the literature on MWI, including a detailed discussion of the B&P aroach

13 4 deeloed by Warrier et al [25]. Chater III introduces concets that underlie Vertex Cloning and gies a detailed mathematical formulation (objectie 1). Chater IV discusses roerties of olyhedra formed by arious B&P formulations (objectie 2). Chater V discusses imlementation issues (objectie 3) and Chater VI analyzes comutational results (objectie 4), comaring the erformance of seeral algorithms for soling the MWI. Finally, Chater VII gies summary and recommendations for future research.

14 5 CHAPTER II LITERATURE REVIEW A solution to the MWI can be obtained as the solution to the maximum weighted clique roblem on the comlementary grah and the literature describes extensie study of both roblems. The solution methods resented in the literature use ariety of aroaches for soling the MWI, which includes B&B [2, 3, 7, 21], imlicit enumeration [9] and standard heuristic methods like genetic algorithms [13] and greedy random adatie search rocedures [10]. Bomze et al [6] gae an extensie surey of algorithms, comlexity and alications of maximum clique roblem. Recently, Carr et al [8] described a branch-and-cut aroach for the MWI. Bazaara et al [5] gae a good descrition of Dantzig-Wolfe decomosition (DWD) for linear rogramming roblems. DWD may be alied to the linear relaxation of an integer rogramming roblem to obtain a bound at each node in the B&B tree in an aroach known as B&P. Oer the last twenty years, B&P has been successfully alied in a wide range of integer rogramming roblems [4, 18, 20, 23, 24, 26]. To aly B&P, integer rogramming roblems must be decomosed into two sets of constraints; those that form sub-roblem(s) and those that are relegated to the RMP. Barnhart et al [4] and Wilhelm [26] roided extensie oeriews of B&P and gae descritions of decomosition methods, and associated imlementation issues. Mehrotra and Trick [18] used B&P to sole the minimum coloring roblem, another imortant grah roblem. The minimum coloring roblem is to find the minimum

15 6 number of colors that allows each ertex to be colored so that the endoints of each edge hae different colors. They used a set coering formulation of the coloring roblem with the objectie of finding the minimum number of maximal indeendent sets such that the union of these sets includes all ertices of the grah. Their RMP consisted of set coering constraints and their (single) sub-roblem inoled finding the maximal indeendent set. Warrier et al s [25] B&P aroach artitions a grah into smaller, ertex-disjoint sub-grahs and soles a MWI on each sub-grah (sub-roblem) to generate columns that are coordinated by a RMP to obtain the MWIS for the original grah. Their aroach artitions the inequalities associated with edge constraints in (1) into two sets; one set, the coordinating set, comrises inequalities associated with edges that connect ertices in different artitions (i.e., x u + x 1 ( u, ) Eˆ ); and the other set, P subroblems, each consisting of inequalities associated with the resectie edges included in a artition (i.e., x u + x 1 ( u, ) E ). They used B&P, forming the RMP (we dulicate their model here) as: Z LP = Max P j λ j ( w x ) (2) = 1 j J P j s.t. λ ( A x ) 1 (3) = 1 j J j J j λ = 1 P (4) j λ 0 P, j (5) j J

16 7 where J denotes the set of integer extreme oints of con( Q B ), V j x is a V - ector that defines extreme oint j J, and λ j is a RMP decision ariable that corresonds to extreme oint J j. Sub-roblem P is formulated as T j j {( ) : } V Z = Max w A x x Q B V where Q = { R+ : x + x 1 ( u, ) E } u α, (6) x and α is an Eˆ -ector of dual ariables associated with constraint (3). They tested two different artitioning rocedures; one artitioned an original grah into chordal sub-grahs and the other used METIS [15, 16, 17], a heuristic that seeks to minimize the number of edges in Ê, while balancing the number of ertices in different artitions, gien the number of artitions. They soled MWI on each chordal subgrah using Frank s algorithm [11]. For soling the NP-Hard MWI osed by each METIS-artitioned sub-grah, they modified the Carraghan and Pardalos [9] algorithm to address weights and sole the MWI in the grah (the original algorithm finds the maximal clique in a grah). We refer to this modified algorithm using the acronym MCP. In addition to ealuating these two methods to artition a grah, they tested with two tyes of RMP formulation and two methods of branching. They tested their methodology with DIMA Challenge Problems [14] and randomly generated -grahs and concluded that the combination of METIS artitioning, RMP formulation in terms of clique inequalities and branching on cliques in B&B tree gae the best results.

17 8 Furthermore, they found that their method outerformed the MCP algorithm for sarse grahs, which are known to be esecially challenging. Subsequently, we refer to this as the Original B&P (OBP) aroach to sole the MWI.

18 9 CHAPTER III VERTEX CLONING APPROACH This chater introduces Vertex Cloning (henceforth referred to as Cloning) and its mathematical formulation. 3.1 Concet Cloning extends the artitioning methods emloyed by Warrier et al [25] by cloning selected ertices with the goal of eliminating edges in set Ê. After using METIS to artition the grah G = ( V, E) into P disjoint sub-grahs G 1,...,G P, each edge e = ( u, ) Eˆ connects ertices in two different artitions ( u V, Vq where, q P, q ) and the associated edge inequality ( x x 1) is included in the RMP. u + Cloning can dulicate ertex u () into artition q ( ) so that edge ( u, ) lies entirely in artition q ( ) and the edge inequality in the RMP can be relaced by an equality x = x ), where w is the clone of u (). Similarly, edge inequalities in the RMP w x u ( can be relaced by relationshis equating the decision ariables associated with a cloned ertex and each of its clones. Cloning is analogous to the cost slitting technique of Lagrange relaxation [19, 22] through which, deending on the structure of the roblem, dulicate ariables can be introduced to imroe bounds. We refer to a ertex that is dulicated as the cloned

19 10 (originating) ertex and any dulicate ertex as a clone. We use the term coies to indicate an original ertex along with its clones. We illustrate Cloning using Figure 1, which deicts a grah comrising 7 ertices and 7 edges. The formulation for the MWI on this grah (as in (1)) can be written as: Max Z IP = x + (7) 1 + x2 + x3 + x4 + x5 + x6 x7 s.t. x + x 1 (8) 1 2 x + x 1 (9) 1 5 x + x 1 10) 2 3 x + x 1 (11) 2 7 x + x 1 (12) 3 4 x + x 1 (13) 3 6 x + x 1 (14) 4 5 x = Z (15) 7 ( x1, x2, x3, x4, x5, x6, x7 ) + Figure 2 shows an arbitrary artitioning with P ={1, 2}, where V =,,, } 1 { and V =,, }. Let G 1 and G 2 reresent the two sub-grahs (artitions), 2 { resectiely, and ˆ {(, ),(, ),(, ) } E = be the set of edges that connects ertices in the artitions. The endoints of all edges e Eˆ comrise the set V ˆ = {,,,, }

20 Fig. 1 Examle grah G. Partition 1 Partition Fig. 2 Vertex disjoint artitions of G.

21 12 Partition 1 Partition Fig. 3 Edge disjoint artitioning of G through ertex cloning. Partition 1 Partition Fig. 4 Edge disjoint artitioning of G by cloning different ertices.

22 13 The OBP reformulates model (7)-(15), creating one sub-roblem with edge inequalities associated with G 1 (i.e., (8) and (11)) and an other sub-roblem with edge inequalities associated with G 2 (i.e., (12) and (14)). The RMP comrises inequalities corresonding to edges e Eˆ (i.e., (9), (10) and (13)). Figure 3 deicts one ossible way to clone ertices so that all edge inequalities in the RMP are relaced with equality constraints. Here, 3 is dulicated (as 8 ) in artition 1 so that the edges, ) and, ) can be included in artition 1 as, ) and ( 2 3 ( 3 6 ( 2 8 6, ), resectiely. Similarly, 1 is cloned as 9 in artition 2 to include edge (, 1 5 ) ( 8 in artition 2 as, ). This cloning rocess results in an edge-disjoint artitioning of ( 5 9 G in which Ê = Ø and equalities x = x and x = x relace corresonding edge inequalities ((9), (10) and (13)) to assure that decision ariables associated with a cloned ertex and each of its clones are equal. Cloning results in ertices, instead of edges, being shared between artitions. Figure 4 demonstrates an alternate way to clone ertices. In this case, three clones (namely 8, 9 and 10 ) are formed (as clones of 5, 6 and 2, resectiely). This alternate cloning adds more ertices into the artitions, making the sub-roblems more challenging to sole and also resulting in a larger RMP. Thus, the aroach should clone a minimum number of ertices to romote tractability. Note that, tyically, only a subset of ertices in Vˆ need be cloned to locate each edge e Eˆ into some artition. In Figure 3, only two ertices from the set Vˆ are cloned and in Figure 4, three ertices are cloned.

23 Formulation We now secialize the MWI to reresent Cloning. Let K V ˆ be the set of cloned ertices and D denote the set of clones corresonding to ertex K Vˆ. Cloning ertex (as w ) relocates a set of edges ( u, ) Eˆ from Ê to artition π u (also, π w = π u ). In artition π u, this relocated edge(s) ( u, ) exists as ( u, w). Note that not all ertices in Vˆ need be cloned (see examle in 3.1). If ertex Vˆ is not cloned, D = and if it is cloned, D gies the set of its clones. Let K V ˆ denote the set of ertices for which D. Cloning increases the number of ertices in the grah to V, where V = V U K D. Cloning adds ertices and edges to certain artitions, changing G ( V, E ) G, ( V E ) =, where includes edges from set is changed to x = { x : } = to V includes V and clones that are added in artition and E as well as relocated edges. Corresondingly, the ector V may now be secialized to reflect Cloning: E. The integer rogramming formulation of the MWI x where D Z MWI = Max wx : xw x = 0 K, w D P V (16) Q V x P and Q = { x B+ : xu + x 1 ( u, ) E }. The formulation gien in (16) can be rewritten as follows:

24 15 D Z MWI = Max P w x (17) s.t. P A x = 0 (18) B x 1 P (19) V x B P (20) where A denotes the matrix of coefficients of decision ariables in equalities (18) and B denotes the matrix of coefficients of decision ariables in inequalities (19). Equalities (18) include an equialence relation between each cloned ertex and each of its clones; and inequalities (19) include edge inequalities in artition P. Inequalities (19) define P disjoint blocks of constraints, one for each artition, forming a block diagonal structure. Alication of DWD to the linear relaxation of (17)- (20) allows each block to be addressed as an indeendent sub-roblem while relegating constraint (18) to the RMP: s.t. j Z = Max λ ( w x ) (21) RMP P j J j J P j J j j λ ( A x ) = 0 (22) j λ = 1 P (23) j λ 0 P, j (24) j J

25 16 where J denotes the set of integer extreme oints of con Q ), ( j x is a V - ector that defines extreme oint j J and λ denotes the RMP decision ariable that corresonds to extreme oint j J. j Sub-Problem P is a MWI of the form: Z j j j {( w A ) x β : Q } = Max cˆ zˆ = Max α x, (25) where α is a ector of dual ariables associated with equality constraints (22) and β is the dual ariable associated with conexity constraint in (23). Otimal extreme oint j in sub-roblem gies ector j x, which is an imroing column if Z > 0. At each iteration, we sole all P sub-roblems and select j x as arg max ( Z ) P to enter the RMP basis. If Z 0 for all P, the current RMP solution is otimal.

26 17 CHAPTER IV ANALYSIS OF BOUNDS In this chater we analyze the olytoe associated with the OBP (gien in (2)-(6)) and Cloning (gien in (21)-(25)) models and their linear relaxations to show that Cloning gies a tighter bound at the root node of B&B tree than that obtained by OBP. Our roof is based on showing that the olytoe associated with Cloning is contained in the olytoe associated with the OBP. To romote simlicity, we resent our discussion in terms of the olytoes associated with decision ariables x. Let S denote the set of feasible integral solutions to (1); C, the conex hull of S ; and L, the olytoe associated with the linear relaxation of (1): S V { Z : x + x 1 ( u, ) E, x {0,1 V } = x + }, u L C = con(s) and V { R : x + x 1 ( u, ) E, 0 x V } = x + 1. u Relatie to the ertex-disjoint artitions formed in the OBP (see Chater II), let S and S denote the set of integral solutions that are feasible relatie to the edge inequalities in E (which constitute block-diagonal set the coordinating set), resectiely: S S P ) and Ê (which constitute V { Z : x + x 1 ( u, ) E, x { 0, } V } = x + 1 and u V { Z : x + x 1 ( u, ) Eˆ, x {0,1 V } = x + }. u

27 18 Similarly, let C = con ( S ) and C = con( S ). Let L denote the olytoe corresonding to the linear relaxation of S for P ; and L, the olytoe associated with the linear relaxation of S. Following their resectie definitions, we hae S C L, S C L and S C L. Noting that E = E U ˆ P E ; and E defines S, C and L ; E defines S, C and L ; Ê defines S, C and L ; we hae S Define olytoe = S IS, C C I P P, and L L I P P R O by substituting (tightening) L, relacing R O I P = L. (26) L with C : = L C. (27) Since S P L and S C L, we may write, S I P S C I P C L I P C I L L, P S C RO L. (28) Cloning relaces eery edge inequality x x 1 (where ( u, ) Eˆ ) in the u + coordinating set (of OBP) by an equality x = xw (ertex in artition π is cloned as w into artition π ) and an inequality corresonding to a clone, x x 1 (associated u u + w with edge ( u, w) in artition π u ). Let L denote the olytoe that is formed by relacing all edge inequalities ( x u + x 1) in L with equalities ( x = xw ) and edge

28 19 inequalities ( x x 1 u + w ). L can be written as intersection of olytoes = L and L, where = L denotes the olytoe associated with the equality constraints that result from cloning ( x = x w ) and L denotes the olytoe comrising edge inequalities ( x x 1), each of which includes a decision ariable associated with a clone: L u + w L V { R : x = x K, w D,0 x 1, V } = = + x, w V { R : x + x 1 ( u, ) Eˆ, K, w D : π = π,0 x 1, V } = + x and w u = = L L L I. u w Note that Cloning increases the number of decision ariables to V so the olytoes L, = L and L are defined in V -dimensional sace. We now roe that L and L are equialent; (i.e., the set of solutions that are feasible with resect to L in terms of the decision ariables that corresond to the original ertices, x : V, is same as those associated with L Proosition 1: L L. Proof: Let X { x V } ). We reresent this equialence by. = : be any ector in L and construct X { x V } = :, comrising a V - sub-ector of ariables x associated with original ertices (which includes all ertices but clones) and a V \ V - sub-ector associated with clones. In articular, for original ertices V, set x = x. For each ertex K Vˆ V, identify each of its clones, w D V \ V and set x w = x. From the construction, it is clear that X is feasible with resect to L.

29 20 It is imortant to note that L contains K clones) than L D more ariables (associated with, tending to increase the dimension of olyhedron L by K D. For K, one equality constraint relates cloned ertex to each of its clones w ( x w = x ) for w D. Since there are exactly K D (linearly indeendent) equality constraints in L, the dimension of L is the same as that of L. L includes more decision ariables but solutions are rojected onto the set of solutions that are feasible with resect to L by the associated equality constraints. Thus, we conclude that L L. Q.E.D. From (27), we hae R O = L I P L I P = con( S ). Let S denote the set P of integral oints that is equialent to the corresonding to set of integral oints V -dimensional sace (i.e., S S written as : R O S in ). Therefore, using L L, R O may be I L con( S ). P Since = = L = L I L, RO L I L I con S P ( ). (29) Relatie to the edge-disjoint artitions formed in Cloning, edge inequalities in E comrise the block-diagonal set P. Let S denote the set of integral solutions that are feasible relatie to block-diagonal set S E for P : V { Z : x + x 1 : ( u, ) E, x { 0, } V } = x + 1 and u

30 21 let C = con ). The block diagonal set P in Cloning ( E ) incororates ( S the inequalities associated with edges in E as well as those associated with clones. L denotes the olytoe corresonding to the inequalities associated with clones. A block diagonal set P incororates a set of inequalities corresonding to clones that are added into. In other words, block diagonal set incororates a subset of inequalities from the set (of inequalities) that defines L (i.e., x u + x w 1 where w D and ( u, ) Eˆ ) for which = π u = π w. Let L denote the olytoe associated with inequalities (corresonding to clones) that are added to artition P such that L I P = L. S consists of integer oints, which are feasible with resect to edge inequalities E as well as inequalities corresonding to integer solutions that are feasible relatie to L L. Let resect to a block-diagonal set in Cloning may be written as: S denote the set of. The feasible integer solutions with = I S S S. (30) = L gies the olytoe associated with the coordinating set in Cloning as it consists of equalities, each of which relates a cloned ertex with one of its clones. Let L denote this olytoe: Let L V { R : x = x K, w D, 0 x 1 V } = = L = x + w. R C denote the olytoe formed by the intersection of L and C :

31 22 R C I = L C. (31) P Proosition 2: R R. C O Proof: From (30), we hae = I S S S, con ( S ) = ( con S I ) S, con ( S ) ( con S ) I ( ) con S, I L con ( S ) con( S ), (as con ) ( S L ), I P I P ( con( S L ) I con ( S ) ), I P I I I con ( S ) con( S ) L, P P I P con I I ( S ) con( S ) L, (as L I P P = L ). From (31), R C = L I P L I P = con ) P ( S ; and substituting for I P con( ), S R C I P ( S L I L Icon S P = L con ) ( ), I I RC L ( ), ( Since L = = L con S P = L ). Using (29), R R. Q.E.D. C O Finally, using S C R O L from (28) and RC RO, we hae S C R C R O L. Let Z L and Z C denote the otimal solution alues obtained by soling the MWI objectie function (1) on olytoes L and C, resectiely.

32 23 Similarly, let Z R and C RO Z denote the otimal solution obtained by soling the MWI on olytoes R C and R O, resectiely. Thus, Z MWI = Z Z Z Z. C R C R O L Proosition 3: In B&P search tree, Cloning gies tighter bound at the root node than the bound obtained by the OBP. Proof: From (27) and (31), we hae R I C = L P and R O L I P P = C. If we aly DWD to the constraint set of R O, the constraints that form L are relegated to form the constraints in the RMP of the OBP model (see (2)-(5)) and those that form C create the constraint set for the sub-roblem (see (6)). Similarly if we aly DWD to the constraint set of R C, the constraints in L form the constraints in the RMP of the Cloning model and those in C creates the constraint set for sub- roblem. Since Z R Z, it imlies that Cloning gies tighter bound at the root node C R O than the bound obtained by the OBP model. Q.E.D. Howeer, should sub-roblems exhibit the Integrality Proerty, (i.e., all extreme oints of L for P are integral), Z R Z R = C O = Z. L Hence, to obtain a tighter bound, it is imeratie that sub-roblems aoid the Integrality Proerty.

33 24 CHAPTER V IMPLEMENTATION ISSUES Cloning inoles two key issues: (a) Selecting ertices to be cloned, (b) Assigning weights to clones. We discuss these issues and roose solutions in this chater. We resent the oerall algorithmic stes inoled in soling the MWI by our B&P aroach and introduce a new concet, Partial Cloning, deeloed to exloit the desirable irtues of both OBP and Cloning aroaches. 5.1 Selecting Vertices for Cloning Each ertex that is cloned increases the size of the artition (i.e., sub-roblem) into which it is cloned as well as the number of equality constraints (in the RMP). Esecially in dense grahs, Cloning may add a large number of ertices, resulting in larger subroblems that are more difficult to sole. Thus, it is imeratie that Cloning dulicate the minimum number of ertices. For examle, in Figure 2, to relace edges ( 2, 3 ) and ( 6, 3 ), either 2 and 6 could be cloned into artition 2, increasing its size by two ertices (and two edges) or 3 could be cloned into artition 1, increasing its size by only one ertex (and, of course, two edges). This issue can be resoled by soling an aroriate set coering roblem. Using binary decision ariables y = 1 if ertex is cloned into artition and y = 0 otherwise, the set coering roblem may be formulated as follows:

34 25 where Q sc m Z = Min y : y Qsc B, ˆ (32) V P m { R : y + y 1 u Eˆ, u V and V } y and m = Vˆ ( P 1). = + uq The set coering roblem is NP-Hard [12], but a near otimal solution would suit our urose so we roose a modified ersion of the greedy set coering heuristic [19] to quickly obtain a solution. We refer to this heuristic as the modified set coering heuristic: Ste 1: i = 0, K =, D = Ste 2: For eery ertex Vˆ ; V = V and E = E for all P. Vˆ, determine N () from Ê. Ste 3: Calculate N ( = max { N ( ) : P, Vˆ, π } ). N ( ) identifies the ertex to be cloned as the one adjacent to the largest number of ertices not in the same artition (i.e., π ) and the artition cloned. Clone ertex U w into artition as ertex w. q into which it would be Ste 4: Udate; D D { }, V V U{ w} E {( u, w) : u N ( )} E U, K K U{}, {( u, ) : u N ( )}, ˆ ˆ E E \ and Vˆ = Vˆ \ if N ( ) = P, π. Ste 5: Reeat Stes 2, 3 and 4 until Vˆ =. Ste 6: For K, w D are the clones of ertex.

35 Assigning Weights Aroriate weights must be assigned to a cloned ertex and its clones. To be an exact coy, a clone should hae the same weight as that of its originating ertex but this would increase the total weight in the grah so that the otimal solution to the MWI on the grah with clones would not be the same as that on the original grah. We imlemented two strategies that result in total weights that are the same in both the original grah and the one that results from cloning. One strategy is to diide the weight of an originating ertex equally among the set of coies. Another, and in fact the simlest, strategy is to assign a null-weight to clones. The chater on comutational ealuation comares the imacts of these strategies on run-time. 5.3 Soling the MWI Cloning may be detailed as follows: Ste 1: Partition an original grah into P artitions using METIS [15, 16, 17]. Ste 2: Aly the modified set coering heuristic to select the set of ertices to be cloned and identify the clones for each. Udate the RMP to include equalities corresonding to equialence relationshis between each originating ertex and its clones. Udate sub-roblems to include clones ( w edge inequalities. D ) and their associated Ste 3: Sole the Cloning formulation utilizing the MCP algorithm to sole subroblems. At each iteration, re-otimize RMP oer known columns and use the resulting dual ariables to define the objectie function coefficients of

36 27 decision ariables in sub-roblems. Use a ool to store the columns generated by the sub-roblems. Maintain reiously generated columns in the ool and otimize oer these known columns before soling sub-roblems in an attemt to consere run-time. Branch on clique inequalities as described in Warrier et al [25]. 5.4 Partial Cloning Warrier et al [25] obsered that the OBP results in large ˆ E so that the RMP may comrise a large number of constraints and require a lengthy solution time. Cloning decreases the number of RMP constraints because the modified set coering heuristic (Section 5.1) seeks the minimum number of ertices to clone. On the other hand, this aroach adds clones to artitions, increasing the size of indiidual sub-roblems and making them more challenging for the MCP algorithm to sole. Hence, Cloning introduces a trade off by which roblem comlexity can be distributed among the RMP and sub-roblems. The sizes of the artitions (sub-roblems) can be controlled to some extent by secifying the number of artitions that METIS is required to deelo. Howeer, the V and E deend on the characteristics of artitions created by METIS and the set of clones rescribed by the modified set coering heuristic. We roose a new aroach to achiee a faorable trade-off between the size (and tightness) of the RMP and the sizes of the sub-roblems. This aroach, which we call Partial Cloning, may not clone all ertices in Ê, erhas retaining some edge

37 28 inequalities in the RMP. We udate ste (2) of the modified set coering heuristic (Section 5.3) to imlement Partial Cloning by setting a threshold (PCThreshold) to affect the ertex selected for cloning. To imlement Partial Cloning, Ste 2 in the heuristic gien in chater 5.1 is udated to be: Ste 2 (udated): If max { ( ) : P, Vˆ }> PCThreshold, continue to Ste 3, N else go to Ste 5. This modification allows the RMP to retain some edge inequalities while including equalities associated with clones. Henceforth, we use Comlete Cloning (CC) to secify the aroach where all the edges in Ê are relocated by cloning and use Partial Cloning (PC) to secify the aroach in which only a subset of edges in Ê are relocated. We set PCThreshold to 1 in our tests so the modified set coering heuristic adds clones corresonding to those ertices Vˆ and artition P, for which ( ) > 1 N for π. If PCThreshold is set to 0, Comlete Cloning results, yielding larger, more sarse sub-roblem that are more challenging for MCP algorithm to sole.

38 29 CHAPTER VI COMPUTATIONAL EVALUATION We comare CC, PC, OBP and MCP comutationally using two tyes of instances: (1) DIMA Instances taken from the Second DIMA Imlementation Challenge [14], and (2) random - grahs: These random grahs are generated by secifying the number of ertices V and alue (robability that edge ( u, ) is included in the grah). We conducted all tests on a Dell PC with a 3.06 GHz Pentium IV rocessor and 512 MB of memory using the Visual C++ enironment and CPLEX 7.1. Preliminary testing of the two Cloning aroaches (CC and PC) each using the two weight-assignment strategies (Chater 5.2) showed that assigning null weights to clones erforms better than assigning each clone the same weight associated with its originating ertex. Hence, we resents results that assign null weights to all clones. We select P based on the criterion that the resulting sub-roblem, after artitioning and cloning, should be less challenging for MCP to sole. Howeer, there is no definite way to ascertain the size of sub-roblems that will result from Cloning. Preliminary tests showed that, for grahs with 100 or more ertices and edge densities less than 40%, P 6 results in sub-roblems that MCP can sole effectiely and for grahs haing edge densities greater than 40%, P = 2 or 3 results in sub-roblems that MCP can sole effectiely.

39 30 The Partial Cloning arameter, PCThreshold, affects the mix of equalities and inequalities in the RMP. We set the default alue of PCThreshold to 1. On some instances, a alue of 1 leads to as many clones as in CC (because for all Vˆ, N ( ) > 1 P, π ). Hence, in these cases, PCThreshold is set to E ˆ / M, where M is equal to K D. ˆ E / M gies the aerage number of ertices that a ertex Vˆ is connected by edges ( u, ) Eˆ. Table 1 comares the erformances of the three B&P aroaches (OBP, CC and PC) and MCP in alication to the DIMA instances. Performance measures include the number of constraints in RMP, otimal solution at root node of the B&B tree, and comutational time(in cu seconds). The first fie columns gie the name of instance; number of ertices, V ; density; Z MWI ; and number of artitions, P. Columns 6-8 gie the number of equality constraints in the RMP for OBP, CC, and PC, resectiely (the number in the braces gie the number of inequalities in the RMP corresonding to edge inequalities). In OBP, RMP comrises only inequalities, and in CC, RMP comrises only equalities. In PC, RMP comrises a mix of inequality and equality constraints. Columns 9-11 gie Z LP (OBP), Z LP (CC) and Z LP (PC), the otimal solution at the root node (of B&B tree) for OBP, CC and PC, resectiely. The otimal solution at root node gies an uer bound on Z MWI. Comutational results confirm that CC and PC gie uer bounds that are tighter than the one that OBP gies and, as exected, Z LP (CC) Z LP (PC) Z LP (OBP).

40 31 Table 1 Performance measures for DIMA instances. Grah V Density Z IP P Number of RMP Z LP Time (in sec) OBP C1 C2 OBP C1 C2 OBP C1 C2 WCP johnson824com (64) 13 1(56) johnson824com (103) 29 25(4) johnson824com (100) 39 29(10) johnson824com (124) 45 39(6) johnson844com (240) 69 59(10) johnson844com (374) (34) johnson844com (381) (57) manna9com (10) 9 1(8) manna9com (16) 14 2(12) manna9com (25) ) manna9com (26) ) manna9com (29) 23 6(17) cfat2001com (8999) 97 29(6253) cfat2001com (12137) (6575) cfat2002com (7952) 97 30(5404) hamming62com (64) 64 0(64) hamming62com (101) 67 27(40) hamming62com (114) 75 33(42) hamming82com (626) - 95(414) johnson1624com (162)

41 32 Z LP In fact, (CC) = for most of the instances (giing an integrality ga of Z MWI 0%). Furthermore, with an increase in P, the bound gets weaken (integrality ga increases) for each of the three B&P aroaches. Figures 5 and 6 shows ariation of Z LP (OBP), (CC) and Z (PC) with increase in Z LP LP P for manna9com and johnson824com resectiely. Columns comare the run times (cu seconds) required by OBP, CC and PC to sole each instance, excluding the times required for artitioning and cloning, which are triial. Column 15 gies the run time required to sole each instance by MCP. A - indicates that the corresonding instance requires more than 12 hours of run time. We found that, as P increases, the run time required by each B&P aroach to sole an instance aries deending uon whether the instance is dense or sarse. For dense instances, run time increases with an increase in P and, for sarse instances run time first decreases and then increases as P increases, so some alue of P gies minimum run-time for sarse grahs. We ary the alue of P for a few reresentatie instances (e.g., manna9com, johnson824 com) to show the ariation in run-time as P increases. For the remaining instances, we tabulate results for those P that gie minimum run-time (for e.g., we set P = 10 for johnson1624com and P = 20 for hamming82com). PC gies quite cometitie results for most of the DIMA instances. Figures 7 and 8 shows ariation of run time for three B&P aroaches with increase in P for manna9com and johnson824com resectiely.

42 33 ZLP P OBP CC PC Fig. 5 Z LP s P for johnson824com. ZLP P OBP CC PC Fig. 6 Z LP s P for manna9com.

43 34 Time (in sec) P OBP CC PC Fig. 7 Run time s P for Johnson 824 com. Time (in sec) P OBP CC PC Fig. 8 Run time s P for manna9com.

44 35 Table 2 reorts alication of the three B&P aroaches to random -grahs, using the same column headings. W0 in the name of instance indicates an un-weighted grah and W1 indicates a weighted grah. Run times reorted in columns of Table 2 show that MCP outerforms all three B&P aroaches on random instances haing densities greater than 40%. For instances with densities below 20%, all three B&P aroaches erform better than MCP. Comaring run times in columns shows that weighted grahs are generally less challenging to sole than un-weighted grahs. Although CC neer gies the best run-time, it gies quite cometitie results for highly dense and highly sarse instances. Furthermore, as obsered in DIMA instances, for all the random grahs, we hae Z LP (CC) Z LP (PC) Z LP (OBP). To gain further insight into the erformance of B&P aroaches for soling the MWI, we comare seeral additional erformance measures in Tables 3 and 4, which relates to the instances reorted in Tables 1 and 2. Columns 1 and 2 in Tables 3 and 4 gie the name of the instance and the number of artitions, P, resectiely. Columns 3-5 gie number of RMP iterations required and columns 6-8 gie number of nodes exlored in the B&B tree to obtain an otimal integral solution by each of the three B&P aroaches. If the number of nodes exlored is zero, the otimal integer solution was obtained at root node of the B&B search tree (i.e., Z LP (CC) = Z MWI ). PC tyically exlores a number of B&B nodes that is between the numbers of nodes required by CC and OBP.

45 36 Table 2 Performance measures for -grahs. Grah V Density Z IP P number of RMP Z LP Run time (in sec) (%) OBP CC PC OBP CC PC OBP CC PC WCP RG_GV1_W1_P (150) (79) RG_GV1_W0_P (349) (125) RG_GV1_W1_P (349) (125) RG_GV1_W0_P (789) - 234(122) RG_GV1_W1_P (789) - 234(122) RG_GV1_W0_P (1231) - 341(105) RG_GV1_W1_P (1231) - 341(105) RG_GV1_W0_P (1517) - 254(14) RG_GV1_W1_P (1517) (14) RG_GV1_W0_P (1089) 49 32(320) RG_GV1_W1_P (1089) 49 32(320) RG_GV1_W0_P (1325) 49 32(399) RG_GV1_W1_P (1325) 49 32(399) RG_GV1_W0_P (2067) (182) RG_GV1_W1_P (2067) (182) RG_GV1_W0_P (1573) 49 37(328) RG_GV1_W1_P (1573) 49 37(328) RG_GV1_W0_P (2444) (148) RG_GV1_W1_P (2444) (148) RG_GV1_W0_P (1875) 50 36(478) RG_GV1_W1_P (1875) 50 36(478) RG_GV1_W0_P (2878) (156) RG_GV1_W1_P (2878) (156) RG_GV1_W0_P (2159) 49 43(235) RG_GV1_W1_P (2159) 49 43(235) RG_GV1_W0_P (3272) (622) RG_GV1_W1_P (3272) (622)

46 37 Table 3 Additional erformance measures for DIMA instances. Grah P number of RMP iterations number of B&B nodes % of time to clone % of time to obtain Z LP % of time to sole sub-roblems OBP CC PC OBP CC PC CC PC OBP CC PC OBP CC PC johnson824com johnson824com johnson824com johnson824com johnson844com johnson844com johnson844com manna9com manna9com manna9com manna9com manna9com cfat2001com cfat2001com cfat2002com hamming62com hamming62com hamming62com hamming82com johnson1624com

47 38 Table 4 Additional erformance measures for random -grahs. Grah P number of RMP iterations number of B&B nodes % of time to clone % of time to obtain Z LP % of time to sole sub-roblems OBP CC PC OBP CC PC CC PC OBP CC PC OBP CC PC RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P RG_GV1_W0_P RG_GV1_W1_P

48 39 Columns 9-10 gie the ercentage of comutational time sent in Cloning relatie to the total run time. Columns gie the ercentage of time utilized to obtain otimal solution at root-node relatie to the total time sent in obtaining Z MWI. Results show that, for CC and PC, more than 90% of run time is sent in obtaining root-node Z LP Z LP solutions, (CC) and (PC). Columns gie the ercentage of time required to sole sub-roblems using the MCP algorithm relatie to the time sent in rescribing an integral otimal solution. Results show that OBP sends a smaller ercentage of run time to sole subroblems than CC and PC. OBP leads to sub-roblems that are less challenging for MCP to sole, but gies a weak Z LP (OBP) bound (see Tables 1 and 2). Because the uer bound is weak, OBP requires exloration of more nodes in the B&B search tree. (see Columns 6-8 in Tables 3 and 4) increasing the number of times the RMP is otimized, and, hence, the total number of RMP iterations (see Columns 3-5 in Tables 3 and 4). As a result, OBP sends most of the time otimizing the RMP and relatiely little time soling sub-roblems. In contrast, both CC and PC send a considerable ercentage of run time soling sub-roblems. Cloning may increase the size of sub-roblems dramatically, making them challenging for MCP but giing tighter bounds. Because Z LP Z LP uer bounds (CC) and (PC) are tight, CC and PC both exlore fewer B&B nodes and, thus, require less run time to otimize. Further, we comare the three B&P aroaches in alication to random -grahs with densities less than 10%. Table 5 gies results for these grahs with the same

49 40 column headings used in Tables 1 and 3. For CC, run time increases raidly with an increase in grah density. Howeer, the (CC) bound is better than (PC) and Z LP (OBP) for all the test cases. Z LP We conjecture that Cloning may work better in alication to instances for which the resulting sub-roblems are less challenging for MCP to sole. Increasing the alue of P may result in desirable sub-roblems but weakens the uer bound and increases the oerall run time by increasing the number of B&B nodes (which increases the time sent in otimizing the RMP). Z LP

50 41 Table 5 Performance measures for sarse random -grahs. Grah V Density Z IP P number of Z LP run time (in sec) RMP constraints (%) OBP CC PC OBP CC PC OBP CC PC WCP RG_GV1_W1_P (30) 25 5(20) RG_GV1_W1_P (70) 61 6(55) RG_GV1_W1_P (108) 91 17(74) RG_GV1_W1_P (150) (79) RG_GV1_W1_P_ (179) (97) RG_GV1_W1_P_ (248) (99) RG_GV1_W1_P_ (273) (110) RG_GV1_W1_P_ (297) - 77(121)