On the Approximability of Multi-level Bottleneck ... · PDF fileOn the Approximability of Multi-level Bottleneck Assignment Problem? Trivikram Dokka Department of Management Science,

On the Approximability of Multi-levelBottleneck Assignment Problem?

Trivikram Dokka

Department of Management Science,Lancaster University Management School.

[email protected].

Abstract. We revisit the multi-level bottleneck assignment problemand study the approximability when there are m > 3 sets. We showthat with respect to the approximability the problem exhibits an easyto hard transition. More specifically, we give a PTAS to the easiest hardcase which is when the alternative edge-sets are complete, and provethat it is NP-hard to approximate the problem to within a ratio O(m

3)

for every fixed m ≥ 3 when each edge-set is arbitrary.Keywords: bottleneck assignment; approximation; computational com-plexity; robustness.

1 Introduction

The following m-dimensional axial assignment problem arises in scheduling androstering applications: Given are m pairwise disjoint sets S1, S2, . . . , Sm each ofcardinality n, and a weight w(s) ∈ N for each s ∈ S where S = ∪mi=1Si. Theset S can be seen as the node-set of an m-partite graph that has a given setof arcs E of the following form: E = {(u, s)| u ∈ Si, s ∈ Si+1}. Thus, an arcin E connects a node from Si with a node from Si+1 (1 ≤ i ≤ m − 1), andthere are no other arcs. An m-tuple D = {s1, s2, . . . , sm} is feasible if it is suchthat si ∈ Si for 1 ≤ i ≤ m, and (si, si+1) ∈ E. The cost of a m-tuple D equalsc(D) =

∑s∈D w(s). The problem is to find a partition of S into n feasible m-

tuples D1, D2, . . . , Dn such that maxjc(Dj) is minimum. We will refer to thispartition of S into {D1, D2, . . . , Dn} as a solution M , and the cost of a solutionM equals c(M) = maxj{c(Dj)|M = {D1, D2, . . . , Dn}}. The problem describedabove is known as the multi-level bottleneck assignment problem (MBA-m).

Literature review and motivation

The multi-level bottleneck assignment problem (MBA) was first introduced andstudied by Carraresi and Gallo [2], motivated by an application in bus driverscheduling. Special cases of the problem have been studied even before [2]. Aparticular important special case which we call complete-MBA, as was referred

? A part of this research was done during the author’s PhD at ORSTAT, KULeuvenand was supported by OT Grant OT/07/015.

2 T. Dokka

to in [5], is when E is complete. The approximability of this special case hasbeen studied by Hsu [9] and by Coffman and Yannakakis [3]. For complete-MBA,Hsu [9] gave an (2− 1

n )-approximation algorithm that runs in O(mnlogn), whileCoffman and Yannakakis [3] gave an O(n2m) ( 3

2−12n )-approximation algorithm.

For the case where m = 3, Hsu gave a 32 -approximation algorithm that runs in

O(nlogn), and a 43 -approximation algorithm that runs in O(n3logn).

Another important problem of which MBA can be seen as a generalizationis the bi-criteria scheduling problem in which one tries to find a schedule withminimum makespan over all flow time optimal schedules on identical machines.This problem was first studied in 1976 by Coffman and Sethi [10] where a 5

4approximation algorithm is given. Eck and Pinedo [6] give a 28

27 approximationfor the two machine case. In more recent work Ravi et al[14] prove a conjecturein [10] for some special cases.

As far as we are aware, all known approximation results except [5] deal withthe case where every pair of nodes of different sets can be joined in a duty, i.e.,the case of complete-MBA. In this paper we deal with a more general setting,namely the case where the edge set between Vi and Vi+1 can be arbitrary (andnot necessarily complete) (1 ≤ i ≤ m − 1). Our main focus is to explore (andextend the work in [5]) the boundary between the easy and hard cases as theedge-sets are relaxed from complete to arbitrary.

Related problems have also been studied from approximation point of view.For example one can see MBA as a generalization of the classical multi-processorscheduling problem, with incompatibilities between jobs. Such problems withincompatible jobs but different from MBA have been studied in Bodlaender etal. [1]. Other types of (three-dimensional) bottleneck assignment problems havebeen studied by Klinz and Woeginger [11] and Goossens et al. [8].

It should be noted that in our problem one can see the edge-set between twonon-adjacent sets is complete. The case when this is not necessarily true ourproblem is equivalent to the k-dimensional matching.

2 Our results

Our results mainly show the change in the approximation hardness of the prob-lem as the edge sets Ei, i = 1, . . . ,m−1, change from being complete to arbitrary.We mainly extend the results in [5] where the case when m = 3 is considered.More specifically for every fixed m ≥ 3:

– We prove the existence of a {(m3 + 1) − ε}-polynomial time approximationalgorithm for MBA-m implies P = NP .

– We give a PTAS when no two adjacent edge sets are both arbitrary i.e., atleast one of Ei and Ei+1 is complete.

In [5] it is shown that complete-MBA-m has PTAS. Therefore our results suggestthat there sharp line when the problem becomes hard with respect to achievableapproximation in polynomial time.

On the Approximability of Multi-level Bottleneck Assignment Problem 3

We also consider the robust version of MBA i.e., when the weights on nodesare not known with certainty. We model the Robust-MBA as a vector assign-ment problem (see Section 4) and note some results which are derived from theapproximability results on MBA-m.

3 Deterministic case

3.1 Inapproximability of Arbitrary case

In [9] it is shown that for complete-MBA the natural sequential heuristic achievesa 2-approximation. It is tempting to believe that this may be true even in thearbitrary case. We show that MBA-m for a fixed m > 3 cannot be approximatedwithin a factor of m

3 + 1 unless P=NP. To do so, we use a traditional technique:we will show that a YES-instance of 3-dimensional matching (3DM) correspondsto an instance of MBA3 with cost 1, whereas a NO-instance corresponds to aninstance of our problem with cost m

3 +1. Then, a polynomial time approximationalgorithm with a worst case ratio strictly less than m

3 + 1 would be able todistinguish the YES-instances of 3DM from the NO-instances, and this wouldimply P = NP.

Let us first recall the 3-dimensional matching problem:Instance: Three sets X = {x1, ..., xq}, Y = {y1, ..., yq}, and Z = {z1, ..., zq},

and a subset T ⊆ X × Y × Z.Question: does there exist a subset T ′ of T such that each element of X∪Y ∪Z

is in exactly one triple of T ′?

Let the number of triples be denoted by |T | = p. Further, let the number oftriples in which element yj occurs, be denoted by #occ(yj), j = 1, . . . , q. Also ifu ∈ X × Y × Z occurs in t ∈ T we write u ∩ t = 1.

Starting from arbitrary instance of 3DM, we now build an instance of MBA-m by specifying Si (i = 1, 2, 3), E, and the weights w. Before we explain theconstruction we first explain the basic building blocks and gadgets which arepieced together to form a MBA-m instance.

Building Sub-blocks

There are two types of nodes in the resulting MBA instance. We will call themmain and dummy nodes.

The main nodes in each set in the constructed MBA instance can be seen aspartitioned into many sub-blocks of nodes. Each sub-block is of cardinality q, p,or d(= p − q). We relate a sub-block of cardinality q to one of the element setsX, Z and refer to such a sub-block as element sub-block. To explicitly refer to anelement sub-block corresponding to the set Z we will call it Zelement sub-block.A sub-block of cardinality p has a node for each triple t ∈ T . We call such asub-block a triple sub-block. There are #occ(yi) nodes in d sub-block for eachyi ∈ Y . We refer to a sub-block of size d as a matching sub-block.

4 T. Dokka

We say an element sub-block (Zelement sub-block) is connected to a triplesub-block when there is an edge between a node in element sub-block (Zelementsub-block ) corresponding to xi ∈ X (zi ∈ Z) and each node correspondingto a t ∈ T in triple sub-block such that xi ∩ t = 1. Similarly, a matchingsub-block is connected to triple sub-block if all nodes corresponding to yi ∈ Yare connected to all nodes corresponding to triples containing yi in the triplesub-block . Hereafter, we simply refer to two blocks as connected when we meanthem connected as explained above.

The role of the dummy nodes and their exact number will be apparent whenwe explain the connections between gadgets in the later sections.

Gadgets

The construction is mainly based on three gadgets G0, G1 and G2. Each ofthese gadgets is an MBA-3 instance. Let V h1 , V h2 , and V h3 be the three node-setsof Gh, h = 0, 1, 2.

There is an element sub-block and a matching sub-block in V 01 , a triple

sub-block in V 02 , a Zelement sub-block and a matching sub-block in V 0

3 . Thetriple sub-block in V 0

2 is connected to the element sub-block and the match-ing sub-block in V 0

1 ; the Zelement sub-block and the matching sub-block in V 03 .

Finally, each node in matching sub-block in V 01 and each node in Zelement

sub-block in V 03 has a weight equal to 1 and all other nodes in G0 are of zero

weight. An illustration of G0 can be seen in Figure 1, where element, matchingand triple sub-blocks are illustrated using triangle, rectangle and oval shapesrespectively.

The structure of G1 is as follows: each of the three sets V 11 , V 1

2 and V 13 have

2pq main nodes. These can viewed as q blocks of nodes with each block having2p nodes. We denote these blocks as b1, b2, . . . bq. Each block of V 1

1 is a union ofan element sub-block, a matching sub-block and a triple sub-block . Each blockin V 1

2 and V 13 contains two triple sub-blocks. We refer to them as head triple

sub-block and tail triple sub-block . Figure 1 gives an illustration of a blockin G1. The edge set between V 1

1 and V 12 is: element sub-block and matching

sub-block within a block bk is only connected to the head triple sub-block in bk,k = 1, . . . , q. The triple sub-block in bk of V 1

1 is only connected to tail triplesub-block in bk of V 1

2 . Note that in the edge set between V 11 and V 1

2 two blocksare not connected i.e., no two nodes not in the same block are connected. Ineach block head triple sub-block in V 1

3 is connected to both triple sub-blocksin V 1

2 , and tail triple sub-block in V 12 is connected both triple sub-blocks in

V 13 . Two blocks in G1 are connected as follows: tail triple sub-block of V 1

3 inbi is connected to the head triple sub-block of V 1

2 in bi+1, i = 1, . . . , q − 1 (SeeFigure 2). Nodes in all sub-blocks have weight equal to 0 except the head triplesub-block in V 1

3 in which all node weights are equal to 1.The gadget G2 is a mirror image of G1 i.e., V 2

1 , V 22 , V 2

3 are identical to V 13 ,

V 12 , V 1

2 respectively and edge sets are as defined for G1. See Figures 2 and 1.Before explaining the main construction of the final MBA instance we will

first explain how similar gadgets are connected to each other. Let A and B be

On the Approximability of Multi-level Bottleneck Assignment Problem 5

A block in G1 A block in G2

G0

Fig. 1. Illustration of G0; a block in G1 and G2, where a triangle, rectangle, ovalshapes indicate q-sub-block, d-sub-block, and p-sub-blocks of nodes. The highlightedsub-blocks have nodes with node weights equal to 1.

1st block

: : : :

Connected to head triple sub-block in second block

Connected to tail triple sub-block in (q-1)th block

qth block

: : : :

Connected to head triple sub-block in second block

Connected to tail triple sub-block in (q-1)th block

Fig. 2. Illustration of connections between blocks in G1 and G2.

6 T. Dokka

two G1 (G2) gadgets then we say A and B are connected as follows: the tail triplesub-block of V 1

3 (V 21 ) in the last block of A is connected to head triple sub-block

of V 12 (V 2

2 ) in the first block of B. We denote this connection as A→ B.

Main Construction

We will now explain the details of connections between many G0, G1, G2

gadgets which together will make an MBA-m instance with fixed m > 3 nodesets. Observe that G0 is identical to the construction used in [5] where the resultis only proved for the case when m = 3. In the resulting MBA instance thefirst three sets are a combination of G0 gadgets which are not connected to eachother, node-sets S3h to S3h−2 are a combination of G1 gadgets when h is evenand a combination of G2 gadgets when h is odd number greater than 1.

Let m = 3u with u ≥ 1. It is easier to understand the construction by meansof induction. That is, suppose we are given a MBA instance I constructed usingG0, G1 and G2 gadgets. We give a procedure to extend the instance I with 3unode sets to an instance with 3u+ 3 node sets by connecting additional gadgetsto I in a specific manner. Hence one can start with m = 3 and extend theinstance to any m ≥ 3. We show the procedure in two parts, in first part we dealwith the case when u is even and then the case when u is odd.

Let u be an even number greater than 1. Then the last three sets i.e., S3u,S3u−1, and S3u−2 are a combination of G1 gadgets.

Let the number of G1 gadgets in last three sets of I be x. This implies thereare xq triple sub-blocks in S3u with nodes in them having a weight equal to 1.

We first create p − 1 additional copies of I and we refer to these copies asI1, . . . , Ip with I1 = I. Each Ii is an exact replica of I with same nodes and edges.Therefore, the new Si, i = 1, . . . , 3u, has the all the nodes in the union of Si’s ineach of the p copies of I. We then create x·p new G2 gadgets. The three new nodesets S3u+1, S3u+2, S3u+3 are a combination of these G2 gadgets. Let us denotethese as G1

2, . . . , Gp2. Then, G1

2, . . . , Gp2 are connected as: G1

2 → G22 → . . .→ Gp2.

Apart from these main nodes there is an Zelement sub-block in each of threenew sets S3u+1, S3u+2, S3u+3. Let us denote these Zelement sub-blocks by ζ1,ζ2, ζ3 for the purpose of explaining their connections with G2 gadgets. Let w2

be the total number of nodes in x · p G2 gadgets. To ensure feasibility we adddummy nodes as follows. We add in each Si, i = 1, . . . , 3u, a total of w2 + qnumber of new dummy nodes. Similarly let w1 be the number of nodes in eachIi, i = 1, . . . , p, then we add a total number of p · w1 dummy nodes to the setsS3u+1, S3u+2, S3u+3. Any two dummy nodes belonging to adjacent node-sets areconnected by an edge and each dummy node has a weight equal to 0.

Let us first explain the way Zelement sub-blocks ζ1, ζ2, ζ3 are connectedto G2’s in S3u+1, S3u+2, S3u+3. The element sub-block ζ1 is connected to thehead triple sub-block of the first block of G1

2. Similarly, element sub-block ζ2

is connected to the tail triple sub-block of the last block of Gp2 and ζ3 is onlyconnected to ζ2. All main nodes nodes in S3u which are not connected to mainnodes in S3u+1 are connected to all dummy nodes in S3u+1. All main nodes in

On the Approximability of Multi-level Bottleneck Assignment Problem 7

S3u+1 are connected to all dummy nodes in S3u. Note that the copies of I arenot connected with each other.

We will now explain the connections between the main nodes in S3u andS3u+1. Recall that each G1 is a union of q blocks each containing two triplesub-blocks. For convenience of explanation we will simply refer to the nodes inS3u as x · q blocks in each Ii, i = 1, . . . , p. Similarly, S3u+1 is a union of x · p · qblocks (not including the dummy nodes and ζ1).

Each node in head triple sub-block in each block of Ih is connected each nodecorresponding the hth triple in T in head triple sub-block of each block in S3u+1.Observe that the total number of nodes in head triple sub-blocks in S3u is x ·p ·qwhich is equal to the total number of blocks in S3u+1.

This completes the description of the new instance with 3u+ 3 node sets.We have the following lemma for a fixed value of u which implies the con-

struction is polynomial in p, q.

Lemma 1. The number of nodes in the constructed instance is polynomial inp, q for each fixed m.

We will now prove the main inapproximability result which has two casescorresponding to when the 3DM instance is a YES or a NO instance. Let us firstconsider the difficult case, that is when the given 3DM instance is a NO-instance.

Lemma 2. If the 3DM instance is a NO-instance then in a (partial) feasiblesolution in S3u, S3u+1, S3u+2, S3u+3, each 4-tuple containing a node from headtriple sub-block of V 1

3 of each block of each G1 in `th copy of I has a cost equalto 2, for some ` ∈ {1, . . . , p}.

Proof. Note that the number of edges between any two triple sub-blocks is equalto p i.e. there is an edge corresponding to each triple in t ∈ T . We will refer tothese edges as triple edges.

To prove the statement we first will prove the following claim: In each blockin G2’s of S3u, S3u+1, and S3u+2, it is true that the node corresponding to the lth

triple in T in head triple sub-block of S3u+1 is matched with a node in elementsub-block in S3u+3. To see this observe first that the set of the triple edgesparticipating in matching between two head triple sub-blocks between S3u+1

and S3u+2 is same in all blocks. It then implies that in each block the sameset of nodes in head triple sub-block in S3u+2 are available to be matched withnodes in element sub-block of S3u+3. To be more precise, suppose T b ⊂ T is theset of triples such that in the bth block the head triple sub-block of S3u+2 thenodes corresponding to T b are matched to nodes in S3u+1 not in the same block.Then we have T b1∪Z = T b2∪Z for any two blocks b1 and b2. Therefore, a subsetof nodes in element sub-block corresponding to the same subset of elements inZ in each block of S3u+3 are matched to nodes in head triple sub-block withinthe same block.

To complete the proof it is enough to note that each node in head triplesub-blocks of S3u in I` are only connected to the nodes in head triple sub-blocks

8 T. Dokka

in S3u+1 which correspond to the `th triple. In other words, each node in headtriple sub-blocks of S3u in I` is matched with a node in Zelement sub-block ofS3u+3 which completes the proof. ut

Lemma 3. If the 3DM instance is a NO-instance then in a feasible solution ineach copy of I there exists a (3u)-tuple with cost equal to u+ 1.

Proof. The proof follows by induction and Lemma 2. ut

Lemma 4. If the 3DM instance is a YES-instance then there exists a feasiblesolution in the corresponding MBA instance with cost equal to 1.

Proof. See Appendix. ut

Let us now consider the case when u is odd. Let u > 1, then the constructionis very similar to that when u is even. Let there be x G2 gadgets in S3u−2,S3u−1, and S3u in the given instance I. We create q − 1 copies of I and threeadditional sets S3u+1, S3u+1, S3u+2 by creating x · q G1 gadgets. To ensure thefeasibility we add three Zelement sub-blocks in S3u+1, S3u+1, S3u+2 and dummynodes are added to all sets as before. The connections in S3u+1, S3u+1, S3u+2

are similar to that in the case when u is even. The main difference is the edge setbetween the S3u and S3u+1. Each node in the element sub-block in each blockof I`, ` = 1, . . . , q, is connected to the node which corresponds to the x` ∈ X inS3u+1. All nodes in S3u except the ones in the element sub-blocks are connectedto all dummy nodes in S3u+1. All main nodes in S3u+1 are connected dummynodes in S3u.

For the case when u = 1, we extend the instance by adding three additionalsets as before by first creating q − 1 additional copies of I (which in this case isG0). Then, we create a G1 gadget. As before, we add dummy nodes to ensurethe feasibility. Then the edge set between S3u and S3u+1 is as follows: each nodein the Zelement sub-block of V 0

3 in `th copy of I, ` = 1, . . . , p, is connected toeach node in head triple sub-block in G1 which correspond to the `th triple inT .

In both cases all nodes in S3u except the ones in the element sub-blocks areconnected to all dummy nodes in S3u+1. All main nodes in S3u+1 are connecteddummy nodes in S3u.

For sake of conciseness we omit the details of Lemmas 2-4 in these cases. Itcan be seen that a similar analysis as in Lemmas 2-4 can be repeated. Hence wehave the following theorem.

Theorem 1. MBA-m cannot be approximated to within a ratio m3 + 1 unless

P=NP.

3.2 PTAS for the easiest hard case

In this section we present a polynomial time approximation scheme for the casewhen at least one of any two adjacent edge sets is complete i.e., either Ei orEi+1 or both are complete.

We use the following famous result in our results of this section:

On the Approximability of Multi-level Bottleneck Assignment Problem 9

Theorem 2 (Lenstra [13], Frank and Tardos [7]). Consider a mixed inte-ger linear program min{cTx|Ax ≥ b and∀i ∈ I : xi ∈ Z} with n variables andm constraints, and where I ⊆ [n] denotes the set of indices of integer variables.Let s denote the binary encoding length of the input. Then, there is an algo-rithm that finds a feasible solution or decides that there is no feasible solution inO(n2.5n+o(n) · s) arithmetic operations.

For simplicity we assume m is an even number and let Ei be arbitrary when iis odd i.e., E1, E3, ...Em−1 are arbitrary. Our scheme for this case uses a similarstrategy to that in the complete case and has two main stages. We pre-processthe given instance to construct a special rounded instance and then give an IPwhich can solved using Lenstra’s algorithm [13] resulting in a solution with costat most (1 + ε)OPT .

We round the weights as follows: Choose an ε > 0 and let W be the largestweight in the given instance. We round each weight w(v) ∈ S in the instance tothe smallest possible multiple of εW . After doing this we will have h = 1

ε + 1types of distinct values for weights 0, εW, 2εW, . . . , 1ε ·εW . Therefore, each set canpartitioned into h blocks of nodes with each block having the nodes with equalweight from the set K = {0, εW, 2εW, . . . , 1ε · εW}. We define a configuration asan m-tuple (f1, f2, . . . , fm) with fi ∈ K, i = 1, . . . ,m, and let C be the set of allconfigurations.

We refer to a block of nodes with rounded weight equal to j ∈ K in Si asSji . Given a R ⊆ Sji we call Shi′ , where Si′ and Si are adjacent node-sets, is theneighbor of R if there is an edge between each node in R to some node in Shi′ .

In addition to the partition of each set according to the rounded weights wefurther use a partition of Sji based on the edge-sets which is done as follows. We

partition the set Sji into rij subsets such that any two subsets r1, r2 ∈ rij satisfy• r1 and r2 are disjoint,• r1, r2 do not have common neighbors .Such a partition for each Sji , j = 1, . . . , h and i1, . . . ,m, can be computed in

O(nmh) time and we denote this partition as P (Sji ). We denote the number of

sets Sji′ which are neighbors of a set r1 ⊆ Sji as NB(r1). The upper bound on thenumber of edges between r1 and NB(r1) is simply equal |r1|. Also observe thatwe will have that the number of matching edges between r1 and Si′\{NB(r1)}is 0.

We will now move to explain the second part where we construct an integerprogram to solve the rounded instance.

Let ca be the total weight of the configuration a ∈ C; pij be the cardinality

of Sji ; Aij is the set of all configurations which contain a node from Sji . Ourinteger program like in the complete case has the following variables:

– xa: an integer variable which measures the number of configurations of typea,

– ya: a binary integer variable which is equal to 1 if xa is positive and 0otherwise,

– w: an integer variable.

10 T. Dokka

min w (1)

s.t. w ≥ caya ∀a ∈ C (2)

xa ≤Mya ∀a ∈ C (3)∑a∈Aij

xa = pij ∀ i = 1, 2, . . . ,m; j ∈ K, (4)∑{a∈C;u,v∈a;u∈r,v∈NB(r)} xa = |r| ∀ r ∈ P (Sji ), i = 1, . . . ,m, (5)∑{a∈C;u,v∈a;u∈r,v/∈NB(r)} xa = 0 ∀ r ∈ P (Sji ), i = 1, . . . ,m, (6)

xa ∈ integer (7)

ya ∈ {0, 1} (8)

Lemma 5. The solution obtained by solving (1)-(8) is a feasible solution to therounded MBA instance.

Proof. See Appendix. ut

4 Robust case

In many situations where the MBA arises it is often the case that the weights onthe nodes are often known only approximately and can take different values. Thisimprecision is mainly due to the lack of full information about the parametersof the problem or their dependence on some uncontrolled events. One approachis when there is no clear characterization of the uncertainty and all possiblescenarios affecting the parameters of the problem are considered. This approachis usually referred to as robust optimization approach [12].

Consider the robust version of MBA where the uncertainties affecting theweights of the nodes are modeled as a discrete set of scenarios. More formally,the robust-MBA is as follows: Given are p scenarios affecting the weights ofthe nodes in MBA-m. With each scenario p is an associated MBA-m instance(Sp, Ep). The quantity spij is weight of jth node in the ith set under scenario p.We assume each Ei is complete under each scenario.

We model the Robust-MBA as a m-dimensional vector assignment problem(BMVA) using a similar notation used in [4] which can be formally definedas follows: Given m disjoint sets S1, . . . , Sm, where each set Sk contains thesame number n of p-dimensional vectors with nonnegative integral components,and by a cost function c(u) : Zp+ → Z+. Thus, the cost function assigns anonnegative cost to each p-dimensional vector. A (feasible) m-tuple is an m-tuple of vectors (u1, u2, . . . , um) ∈ S1×S2× . . .×Sm, and a feasible assignmentfor S ≡ S1 × . . . × Sm is a set A of n feasible m-tuples such that each elementof S1 ∪ . . . ∪ Sm appears in exactly one m-tuple of A.

Let us denote the operator ∨ as follows: for every pair of vectors u, v ∈ Zp+,u ∨ v = max((u1 + v1), (u2 + v2), . . . , (up + vp)). Now, the cost of an m-tuple(u1, . . . , um) is defined as c(u1 ∨ . . . ∨ um) and the cost of a feasible assignmentA is the sum of the costs of its m-tuples: c(A) =

∑(u1,...,um)∈A c(u

1 ∨ . . .∨ um).

On the Approximability of Multi-level Bottleneck Assignment Problem 11

The vector assignment problems have recently been studied in [4] where theobjective is different from robust-MBA.

Lemmas 6-7 follow from the results on deterministic case and we omit theproofs in this version.

Lemma 6. BMVA with m = 2 is polynomial time solvable.

Lemma 7. If p is fixed then there exists a PTAS for BMVA.

Lemma 8. BMVA is hard to approximate within m3 + 1 in polynomial time

unless P=NP.

Proof. See Appendix. ut

Acknowledgements: We thank Frits C.R. Spieksma for stimulating discus-sions.

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12 T. Dokka

5 Appendix

Proof of Lemma 4

Proof. It is enough to prove that when the given 3DM instance is a YES instancethe cost of partial feasible solution in the MBA-4 instance constituting S3u,S3u+2, S3u+2 and S3u+3 is equal to 1. Then the statement follows by induction.

Since it is a YES instance of 4DM, there exists a solution such that no nodesin the element sub-block in each block of the S3u+3 are connected to nodes ofhead triple sub-block within the same block of S3u+1, and hence not connectedto any main nodes in S3u. The statement now follows by induction. ut

Proof of Lemma 5

Proof. (2)-(4) ensure that∑a∈C xa = n and (5)-(6) ensure that in each edge-set

each node is only matched with one of its neighbour. ut

Proof of Lemma 8

Proof. The proof is by a simple reduction from MBA-m. Let I be a MBA-minstance then BMVA instance can be constructed as follows:

– there is a set for each Si ∈ S in I,– there is a vector for each node in S in I,– length of each vector i.e., p = n2 + 1,– the value in the position h in vectors i ∈ Si and j ∈ Si+1, i = 2, . . . ,m, is

equal to 1 if there is an edge between the corresponding nodes in I,– the value in the (n2 + 1)th in each vector is equal to the cost of weight of the

corresponding node in I.

It can be seen that if there exists a solution with cost x to the constructed BMVAinstance then there exists a solution with cost x to the corresponding MBA-minstance whenever such an instance is constructed as in Section 3.1. ut