A Decomposition of the Max-min Fair Curriculum-based Course Timetabling Problem
aa r X i v : . [ c s . A I] A ug A Decomposition of the Max-min FairCurriculum-based Course Timetabling Problem ∗ The Impact of Solving Subproblems to Optimality
Moritz M ¨uhlenthaler Rolf WankaDepartment of Computer ScienceUniversity of Erlangen-Nuremberg, Germany { moritz.muehlenthaler,rolf.wanka } @cs.fau.de Abstract
We propose a decomposition of the max-min fair curriculum-based course timetabling(MMF-CB-CTT) problem. The decomposition models the room assignment subproblem asa generalized lexicographic bottleneck optimization problem (LBOP). We show that the gen-eralized LBOP can be solved efficiently if the corresponding sum optimization problem can besolved efficiently. As a consequence, the room assignment subproblem of the MMF-CB-CTTproblem can be solved efficiently. We use this insight to improve a previously proposed heuris-tic algorithm for the MMF-CB-CTT problem. Our experimental results indicate that using thenew decomposition improves the performance of the algorithm on most of the 21 ITC2007 testinstances with respect to the quality of the best solution found. Furthermore, we introduce ameasure of the quality of a solution to a max-min fair optimization problem. This measurehelps to overcome some limitations imposed by the qualitative nature of max-min fairness andaids the statistical evaluation of the performance of randomized algorithms for such problems.We use this measure to show that using the new decomposition the algorithm outperforms theoriginal one on most instances with respect to the average solution quality.
We consider a decomposition approach to a variant of the curriculum-based course timetabling(CB-CTT) problem. The CB-CTT problem has been proposed in the context of the timetablingcompetition ITC2007 [21] and has since then received a great deal of attention in the timetablingcommunity. The CB-CTT problem models the task of assigning timeslots and rooms to courses in ∗ Research funded in parts by the School of Engineering of the University of Erlangen-Nuremberg. fair timetables and thus improve the overallstakeholder satisfaction. The underlying fairness concept is (lexicographic) max-min fairness.It is a common technique to decompose the CB-CTT problem into subproblems which areeasier to handle individually [17, 19]. The usual approach is to perform room and timeslot assign-ments separately, but other approaches have been explored as well [7]. The CB-CTT problem canbe decomposed into a bounded list coloring problem that models the timeslot assignment and, foreach timeslot, a linear sum assignment problem (LSAP) for assigning the courses in this timeslot torooms [8, 17]. Unfortunately, there are dependencies between LSAPs for different timeslots, so anoptimal room assignment can only be obtained for a single timeslot, while the rest of the timetableremains fixed. We show that for an analogous decomposition of the MMF-CB-CTT problem, theroom assignment subproblem for a single timeslot can also be solved efficiently by modeling it asa generalized lexicographic bottleneck optimization problem (GLBOP), which is a generalizationof the lexicographic bottleneck optimization problem (LBOP) from [4]. We show that the GLBOPcan be solved efficiently if the corresponding sum optimization problem can be solved efficiently.Furthermore, we propose a new measure for the quality of a solution to an optimization problemwith a max-min fairness (lexicographic bottleneck) objective such as the MMF-CB-CTT problem.This measure helps to overcome some limitations imposed by the qualitative nature of max-minfairness. We use this measure to determine the average solution quality of a randomized algorithmfor the MMF-CB-CTT problem.We evaluate the quality of the timetables produced by the algorithm M AX M IN F AIR
SA from [22],with and without the new decomposition. In the original algorithm the room assignment subprob-lem was modeled as an LSAP. However, an optimal solution to the LSAP is not necessarilyoptimal for corresponding the room assignment subproblem of the MMF-CB-CTT problem. Ourexperiments indicate that making use of the new decomposition improves the best produced by thealgorithm on 18 out of 21 CB-CTT instances from the ITC2007 competition. We use the afore-mentioned measure to show that the new decomposition yields in an improved average solutionquality for 16 out of 21 instances. According to the Wilcoxon rank-sum test (one-sided, signifi-cance level 0 .
01) M AX M IN F AIR
SA using the new decomposition is significantly better than theoriginal approach on 12 instances.The remainder of this work is organized as follows: In Section 2, we provide relevant back-ground on the CB-CTT and MMF-CB-CTT problem formulations, as well as max-min fairnessand the assignment problem. In Section 3, we introduce the GLBOP and the decomposition of2he MMF-CB-CTT problem. We propose the measure of the max-min fairness of an allocation inSection 4. Section 5 presents our evaluation of the performance impact of the decomposition forMMF-CB-CTT problems. Section 6 concludes the paper.
In this section we will provide some relevant background on fair resource allocation, universitycourse timetabling and the assignment problem.
Fairness comes into play when scarce resources are distributed over a set of stakeholders with de-mands. The topic of fairness and how to measure it has received great attention for example ineconomics, where the distribution of wealth and income is of interest [11]. In computer science,fairness aspects have been studied for example in the design of network communication protocols,in particular in the context of bandwidth allocation and traffic shaping [18, 13]. Fairness aspectshave been addressed explicitly for example for various kinds of scheduling problems, includingpersonnel scheduling [29], sports scheduling [27], course scheduling [22] and aircraft schedul-ing [30]. In the context of resource allocation allocation in general, fairness has been studiedin [3, 24]. Approximation algorithms for fair optimization problems have been studied in [15, 16].In the next sections of this work we will deal with a fair variant of a university course timetablingproblem from [22] that builds on the notion of max-min fairness . Consider the problem of allocat-ing resources to n stakeholders. A resource allocation induces an allocation vector x = ( x , . . . , x n ) ,where x i , 1 ≤ i ≤ n , corresponds to the amount of resources allocated to stakeholder i . Later on, wewill deal with minimization problems almost exclusively, so unless stated otherwise, we assumethat we allocate to the stakeholders a cost, penalty, or some similar quantity that is to be mini-mized. In this contexct, an allocation is max-min fair , if the cost that the worst-off stakeholder hasto cover is minimal, and under this condition, the second worst-off stakeholder covers the minimalcost, and so forth. This concept can be formalized as follows: Let ~ x denote the sequence containingthe entries of the allocation x sorted in non-increasing order. For allocation two vectors x and y , x isat least as fair as y , denoted by x (cid:22) y , if ~ x (cid:22) lex ~ y , where (cid:22) lex is the usual lexicographic comparison.Now, an allocation x is max-min fair, if x (cid:22) y for all feasible allocations y .Max-min fairness enforces an efficient resource usage to some extent, since an improved re-source utilization is accepted to the benefit of a stakeholder as long as it is not at the expense ofanother stakeholder who is worse-off. Hence, a max-min fair allocation is Pareto-optimal. Onelimitation of max-min fairness is that the concept is purely qualitative, i. e., given two allocations x and y , max-min fairness just determines which of the two is fairer, but not by how much. Inorder to aid the statistical evaluation of the performance of algorithms for max-min fair resourceallocation problems, in Section 4 we will introduce a metric for the difference in quality betweentwo allocation vectors which is compatible with the (cid:22) -relation.3 .2 Curriculum-based Course Timetabling The academic course timetabling problem captures the task of assigning a set of courses to roomsand timeslots in the setting of a university. In Section 3 we will focus on decompositions oftwo particular variants of the academic course timetabling problem: the curriculum-based coursetimetabling (CB-CTT) problem from track three of the second international timetabling compe-tition [10], and its max-min fair version, MMF-CB-CTT, proposed in [22]. The CB-CTT for-mulation has attracted a great deal of interest in the research community and the competition in-stances are popular benchmarking instances for comparing different algorithms [19, 17, 28]. TheMMF-CB-CTT problem differs from the basic CB-CTT formulation only with respect to the ob-jective function. We will now introduce some terminology and state definitions relevant to the latersections of this work.A CB-CTT instance consists of a set of courses, a set of curricula, a set of rooms, a set ofteachers and a set of days. Each day is divided into a fixed number of timeslots; a day togetherwith a timeslot is referred to as a period . A period together with a room is called a resource . Eachcourse consists of a set of events that need to be scheduled. A course is taught by a teacher and hasa fixed number of students attending it. A course can only be taught in certain available periods.Each curriculum is a set of courses, no two of which may be taught in the same period. Each roomhas a capacity , a maximum number of students it can accommodate. A solution to a CB-CTTinstance is a timetable , i. e., an assignment of the courses to the resources subject to a number ofhard and soft constraints. A timetable that satisfies all hard constraints is feasible .Later on, we will deal exclusively with feasible timetables, so we will not cover the evaluationof hard constraints at all (please refer to [10] a detailed description). However, some understandingof the soft constraint evaluation will be helpful later on, so we will touch on this very briefly. TheCB-CTT problem formulation features the following soft constraints:S1
RoomCapacity : Each lecture should be assigned to a room of sufficient size.S2
MinWorkingDays : The individual lectures of each course should be distributed over a certainminimum number of days.S3
IsolatedLectures : For each curriculum, all courses in the curriculum should be scheduled inadjacent periods.S4
RoomStability : The lectures of each course should be held in the same room.The violation of a soft constraint results in a “penalty” for the timetable. The total penalty of atimetable t is aggregated by the objective function c which just sums the penalties for the individualsoft constraint violations: c ( t ) = (cid:229) ≤ i ≤ c S i ( t ) , (1)where c S1 , . . . , c S4 are the penalties determined by the soft constraints (S1)–(S4). The relativeimportance of the different soft constraints is set by a weight factor for each soft constraint. Sincethe weights will be of no relevance to our arguments later on, we assume that appropriate weightinghas been applied within c S1 , . . . , c S4 . A detailed specification of the penalty functions can be foundin [10]. 4 efinition 1 (CB-CTT Problem) . Given a CB-CTT instance I, find a feasible timetable t such thatc ( t ) is minimal. A max-min fair variant of the CB-CTT problem was defined in [22]. Given a CB-CTT instancewith curricula u , . . . , u k . The allocation vector of a timetable t is given by: A ( t ) = ( c ( u , t ) , c ( u , t ) , . . . , c ( u k , t )) , (2)where c ( u j , t ) = (cid:229) ≤ i ≤ c S i ( u j , t ) , i ∈ { , . . . , } , is the CB-CTT objective function restricted tothe events of the courses in curriculum u j , j ∈ { , . . . , k } . Definition 2 (MMF-CB-CTT Problem) . Given a CB-CTT instance I, find a feasible timetable t such that A ( t ) is max-min fair. The assignment problem is a classical problem in combinatorial optimization which appears inmany applications, for example personnel scheduling, job scheduling and object tracking, just toname a few. For a comprehensive overview of the body of research on the assignment problem andthe applications see [5, 26]. In CB-CTT problem for example, the assignment problem appears as asubproblem [17, 20]. There exist polynomial-time algorithms for many variants of the assignmentproblem.Let A = B = { , . . . , n } , for some n ∈ N . An assignment of the elements of A to the elementsof B is a bijection s : A → B . Typically, assignment problems are optimization problems, i. e.,among all bijections from A to B , we are looking for one that is optimal with respect to a certainobjective function. In the context of (fair) curriculum-based course timetabling, we are in particularinterested in two variants of the assignment problem, namely the linear sum assignment problem(LSAP) and the lexicographic bottleneck assignment problem (LBAP). Definition 3 (LSAP) . Given a cost function c : A × B → N , find a bijection s : A → B such that (cid:229) ni = c ( i , s ( i )) is minimal. There exist various algorithms for solving LSAPs efficiently, including the well-known Hun-garian algorithm [25, p. 248ff] and network flow algorithms [12]. In the following, let T LSAP ( n ) bethe time complexity of solving an LSAP instance with | A | = | B | = n .When solving MMF-CB-CTT problems using the decomposition proposed in the next section,the task of finding max-min fair assignments occurs as a subproblem. An assignment s : A → B is called max-min fair, if for any assignment s ′ : A → B we have ~ c ( s ) (cid:22) ~ c ( s ′ ) , where ~ c ( s ) =( c ( i , s ( i ))) i = ,..., n and (cid:22) is the max-min fair comparison from Section 2.1. Definition 4 (LBAP) . Given a cost function c : A × B → R , find a max-min fair bijection s : A → B. A LBAP can be transformed into a LSAP by scaling the cost values appropriately. This re-sults in an exponential blow-up of the cost values, which may be undesirable in practical appli-cations [4]. Alternatively, an LBAP can be reduced to a lexicographic vector assignment, which5elongs to the class of algebraic assignment problems [6]. Using this reduction, a given LBAP witha cost function c can be solved in time O ( kn ) , where k is the number of distinct values attained by c [9]. The reduction is straightforward: For each j ∈ { , . . . , k } , let e j ∈ N k be the vector whose j -thcomponent is 1 and all other components are 0. The cost function c is replaced by a vector-valuedfunction c ′ : A × B → N k such that c ′ ( a , s ( a )) = e j if c ( a , s ( a )) is the j -th largest value attainedby c . An assignment s that yields a lexicographically minimal cost vector (cid:229) ni = c ′ ( i , s ( i )) is anoptimal solution to the corresponding LBAP. It is a common approach to decompose the CB-CTT problem in a way that room and timeslotassignment is preformed separately, see for example [17, 20]. For courses in a single timeslot, anoptimal room assignment can be determined by solving a LSAP instance. In this section, we estab-lish a similar result for the MMF-CB-CTT problem: An optimal room assignment for the coursesin a single timeslot can be determined by solving a (generalized) LBOP instance. In the spiritof Benders decomposition approach [2], we further consider decompositions of sum optimizationproblems into master and subproblems such that the subproblems can be solved efficiently. We de-rive sufficient conditions under which such a decomposition of a sum optimization problem carriesover to its generalized lexicographic bottleneck counterpart.In the following, we consider a generalization of the LBOP in [4]. Let E = { e , . . . , e m } be theground set and let S = { S , S , . . . } be the set of feasible solutions. We assume that each solution S ∈ S contains exactly n items from the ground set, that is S ⊆ E and | S | = n . Furthermore, let N ⊂ N be a finite set of the natural numbers. The weight function w : S → M ( N ) assigns to afeasible solution a weight, that is a finite multiset of the numbers in N . The weight of a solution S ∈ S is the disjoint union of the individual weights, W ( S ) = [ · e ∈ S w ( e ) . The weights of two solutions S , S ∈ S can be compared by arranging the items in W ( S ) and W ( S ) in non-increasing order and performing a lexicographic comparison of the sorted sequences.We can essentially the comparison (cid:22) from Section 2.1, but the sorted sequences do not necessarilyhave the same length. Let Seq ( N ) be the set of finite sequences on the alphabet N that are arrangedin non-increasing order. For two sequences s , s ′ ∈ Seq ( N ) , s (cid:22) lex s ′ iff one of the following is true:i) s = s ′ , ii) s is a prefix of s ′ , iii) there is a decomposition s = zuv , s ′ = zuw such that z is a maximalprefix of s and s ′ , u , v ∈ N and u < v . Due to the conceptual similarity, we will use the symbol (cid:22) for the comparison with respect to max-min fairness in the new setting as well. The GLBOP is thefollowing problem: min (cid:22) W ( S ) s.t. S ∈ S . (GLBOP)Note that the minimum weight is determined according to the comparison (cid:22) . If | w ( e ) | = e ∈ E then we have a LBOP as defined in [4]. 6onsider a weight function of the form W ′ : S → N . Then the min-sum optimization problem(SOP) is the following problem: min ≤ (cid:229) e ∈ S W ′ ( e ) s.t. S ∈ S . (SOP)Let T n , m be the time required for solving (SOP). Theorem 1.
A GLBOP instance can be solved in time O ( | N | · T n , m ) .Proof. Following the vectorial approach of Della Croce et al. in [9], reduce the GLBOP to a lexi-cographic vector optimization problem (LVOP). Let t = | N | and for 1 , ≤ i ≤ t , let t i denote the i -thlargest item in N . Let the function f : M ( N ) → N t assign to a multiset v of the items N a vector ( v , . . . , v t ) ∈ N t such that v j = mult v ( t j ) , j = , . . . , t , where mult v : N → N , and for each a ∈ N , mult v ( a ) is the multiplicity of a in v . Now, we have tosolve the following problem: min (cid:22) lex (cid:229) e ∈ S f ( W ( e )) s.t. S ∈ S . (LVOP)We show that a solution S ∈ S is an optimal solution to (LVOP) if and only if it is an optimalsolution to (GLBOP). Suppose for a contradiction that S ∈ S is an optimal solution to (LVOP)that is not optimal to (GLBOP). Then there is a solution S ′ ∈ S such that W ( S ′ ) ≺ W ( S ) . As aconsequence, f ( W ( S ′ )) ≺ lex f ( W ( S )) by the construction of the cost vectors above. The “only if”part can be shown analogously. This is a contradiction to the optimality of S ′ . The problem (LVOP)can be solved in time O ( t · T n , m ) because each elementary operation involving the weights is nowperformed on a vector of length t .As noted by Della Croce et al. in [9], the vectorial approach is essentially cost scaling. Theconstruction of the cost vectors above enables us to naturally handle multisets as cost values. Wewill see shortly that in the context of the MMF-CB-CTT problem, if an item of the ground set hasa weight of cardinality k , then choosing this item to be part of the solution concerns k differentstakeholders.Consider the following decomposition of the MMF-CB-CTT problem: We isolate, for a singleperiod, the assignment of courses to rooms, from the rest of the problem. So, the task is to findan optimal room assignment for a given period, assuming that the rest of the timetable is fixed.Optimizing the rest of the timetable can be considered the master problem corresponding to theroom assignment subproblem for a particular period. The room assignment subproblem of theCB-CTT problem is a LSAP which can be solved efficiently (see Section 2.3). Our goal is to showthat the room assignment subproblem of the MMF-CB-CTT problem can also be solved efficiently.In the following, let I be a MMF-CB-CTT instance, where C is the set of courses, R is the set ofrooms, P is the set of periods, and U ⊆ P ( C ) are the curricula. By C p we denote the set of coursesscheduled in the period p ∈ P . Please note that C p is determined by the solution of the masterproblem. Furthermore, let U e = { u ∈ U | e ∈ u } .7 heorem 2. The room assignment subproblem of the MMF-CB-CTT problem is a GLBOP.Proof.
We construct a GLBOP that models the room assignment subproblem for a fixed period p .We can assume that | C p | = | R | since if not, we can add suitables dummy nodes to either C p or R .Let G = ( C p ∪ R , E ) be a complete bipartite graph, where E = {{ e , r } | e ∈ C , r ∈ R } . The groundset of the GLBOP is E and the feasible solutions S are all perfect matchings in G . Assigning acourse e ∈ C p to a room r ∈ R completes the timetable t from the perspective of all curricula in U e and therefore determines their cost entries c ( u , t ) for each u ∈ U e . We denote the cost entry c ( u , t ) ,which can be determined after the assignment of e to r , by c e → r ( u ) . Thus, the weight w ( e , r ) of anitem { e , r } ∈ E is the multiset w ( e , r ) = [ · u ∈ U e { c e → r ( u ) } . The costs entries of the curricula that do not contain any course in C p are not altered and haveno influence on the optimality of a particular room assignment. Therefore, the room assignmentsubproblem of the MMF-CB-CTT problem can be written as:min (cid:22) [ · { e , r }∈ S w ( e , r ) s. t. S ∈ S , which is a GLBOP.Note that according to the MMF-CB-CTT problem formulation, a course can be assigned toany of the rooms. If desired, room availabilities can be added to the model in a straightforwardmanner: If a room r is unavailable for a particular course e , then the edge { e , r } in the ground setis assigned a suitably large weight.Figure 1 shows an example of a simple room assignment subproblem of the MMF-CB-CTTproblem modelled as a GLBOP. There is only a single period p , two courses, C = C p = { c , c } ,and two rooms R = { r , r } . However, the course e is in two curricula and thus determines thecost entries of two stakeholders in the overall allocation vector. The cost on each edge connectedto e shows the costs generated for each of the two curricula when assigning e to r or r . Fig-ures 1b and 1c are LBAP instances that reflect only costs for one of the two curricula of e . Theassignments highlighted are both optimal solutions to the individual LBAPs, but none of them isoptimal for the room assignment shown in Figure 1a. Corollary 1.
For a given period p, the room assignment subproblem can be solved in time O ( | U | · T LSAP ( n )) , where n = max {| C p | , | R |} .Proof. Problem (3) is an assignment problem, just as the room assignment subproblem of theCB-CTT problem. Only the objective function is different. Hence, combining Theorems 1 and 2yields the result.
Remark 1.
As noted by Lach and L¨ubbecke in [17], the room stability constraint (S4) introducesdependencies between room assignments in different timeslots, which prevents us from extendingthe decomposition to more than a single period. r e e (a) Room assignment subproblem:optimal cost 7, 5, 4 r r e e (b) LBAP 1: optimal cost 7,5 r r e e (c) LBAP 2: optimal cost 6,5 Figure 1: A room assignment problem example with two courses, e and e , and two rooms r and r . The dashed edges are optimal assignments. The shown optimal solutions of the two LBOPs (b)and (c) are not optimal for the GLBOP (a). Remark 2.
In the general case, whenever there is a decomposition of (SOP) into a master problemand a subproblem, such that the subproblem can be solved efficiently, then the subproblem of thecorresponding max-min fair optimization problem can be solved efficiently if it is a GLBOP. Thisobservation may be useful when turning a problem of the form (SOP) into a LBOP.
In Section 5 we will provide experimental evidence that solving the room assignment sub-problem to optimality is useful for improving the performance of a heuristic algorithm for theMMF-CB-CTT problem.
When dealing with randomized optimization algorithms, one can employ a wealth of statisti-cal tools to extract meaningful information about algorithms’ absolute and relative performance.These tools include statistical tests such as the Wilcoxon rank-sum test and measures such as themean quality of the solutions, the standard deviation, the median quality, the quality of the best andworst solutions, and so on. Due to the qualitative nature of max-min fairness, so far only statisticaltools based on ranking can be used for evaluating randomized max-min fair optimization algo-rithms. In this section we propose a novel approach to partially overcome this limitation. Similarto the problem (GLBOP) from the previous section, we consider combinatorial problems such thatthe cost of a feasible solution is a finite multiset. The main idea is to construct an isomorphismfrom the cost multisets ordered by ≺ , to an interval of the natural numbers ordered by the usual < relation. Using this isomorphism, we can perform all operations on natural numbers, and retrievethe corresponding cost multiset. This means that if we have a set of allocation vectors, we can de-termine for example an allocation vector close to the average allocation. Thus, in our experimentsin the next section we will be able to compare the average solution quality of two algorithms forthe MMF-CB-CTT problems. 9et k ∈ N and N = { , . . . , k } . Further, let Seq n ( N ) denote the non-increasing sequences oflength n over the alphabet ( N , < ) . Lemma 1.
Let rank : Seq n ( N ) → N be a mapping such that for any s ∈ Seq n ( N ) , s = x , . . . , x n , rank ( s ) = n (cid:229) i = (cid:18) n + x i − ix i − (cid:19) . (3) Then rank is an isomorphism ( Seq n ( N ) , ≺ lex ) → ( N , < ) .Proof. ( Seq n ( N ) , ≺ lex ) is a linearly ordered set with least element ( , . . . , ) . Since Seq n ( N ) islinearly ordered by ≺ lex , there is a unique number r s for each s ∈ Seq n ( N ) , which is the cardinalityof the set { s ′ ∈ Seq n ( N ) | s ′ ≺ lex s } . Thus, the function mapping each s ∈ Seq n ( N ) to r s is a bijectivemapping and it is order-preserving as required. It remains to be shown that rank ( s ) computes r s for all s ∈ Seq n ( N ) .Let s = ( x , . . . , x n ) ∈ Seq n ( N ) . The value r s can be determined by the following recursion: r x ,..., x n = r x ,..., x n + r x , ,..., . (4)This recursion separately counts the non-increasing sequences { s ′ ∈ Seq n ( N ) | ( x , , . . . , ) (cid:22) lex s ′ ≺ lex s } and { s ′ ∈ Seq n ( N ) | s ′ ≺ lex ( x , , . . . , ) } . The number of sorted sequences of length n over an ordered alphabet of size k is (cid:0) n + kk (cid:1) . Thus, rx , , . . . , | {z } length n = (cid:18) n + x − x − (cid:19) . In particular, for s ∈ Seq ( N ) we have r x = (cid:0) xx − (cid:1) = x . Unfolding the recursion (4) yields (3).Therefore, rank ( s ) computes r s for each s ∈ Seq n ( N ) .The argument above can be extended to non-increasing sequences of length at most n by choos-ing as alphabet N ∪ {− ¥ } and constructing from each sequence of length less than n a sequence oflength n by padding it with − ¥ . Please note that the alphabet can be any finite totally ordered set ( A , < A ) , since it is isomorphic to { , . . . , k } for some k ∈ N .Let M n ( N ) denote the finite multisets of cardinality n over a finite alphabet ( N , < ) . Similarto the problem (GLBOP), consider an instance of some combinatorial minimization problem withfeasible solutions S and a cost function of the form w : S → M n ( N ) . Let ~ x denote the non-increasing sequence of length n containing the items of x ∈ M n ( N ) . Theorem 3.
The mapping r : ( M n ( N ) , ≺ ) → ( N , < ) r ( x ) rank ( ~ x ) , is an isomorphism. roof. Each multiset x ∈ M n ( N ) can be represented in a unique way as a non-increasing sequenceof length n . Thus, we can apply the isomorphism rank from Lemma 1 to ~ x .We establish a similar result for maximization problems. In this context, the fairness of two al-locations x , y ∈ M n ( N ) can be compared as follows: x is fairer than y , denoted by x ≺ max y , if ~ y ≺ lex ~ x , where ~ x denotes the non-decreasing sequence of the items in x . Let m = max { N } , . . . , max { N } ∈ Seq n ( N ) , and let a − b denote the element-wise substraction of two sequences a , b ∈ Seq n ( N ) . Fur-thermore, let ~ x denote the non-decreasing sequence of length n containing the items of x ∈ M n ( N ) . Theorem 4.
The mapping r : ( M n ( N ) , ≺ max ) → ( N , < ) r ( x ) rank ( m − ~ x ) , is an isomorphism. In this section we are going to present experimental evidence for the usefulness decompositionpresented in Section 3. We compare the performance of two randomized heuristic algorithmsfor the MMF-CB-CTT problem, both of which are based on the algorithm M AX M IN F AIR
SAfrom [22]. The first algorithm is the one that performed best in [22]. It uses a decomposition ofthe MMF-CB-CTT problem that is similar to the one presented in this work, but models the roomassignment subproblem as LSAP. Thus, we will refer to this algorithm by MMF SA LSAP. Thesecond algorithm, MMF SA GLBOP, uses the MMF-CB-CTT decomposition from Section 3 andthus solves (GLBOP) to obtain an optimal room assignment for a given period. Apart from howthe room assignment subproblem is solved, the the two algorithms are identical. We compare bothalgorithms with respect to the best solutions they produce as well as the average solution qualityper instance. In order to determine average results, we use the isomorphism r from Theorem 3 asdescribed in the previous section. Our results indicate that the algorithm which solves the GLBOProom assignment subproblem significantly outperforms the other one.The algorithm M AX M IN F AIR
SA is a variant of simulated annealing (SA) [14] which hasbeen tailored to the MMF-CB-CTT problem. Simulated annealing iteratively generates new can-didate solutions and keeps (or accepts ) a new solution if it is better. If the new solution is worse,then it is accepted with a certain probability which depends on the temperature J . There arethree crucial design choices when adapting simulated annealing to a particular problem: The cool-ing schedule, the acceptance criterion, and the neighborhood structure. In both algorithms underconsideration, MMF SA GLBOP and MMF SA LSAP, we use the standard geometric coolingschedule, which lets the temperature decay exponentially from a given J max to a given J min . Bothalgorithms use the acceptance criterion based on the component-wise energy difference, whichperformed best in a comparison of different acceptance criteria in [22]. Also, both algorithms use11 neighborhood structure based on the well-known Kempe-move. A Kempe-move swaps a subsetof the events assigned to two given periods such that the conflict constraints between the eventsare not violated. The difference between both algorithms is how the room-assignment subprob-lems are solved after performing a Kempe-move: MMF SA LSAP solves two LSAPs in order toassign rooms efficiently, while MMF SA GLBOP solves two GLBOPs. Thus, the second roomassignment performed by MMF SA GLBOP is optimal assuming that the rest of the timetable isfixed. Our evaluation shows that this is beneficial for the overall algorithm performance.In order to compare the two algorithms, we performed 50 independent runs for each algorithmon the 21 CB-CTT instances from track three of the timetabling competition ITC2007 [10]. Ineach run, 10 iterations of the simulated annealing procedure were performed. We did not tweakthe temperature-related parameters of the algorithms extensively, but determined experimentallythat J max = J min = .
01, as suggested in [22] work well. Table 1 shows the best allocationvectors obtained by MMF SA LSAP and MMF SA GLBOP on the 21 CB-CTT instances. Toaid the presentation of the results, Table 1 shows the allocation vectors in a compressed form:The penalty values are sorted in non-increasing orders the multiplicities of the values are shownas exponents. For example, a table entry of 5 comp04 and comp18 . In addition to the best allocation vectors, Table 1 shows the averageallocations over the 50 runs for each instance. The average allocations have been computed using r from Section 4: The allocation vectors were mapped to the natural numbers, then the averagewas calculated and rounded to the nearest integer. Finally, the result was mapped back to thecorresponding equivalence class of allocation vectors. A comparison of the average allocationsshows that in this respect, MMF SA GLBOP outperforms MMF SA LSAP on 16 instanceswhile it is beaten on the instances comp05 , comp08 , comp15 and comp21 .We also performed the one-sided Wilcoxon rank-sum test with a significance level of 0.01. Ac-cording to the test, MMF SA GLBOP yields significantly better results than MMF SA LSAP oninstances comp02 , comp06 , comp07 , comp08 , comp10 , comp13 , comp14 , comp17 , comp18 , comp19 , comp20 and comp21 . In contrast, MMF SA LSAP is not significantly better than MMF SA GLBOPon any of the 21 instances with this significance level. The results of the Wilcoxon test are consis-tent with the data in Table 1.In contrast to experimental setup in [22], we did not use a timeout, but set a fixed numberof iterations for the direct comparison of MMF SA GLBOP and MMF SA LSAP. The reasonfor this decision is that in practice, solving (GLBOP) takes significantly more time than solvinga LSAP. The increase in runtime is due to overhead required by construction of the cost vectors( O ( | U | ) ) and the O ( | U | ) factor for solving the linear vector assignment problem (see Corollary 1).Since we are mainly interested in the implications of modelling the room assignment subproblemas a GLBOP instead of an LSAP, both algorithms should be able to solve a similar number ofroom-assignment subproblems. From the data shown in Table 1 we can conclude, that using the de-composition presented in Section 3 is clearly the smarter choice, since, after solving equally many12able 1: Comparison of the best and average objective values of the solutions found by MMF SA LSAP and MMF SA GLBOPon the 21 CB-CTT instances from [21] for 50 independent runs per instance and per algorithm. For each instance, the best resultsand best average results are marked in bold face. MMF SA GLBOP MMF SA LSAPInstance best average best average comp01.ectt , , , , , , , , , , , , , , comp02.ectt , , , , , , , , , , , , , , , comp03.ectt , , , , , , , , , , , , , , , , , , , comp04.ectt , , , , , , , , , , , , , , , , , comp05.ectt , , , , , , . . . , , , , , , . . . , , , , , , . . . , , , , , , . . . comp06.ectt , , , , , , , , , , , , , , , , , , , , , comp07.ectt , , , , , , , , , , , , , , , , , comp08.ectt , , , , , , , , , , , , , , , , , , comp09.ectt , , , , , , , , , , , , , , , , , , , , comp10.ectt , , , , , , , , , , , comp11.ectt comp12.ectt , , , , , , . . . , , , , , , . . . , , , , , , . . . , , , , , , . . . comp13.ectt , , , , , , , , , , , , , , , , , , comp14.ectt , , , , , , , , , , , , , , , , , , comp15.ectt , , , , , , , , , , , , , , , , , , , , , comp16.ectt , , , , , , , , , , , , , , , , , comp17.ectt , , , , , , , , , , , , , , , , , , , , , , comp18.ectt , , , , , , , , , , , , , , , , , , , , , , , comp19.ectt , , , , , , , , , , , , , , , , , , , , comp20.ectt , , , , , , , , , , , , , , , , , comp21.ectt , , , , , , , , , , , , , , , , , , , , , , , , ubproblems, it produces superior results compared to the approach from [22]. However, if weemploy the timeout as it was required for the ITC2007 competition (see [10]), MMF SA LSAPis the better choice, because it can perform significantly more iterations within the given timeout. In this work we proposed a decomposition of the MMF-CB-CTT problem from [22]. The de-composition models the room assignment subproblem as a assignment problem with a generalizedlexicographic bottleneck objective. Using this decomposition, the room assignment subproblemcan be solved in polynomial time. We use this result to improve the performance of the M AX M IN -F AIR
SA algorithm proposed in [22], which originally modelled the room assignment as an LSAP.In our experiments we compare the performance of two variants of the M AX M IN F AIR
SA algo-rithm, wich differ with respect to how the room assignment subproblem is performed. Our re-sults indicate that using the decomposition proposed in Section 3 improves the performance ofthe M AX M IN F AIR
SA algorithm on most of the ITC2007 benchmark instances. Furthermore, weproposed a measure for quantifying how fair a timetable is with respect to max-min fairness. Usingthis measure helps to apply statistical methods in the analysis of the performance of randomizedoptimization algorithms for optimization problems with a bottleneck objective. In particular, itenables us to compare the average solution quality of the two variants of the M AX M IN F AIR
SAalgorithm. The results indicate that using the new decomposition, M AX M IN F AIR
SA producesbetter timetables on average for 16 out of 21 instances.
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