Assignment Problems y Eranda C ela Abstract Assignment problems arise in dierent situations where we have to nd an optimal way to assign n ob jects to m other ob jects in an injective fashion Dep ending on the ob jective we want to optimize we obtain dierent problems ranging from linear assignment problems to quadratic and higher dimensional assignment problems The assignment problems are a well studied topic in combinatorial opti mization These problems nd numerous application in pro duction planning telecommunication VLSI design economics etc We intro duce the basic problems classied into three groups linear as signment problems three and higher dimensional assignment problems and quadratic assignment problems and problems related to it For each group of problems we mention some applications show some basic prop erties and de scrib e briey some of the most successful algorithms used to solve these prob lems Intro duction Assignment problems deal with the question how to assign n ob jects to m other ob jects in an injective fashion in the b est p ossible way An assignment problem is completely sp ecied by its two comp onents the assignments which represent the underlying combinatorial structure and the ob jective function to b e optimized which mo dels the b est p ossible way In the classical assignment problem one has m n and most of the problems with m n can b e transformed or are strongly related to analogous problems with m n Therefore we will consider m n through the rest of this chapter unless otherwise sp ecied This research has b een supp orted by the Sp ezialforschun gsb erei ch F Optimierung und Kontrolle Pro jektb ereich Diskrete Optimierung y Technical University Graz Institute of Mathematics Steyrergasse A Graz Austria Email celaoptmathtugrazacat From the mathematical p oint of view an assignment is a bijective mapping of a nite set N f ng into itself ie a permutation assigning some j i to each i N The set of all p ermutations assignments of n items will b e denoted by S and has n elements Every p ermutation of the set N f ng corresp onds n uniquely to a permutation matrix X x with x for j i and x ij ij ij for j i Thus a p ermutation matrix X x can b e dened as a matrix which ij fullls the following conditions socalled assignment constraints n X x for all j n ij i=1 n X x for all i n ij j =1 x f g for all i j n ij By replacing the conditions x f g by x in we get a doubly stochastic ij ij matrix The set of all doubly sto chastic matrices forms the assignment polytope P Due to a famous result of Birkho the assignment p olytop e P is the A A convex hull of all assignments or equivalently every doubly sto chastic matrix can b e written as convex combination of p ermutation matrices The concept of an assignment is strongly related to another well known concept in graph theory and in combinatorial optimization matching in bipartite graphs A bipartite graph G is a triple V W E where the vertex sets V and W have no vertices in common and the edge set E is a set of pairs i j where i V and j W A subset M of E is called a matching if every vertex of G is incident with at most one edge from M The cardinality of M is called cardinality of the matching The maximum matching problem asks for a matching with as many edges as p ossible A matching M is called a perfect matching if every vertex of G is incident with exactly one edge from M Evidently every p erfect matching is a maximum matching A p erfect matching in a bipartite graph G V W E with V fv v v g 1 2 n W fw w w g can b e represented by a p ermutation of f ng such 1 2 n M that i j if and only if v w M Hence a p erfect matching in a bipartite M i j graph is an assignment p Hop croft and Karp gave an O jE j jV jalgorithm which constructs a p erfect p jV jjE j algorithm for matching if it exists Even and Tarjan gave an O the maximum ow problem on unit capacity simple networks algorithm which can also b e applied to nd a matching of maximum cardinality in a bipartite graph p 15 Alt et al gave an O jV j jE j log jV j implementation for the Hop croft Karp algorithm Based on ideas similar to those in Hop croft and Karp a fast randomized MonteCarlo algorithm is given by Mulmuley et al This algorithm nds a p erfect matching at costs of a single matrix inversion The reader is referred to the bibliography in Burkard and C ela for further reference p ointers related to algorithms for cardinality matching problems Linear Assignment Problems The linear assignment problem LAP is one of the oldest and most studies prob lems in combinatorial optimization Many dierent algorithms have b een develop ed to solve this problem Also other asp ects of the problem as the asymptotic b ehav ior or sp ecial cases have b een thoroughly investigated The reader is referred to DellAmico and Martello for a comprehensive annotated bibliography and to Burkard and C ela for a recent review on assignment problems Problem denition and applications Recall the original mo del where n items are to b e assigned to n other ob jects in the b est p ossible way Let c b e the cost incurred by the assignment of ob ject i ij to ob ject j We are lo oking for an assignment which minimizes the overall cast P n c Thus the linear assignment problem LAP is given as follows i(i) i=1 n X c min i(i) S n i=1 where S is the set of p ermutations of f ng Based on the description of n the set of all assignments see Section the LAP can also b e formulated as follows P min c x over all matrices X x which fulll ij ij ij ij Due to Birkho s result we can relax the conditions x f g to x and ij ij obtain the linear programming formulation of the LAP Any basic solution of this linear program corresp onds to a p ermutation matrix n X min c x ij ij i=1 n X x j n ij LP i=1 n X x i n ij j =1 x i j n ij As we will mention in the next section many algorithms for the LAP are based on linear programming techniques and consider often the dual linear program n n X X v u max j i j =1 i=1 u v c i j n i j ij u v IR i j n i j where u and v i j n are dual variables i j Among the numerous applications of the LAP the socalled p ersonnel assignments are the most typical In the p ersonnel assignment we want to assign p eople to ob jects eg jobs machines ro oms to other p eople etc Each assignment has a cost and we want to make the assignment so as to minimize the overall sum of the costs For example one company might want to assign graduates to vacant jobs In this case the cost c is given by c p where p is the prociency index ij ij ij ij for placing candidate i to job j and the goal is to assign each candidate i to some P vacancy i such that the overall cost c is minimized or equivalently the i(i) i P p is maximized overall prociency i(i) i There are many other applications of the linear assignment problem eg in lo cating and tracing ob jects in space scheduling on parallel machines inventory planning vehicle and crew scheduling wiring of typ ewriters etc The reader is referred to Ahuja et al and Burkard and C ela for a detailed description of some applications of the LAP and literature p ointers to other applications Algorithms for the LAP The LAP can b e solved eciently and the design of ecient solution metho ds for this problem has b een an ob ject of research for many years There exists an amazing amount of algorithms sequential and parallel for the LAP ranging from primal dual combinatorial algorithms to simplexlike metho ds The worstcase complexity 3 of the b est sequential algorithms for the LAP is O n where n is the size of the problem From the computational p oint of view very large scale dense assignment 6 problems with ab out no des can b e solved within a couple of minutes by sequential algorithms see Lee and Orlin There is a numb er of survey pap ers and b o oks on algorithms among others Derigs DellAmico and Toth and the b o ok on the rst DIMACS challenge edited by Johnson and McGeo ch Among pap ers rep orting on computational exp erience we mention Carpaneto et al Lee and Orlin DellAmico and Toth and some of the pap ers in Johnson and McGeo ch Most sequential algorithms for the LAP can b e classied into primaldual algorithms and simplexbased algorithms Primaldual algorithms work with a pair consisting of an infeasible solution x i j n of LP called primal solution and a feasible ij solution u v i j n of the dual called dual solution These solutions i j fulll the complementarity slackness conditions x c u v for i j n ij ij i j These solutions are up dated iteratively until the primal solution b ecomes feasible while keeping the complementary slackness conditions fullled and the dual solution feasible At this p oint the primal solution would b e optimal according to duality theory Dierent primaldual algorithms dier on the way they obtain a starting pair of a primal and a dual solution fullling the conditions describ ed ab ove and the way the solutions are up dated A starting dual solution can b e obtained as in the Hungarian metho d by setting u min fc j ng for i n and then i ij v minfc u i ng for j n An infeasible primal starting solution j ij i could b e given by a matching of maximal cardinality in the bipartite graph G V W E where V W f ng and E fi j c c u v g ij ij i j Then set x if i j is an edge of the matching and x otherwise Clearly ij ij the pair of solutions obtained