Some recent results in the analysis of greedy algorithms for assignment problems

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems

Bounds on Integrals with Respect to Multivariate Copulas

Finding upper and lower bounds to integrals with respect to copulas is a quite prominent problem in applied probability. In their 2014 paper, Hofer and Iaco showed how particular two dimensional copulas are related to optimal solutions of the two dimensional assignment problem. Using this, they managed to approximate integrals with respect to two dimensional copulas. In this paper, we will further illuminate this connection, extend it to d-dimensional copulas and therefore generalize the method from Hofer and Iaco to arbitrary dimensions. We also provide convergence statements. As an example, we consider three dimensional dependence measures

Weak Monge arrays in higher dimensions

AbstractAn n × n matrix C is called a weak Monge matrix if cii + crs ⩽is + cri for all 1 ⩽ i ⩽ r, s ⩽ n. It is well known that the classical linear assignment problem is optimally solved by the identity permutation if the underlying cost-matrix fulfills the weak Monge property.In this paper we introduce d-dimensional weak Monge arrays, (d ⩾ 2), and prove that d-dimensional axial assignment problems can be solved efficiently whenever the underlying cost-array fulfills the d-dimensional weak Monge property. Moreover, it is shown that all results also carry over into an abstract algebraic framework. Finally, the problem of testing whether or not a given array can be permuted to become a weak Monge array is investigated

Computational problems without computation

Problemen uit de discrete wiskunde lijken op het eerste gezicht vaak erg simpel. Ze kunnen meestal gemakkelijk en zonder gebruik te maken van wiskundige begrippen worden geformuleerd. Toch komt het vaak voor dat zo’n ogenschijnlijk eenvoudig probleem nog open is of dat er, zoals bij het handelsreizigersprobleem, wel een oplossing gegeven kan worden,maar alleen een die onbruikbaar is omdat de rekentijd bij grotere getallen te snel groeit. In dit artikel, gebaseerd op zijn voordracht op het NMC 2002, kijkt Gerhard Woeginger naar de tegenovergestelde situatie. Hij introduceert allerlei discrete\ud problemen die onoplosbaar lijken, maar waarvoor er een simpele oplossing bestaat

Submodular linear programs on forests

A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed

On Monge sequences in d-dimensional arrays

AbstractLet C be an n × m matrix. Then the sequence j:= ((i1, j1), (i2, j2), …, (inm, jnm)) of pairs of indices is called a Monge sequence with respect to the given matrix C if and only if, whenever (i, j) precedes both (i, s) and (r, j) in j, then c[i, j] + c[r, s] ≤ c[i, s] + c[r, j]. Monge sequences play an important role in greedily solvable transportation problems. Hoffman showed that the greedy algorithm which maximizes all variables along a sequence j in turn solves the classical Hitchcock transportation problem for all supply and demand vectors if and only if j is a Monge sequence with respect to the cost matrix C. In this paper we generalize Hoffman's approach to higher dimensions. We first introduce the concept of a d-dimensional Monge sequence. Then we show that the d-dimensional axial transportation problem is solved to optimality for arbitrary right-hand sides if and only if the sequence j applied in the greedy algorithm is a d-dimensional Monge sequence. Finally we present an algorithm for obtaining a d-dimensional Monge sequence which runs in polynomial time for fixed d