Linear Assignment Problems and Extensions

Question

1. What are the contributions in this paper?

2. How can a shortest path be computed in O(n2) time?

3. What are the two density constraints in a flow in a network?

4. What is the definition of perfect matching in a graph?

Accepted Answer

This paper aims at describing the state of the art on linear assignment problems ( LAPs ).. The authors consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour.. Further, the authors describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e. g. tracking of moving objects in the space.. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed.. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D. -Z.

Accepted Answer

The single-source shortest paths can be computed in O(n2) time in a graph with nonnegative weights, e.g. by applying the Dijkstra algorithm implemented with Fibonacci heaps, see Fredman and Tarjan [80].

Accepted Answer

A flow in network N is a function f :A→ IR with∑x∈V ∪Wf(x, y) = ∑x∈V ∪Wf(y, x) , for all y ∈ V ∪W (2)0 ≤ f(x, y) ≤ c(x, y), for all (x, y) ∈ A (3)Equalities (2) and (3) are called flow conservation constraints and capacity constraints,respectively.

Accepted Answer

(Matchings and perfect matchings in directed bipartite graphs are defined as matchings and perfect matchings of the graphs obtained from the digraphs by removing the orientation of the edges, respectively.)

Accepted Answer

A characteristic feature of the algorithm is that Ẽ grows monotonically in the course of the algorithm, i.e., if an edge becomes dual feasible at a certain iteration it remains feasible in the subsequent iterations until the algorithm stops.

Accepted Answer

The coordination in the asynchronous algorithm is realized by maintaining a master copy of a pair of primal and dual solutions which can be accessed by all processors.

Accepted Answer

A randomized algorithm to decide whether G has a perfect matching can be obtained by generating xij uniformly at random from {1, 2, . . . , 2|E|} and by computing the determinant of the corresponding Tutte matrix A(x), and eventually repeating this process a fixed number of times.

Accepted Answer

A feasible solution of the above problem will be a three dimensional permutation array , i.e., a 0-1 array that satisfies the above assignment constraints.

Accepted Answer

The cost matrix of the k-th subproblem is defined by the first k rows and the first k columns of C. Balinski’s algorithm is used to solve the k-th subproblem, i.e., to produce a tree corresponding to a dual feasible basis with signature (2, . . . , 2, 1).

Accepted Answer

The good behavior of the heuristic relies on the fact that the sparse subgraph contains with high probability a cheap perfect matching, i.e., a good solution of the LSAP instance.

Accepted Answer

As pointed out by Burkard, Hahn, and Zimmermann [41], sum and bottleneck assignment problems are just special cases of a more general problem, the so-called algebraic assignment problem which can also be solved efficiently.

Accepted Answer

This evaluation implies that the probability of finding an optimal assignment after row/column reductions tends to 0 as n tends to infinity.

Accepted Answer

Parallel implementations of primal simplex-based algorithmsMost of the computation load of primal simplex-based algorithms concerns the computation of the new edge to be included in the basic solution.

Accepted Answer

Consider the class G(n, d) of directed bipartite graphs with n vertices in each part, where each vertex has out-degree d, and denote by P (n, d) the probability that a graph chosen uniformly at random from G(n, d) contains a perfect matching.

Accepted Answer

It can be shown that a spanning tree T which defines a dual feasible basis and whose row signature contains only one 1 and the other entries equal to 2 is an optimal basis.

Accepted Answer

If xij and ui, vj are optimal primal and dual solutions of the 1-relaxed problem with costs c′ij = (n + 1)cij , then xij and ui n+1 ,vj n+1 , is an optimal pair of primal and dual solutions for the 1 n -relaxed problem with costs cij .

Accepted Answer

The first heuristic which produced a solution with expected value within a constant factor from the optimum value, was proposed by Avis and Lai [12].

Linear Assignment Problems and Extensions

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Most frequently asked questions

1. What are the contributions in this paper?

2. How can a shortest path be computed in O(n2) time?

3. What are the two density constraints in a flow in a network?

4. What is the definition of perfect matching in a graph?

5. What is the characteristic of the algorithm?

6. How does the coordination in the asynchronous algorithm work?

7. How many times can a randomized algorithm determine whether a graph has a perfect matching?

8. What is the possible solution to the axial 3-dimensional assignment problem?

9. What is the cost matrix of the k-th subproblem?

10. What is the behavior of the heuristic?

11. What is the meaning of sum and bottleneck assignment problems?

12. What is the probability of finding an optimal assignment after row/column reductions?

13. What is the important aspect of the computation load of primal simplex-based algorithms?

14. What is the probability of a perfect matching in a graph?

15. What is the way to show that a spanning tree T is feasible?

16. What is the optimal solution of the 1-relaxed problem with costs cij?

17. What is the first heuristic for the LSAP?

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Citations

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Analyzing the Sensitivity of the Optimal Assignment in Probabilistic Multi-Robot Task Allocation

A new approach to population sizing for memetic algorithms: A case study for the multidimensional assignment problem

References

A note on two problems in connexion with graphs

The Hungarian method for the assignment problem

Reducibility Among Combinatorial Problems.

Reducibility Among Combinatorial Problems

Linear Programming and Extensions

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