32 research outputs found
An asymptotical study of combinatorial optimization problems by means of statistical mechanics
AbstractThe analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study. Simulated annealing, a metaheuristic for combinatorial optimization problems, is based on this analogy. In this paper we show how a statistical mechanics formalism can be utilized to analyze the asymptotic behavior of combinatorial optimization problems with sum objective function and provide an alternative proof for the following result: Under a certain combinatorial condition and some natural probabilistic assumptions on the coefficients of the problem, the ratio between the optimal solution and an arbitrary feasible solution tends to one almost surely, as the size of the problem tends to infinity, so that the problem of optimization becomes trivial in some sense. Whereas this result can also be proven by purely probabilistic techniques, the above approach allows one to understand why the assumed combinatorial condition is essential for such a type of asymptotic behavior
Linearizable special cases of the QAP
We consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold. We provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP, and of the linearizable instances of the traveling salesman QAP. As a by-product, this yields a new well-solvable special case of the weighted feedback arc set problem
Integrating multiple sources of ordinal information in portfolio optimization
Active portfolio management tries to incorporate any source of meaningful
information into the asset selection process. In this contribution we consider
qualitative views specified as total orders of the expected asset returns and
discuss two different approaches for incorporating this input in a
mean-variance portfolio optimization model. In the robust optimization approach
we first compute a posterior expectation of asset returns for every given total
order by an extension of the Black-Litterman (BL) framework. Then these
expected asset returns are considered as possible input scenarios for robust
optimization variants of the mean-variance portfolio model (max-min robustness,
min regret robustness and soft robustness). In the order aggregation approach
rules from social choice theory (Borda, Footrule, Copeland, Best-of-k and MC4)
are used to aggregate the total order in a single ``consensus total order''.
Then expected asset returns are computed for this ``consensus total order'' by
the extended BL framework mentioned above. Finally, these expectations are used
as an input of the classical mean-variance optimization. Using data from
EUROSTOXX 50 and S&P 100 we empirically compare the success of the two
approaches in the context of portfolio performance analysis and observe that in
general aggregating orders by social choice methods outperforms robust
optimization based methods for both data sets
The Wiener maximum quadratic assignment problem
We investigate a special case of the maximum quadratic assignment problem
where one matrix is a product matrix and the other matrix is the distance
matrix of a one-dimensional point set. We show that this special case, which we
call the Wiener maximum quadratic assignment problem, is NP-hard in the
ordinary sense and solvable in pseudo-polynomial time. Our approach also yields
a polynomial time solution for the following problem from chemical graph
theory: Find a tree that maximizes the Wiener index among all trees with a
prescribed degree sequence. This settles an open problem from the literature.Comment: 11 pages, no figure
The multi-stripe travelling salesman problem
In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q ≥ 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP
Heuristics for Biquadratic Assignment Problems and their Computational Comparison
The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). As for any hard optimization problem also for BiQAP, a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which, possibly, yield a good trade off between the solution quality and the time and memory requirements. In this paper we describe several heuristics for BiQAPs, in particular pair exchange algorithms (improvement methods) and variants of simulated annealing and taboo search. We implement these heuristics as C codes and analize their performance
An Asymptotical Study of Combinatorial Optimization Problems By Means of Statistical Mechanics
. The analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study. Simulated annealing, a metaheuristic for combinatorial optimization problems, is based on this analogy. In this paper we use the statistical mechanics formalism based on the above mentioned analogy to analyze the asymptotic behavior of a special class of combinatorial optimization problems characterized by a combinatorial conditions which is well known in the literature. Our result is analogous to results of other authors derived by purely probabilistic means: Under natural probabilistic conditions on the coefficients of the problem, the ratio between the optimal value and size of a feasible solution approaches almost surely the expected value of the coefficients, as the size of the problem tends to infinity. Our proof shows clearly why the above mentioned combinatorial condition which characterizes the class of investigated problems is essential. Keywords: combinato..