On the approximability of robust spanning tree problems

arXiv:1004.2891v1 [cs.CC] 16 Apr 2010 On the approximability of robust spanning tree problems Adam Kasperski Institute of Industrial Engineering and Management, Wroc law University of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroc law, Poland, adam.kasperski@pwr.wroc.pl Pawe l Zieli´nski Institute of Mathematics and Computer Science Wroc law University of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroc law, Poland, pawel.zielinski@pwr.wroc.pl Abstract In this paper the minimum spanning tree problem with uncertain edge costs is dis- cussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within O(log1−ǫ n) for any ǫ > 0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(log n) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approxima- tion algorithms with performance ratio of O(log2 n) for min-max and 2-stage min-max problems are also proposed. Keywords: Combinatorial optimization; Approximation; Robust optimization; Two-stage optimization; Computational complexity 1 Introduction The usual assumption in combinatorial optimization is that all input parameters are precisely known. However, in real life this is rarely the case. There are two popular optimization settings of problems for hedging against uncertainty of parameters: stochastic optimization setting and robust optimization setting. In the stochastic optimization, the uncertainty is modeled by specifying probability dis- tributions of the parameters and the goal is to optimize the expected value of a solution built (see, e.g., [7, 22]). One of the most popular models of the stochastic optimization is a 2-stage model [7]. In the 2-stage approach the precise values of the parameters are specified in the first stage, while the values of these parameters in the second stage are uncertain and are specified by probability distributions. The goal is to choose a part of a solution in the first stage and complete it in the second stage so that the expected value of the obtained solu- tion is optimized. Recently, there has been a growing interest in combinatorial optimization problems formulated in the 2-stage stochastic framework [9, 10, 12, 16, 21]. In the robust optimization setting [17] the uncertainty is modeled by specifying a set of all possible realizations of the parameters called scenarios. No probability distribution in the scenario set is given. In the discrete scenario case, which is considered in this paper, we 1 define a scenario set by explicitly listing all scenarios. Then, in order to choose a solution, two optimization criteria, called the min-max and the min-max regret, can be adopted. Under the min-max criterion, we seek a solution that minimizes the largest cost over all scenarios. Under the min-max regret criterion we wish to find a solution which minimizes the largest deviation from optimum over all scenarios. A deeper discussion on both criteria can be found in [17]. The minmax (regret) versions of some basic combinatorial optimization problems with discrete structure of uncertainty have been extensively studied in the recent literature [2, 3, 14, 19]. Furthermore, both robust criteria can be easily extended to the 2-stage framework. Such an extension has been recently done in [8, 16]. In this paper, we wish to investigate the min-max (regret) and min-max 2-stage versions of the classical minimum spanning tree problem. The classical deterministic problem is formally stated as follows. We are given a connected graph G = (V, E) with edge costs ce, e ∈E. We seek a spanning tree of G of the minimal total cost. We use Φ to denote the set of all spanning trees of G. The classical deterministic minimum spanning tree is a well studied problem, for which several very efficient algorithms exist (see, e.g., [1]). In the robust framework, the edge costs are uncertain and the set of scenarios Γ is defined by explicitly listing all possible edge cost vectors. So, Γ = {S1, . . . , SK} is finite and contains exactly K scenarios, where a scenario is a cost realization S = (cS e )e∈E. In this paper we consider the unbounded case, where the number of scenarios is a part of the input. We will denote by C∗(S) = minT∈Φ P e∈T cS e the cost of a minimum spanning tree under a fixed scenario S ∈Γ. In the Min-max Spanning Tree problem, we seek a spanning tree that minimizes the largest cost over all scenarios, that is OPT1 = min T∈Φ max S∈Γ X e∈T cS e . (1) In the Min-max Regret Spanning Tree, we wish to find a spanning tree that minimizes the maximal regret: OPT2 = min T∈Φ max S∈Γ (X e∈T

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