This paper presents the first convergence result for random search algorithms to a subset of the Pareto set of given maximum size k with bounds on the approximation quality. The core of the algorithm is a new selection criterion based on a hypothetical multilevel grid on the objective space. It is shown that, when using this criterion for accepting new search points, the sequence of solution archives converges with probability one to a subset of the Pareto set that epsilon-dominates the entire Pareto set. The obtained approximation quality epsilon is equal to the size of the grid cells on the finest level of resolution that allows an approximation with at most k points within the family of grids considered. While the convergence result is of general theoretical interest, the archiving algorithm might be of high practical value for any type iterative multiobjective optimization method, such as evolutionary algorithms or other metaheuristics, which all rely on the usage of a finite on-line memory to store the best solutions found so far as the current approximation of the Pareto set.
Despite the existence of suitable approximation concepts, investigations on the of particular algorithms towards such approximation sets, that is, their ability to obtain a suitable Pareto set approximation in the limit, have remained rare. In @cite_15 @cite_1 the stochastic search procedure proposed by earlier by @cite_7 was analyzed and proved to converge to an @math -Pareto set with @math in case of a finite search space. Obviously, the solution set maintained by this algorithm might in the worst case grow as large as the Pareto set @math itself. Thus, a different version with bounded memory of at most @math elements was proposed and shown to converge to some subset of @math of size at most @math , but no guarantee about the approximation quality could be given. Similar results were obtained by @cite_8 for continuous search spaces.
@cite_15 @cite_1 @cite_7 @cite_8
2077178480 2808191361 2165626989 2001663593
We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple ((s_i, _i) ) consisting of a point (s_i ) in a (d )-dimensional space and a probability ( _i ) determining its existence. The problem is interesting and non-trivial even in (1 )-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity ( (n^2) ). We then show that under more natural and less adversarial conditions, the size of the (1 )-dimensional LVD is significantly smaller: (1) ( (k n) ) if the input has only (k ) distinct probability values, (2) (O(n n) ) on average, and (3) (O(n n ) ) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (LNN) search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models, as well as worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries. We consider the over-time version of the Max-Min Fair Allocation problem. Given a time horizon @math , with at each time t a set of demands and a set of available resources that may change over the time defining instance @math , we seek a sequence of solutions @math that (1) are near-optimal at each time t , and (2) as stable as possible (inducing small modification costs). We focus on the impact of the knowledge of the future on the quality and the stability of the returned solutions by distinguishing three settings: the off-line setting where the whole set of instances through the time horizon is known in advance, the on-line setting where no future instance is known, and the k -lookahead setting where at time t , the instances at times @math are known. We first consider the case without restrictions where the set of resources and the set of agents are the same for all instances and where every resource can be allocated to any agent. For the off-line setting, we show that the over-time version of the problem is much harder than the static one, since it becomes @math -hard even for families of instances for which the static problem is trivial. Then, we provide a @math -approximation algorithm for the off-line setting using as subroutine a ρ-approximation algorithm for the static version. We also give a @math -competitive algorithm for the online setting using also as subroutine a ρ-approximation algorithm for the static version. Furthermore, for the case with restrictions, we show that in the off-line setting it is possible to get a polynomial-time algorithm with the same approximation ratio as in the case without restrictions. For the online setting, we prove that it is not possible to find an online algorithm with bounded competitive ratio. For the 1-lookahead setting however, we give a @math -approximation algorithm using as subroutine a ρ-approximation algorithm for the static version. Search algorithms for Pareto optimization are designed to obtain multiple solutions, each offering a different trade-off of the problem objectives. To make the different solutions available at the end of an algorithm run, procedures are needed for storing them, one by one, as they are found. In a simple case, this may be achieved by placing each point that is found into an "archive" which maintains only nondominated points and discards all others. However, even a set of mutually nondominated points is potentially very large, necessitating a bound on the archive's capacity. But with such a bound in place, it is no longer obvious which points should be maintained and which discarded; we would like the archive to maintain a representative and well-distributed subset of the points generated by the search algorithm, and also that this set converges. To achieve these objectives, we propose an adaptive archiving algorithm, suitable for use with any Pareto optimization algorithm, which has various useful properties as follows. It maintains an archive of bounded size, encourages an even distribution of points across the Pareto front, is computationally efficient, and we are able to prove a form of convergence. The method proposed here maintains evenness, efficiency, and cardinality, and provably converges under certain conditions but not all. Finally, the notions underlying our convergence proofs support a new way to rigorously define what is meant by "good spread of points" across a Pareto front, in the context of grid-based archiving schemes. This leads to proofs and conjectures applicable to archive sizing and grid sizing in any Pareto optimization algorithm maintaining a grid-based archive. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 .