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Randomized Local Search, Evolutionary Algorithms, and the Minimum Spanning Tree Problem
 IN PROC. OF GECCO ’04
, 2004
"... Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization. The analysis ..."
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Cited by 79 (31 self)
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Randomized search heuristics, among them randomized local search and evolutionary algorithms, are applied to problems whose structure is not well understood, as well as to problems in combinatorial optimization. The analysis
An analysis on recombination in multiobjective evolutionary optimization
 In Proceedings of the 13th ACM Annual Conference on Genetic and Evolutionary Computation (GECCO’11
, 2011
"... Evolutionary algorithms (EAs) are increasingly popular approaches to multiobjective optimization. One of their significant advantages is that they can directly optimize the Pareto front by evolving a population of solutions, where the recombination (also called crossover) operators are usually emp ..."
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Cited by 8 (5 self)
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Evolutionary algorithms (EAs) are increasingly popular approaches to multiobjective optimization. One of their significant advantages is that they can directly optimize the Pareto front by evolving a population of solutions, where the recombination (also called crossover) operators are usually employed to reproduce new and potentially better solutions by mixing up solutions in the population. Recombination in multiobjective evolutionary algorithms is, however, mostly applied heuristically. In this paper, we investigate how from a theoretical viewpoint a recombination operator will affect a multiobjective EA. First, we employ artificial benchmark problems: the Weighted LPTNO problem (a generalization of the wellstudied LOTZ problem), and the wellstudied COCZ problem, for studying the effect of recombination. Our analysis discloses that recombination may accelerate the filling of the Pareto front by recombining diverse solutions and thus help solve multiobjective optimization. Because of this, for these two problems, we find that a multiobjective EA with recombination enabled achieves a better expected running time than any known EAs with recombination disabled. We further examine the effect of recombination on solving the multiobjective minimum spanning tree problem, which is an NPHard problem. Following our finding on the artificial problems, our analysis shows that recombination also helps accelerate filling the Pareto front and thus helps find approximate solutions faster.
Reconfiguration of Dominating Sets
, 2014
"... We explore a reconguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions co ..."
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Cited by 4 (3 self)
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We explore a reconguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of Dk(G), the graph consisting of a vertex for each dominating set of size at most k and edges speci ed by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that D (G)+1(G) is not necessarily connected, for (G) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded treewidth, or bpartite for b 3. Moreover, we construct an infinite family of graphs such that D
(G)+1(G) has exponential diameter, for
(G) the minimum size of a dominating set. On the positive side, we show that Dn(G) is connected and of linear diameter for any graph G on n vertices with a matching of size at least + 1.
TreeWeighted Neighbors and Geometric k Smallest Spanning Trees
, 1992
"... We compute the k smallest spanning trees of a point set in the planar Euclidean metric in time O(n log n log k +k min(k,n) 1/2 log(k/n)), and in the rectilinear metrics in time O(n log n + n log log n log k + k min(k,n) 1/2 log(k/n)). In three or four dimensions our time bound is O(n 4/3+# ..."
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Cited by 1 (1 self)
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We compute the k smallest spanning trees of a point set in the planar Euclidean metric in time O(n log n log k +k min(k,n) 1/2 log(k/n)), and in the rectilinear metrics in time O(n log n + n log log n log k + k min(k,n) 1/2 log(k/n)). In three or four dimensions our time bound is O(n 4/3+# + k min(k,n) 1/2 log(k/n)), and in higher dimensions the bound is O(n 22/(#d/2#+1)+# + kn 1/2 log n). 1 Introduction The k smallest spanning tree problem for graphs has been studied extensively [5, 6, 7, 8, 9, 10, 11, 12], but it was only recently that the author introduced the corresponding geometric problem [7]. In this problem, one is given a point set as input, and one must construct k di#erent spanning trees that have the minimum total edge lengths among all possible spanning trees of the set. The trees need not be edgedisjoint. This problem is to be distinguished from the much harder one of finding the k smallest distinct spanning tree weights [12]. Since the geometric pro...
Several geometric data structures for objects in the plane can be constructed using persistent binary search trees. (I know, I promised I wouldn't do much geometry, but this was sort of irresistible.)
"... ach insertion or deletion crates a new version of T y ; we attach each version to the corresponding leaf in T x . The overall preprocessing time is O(n log n), and the overall space is O(n). To answer a query for point (a; b), we search for a in T x to nd the correct version of T y , and then sear ..."
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ach insertion or deletion crates a new version of T y ; we attach each version to the corresponding leaf in T x . The overall preprocessing time is O(n log n), and the overall space is O(n). To answer a query for point (a; b), we search for a in T x to nd the correct version of T y , and then search that version of T y for the successor of b. The query time in each tree is O(log n), so the overall query time is also O(log n). (b) Modify the previous data structure to store a set of n disjoint nonhorizontal segments, with the same preprocessing, space, and query bounds. [Hint: Use a persistent kinetic binary search tree.] Solution: A fulledged kinetic data structure is overkill here. The only necessary change from part (a) is that we store the line equations of segments in T y . Whenever we search T y , either to locate a point (a; b) or to insert a new segment with left endpoint (a; b), we replace every direct comparison with b with a test whether (a; b) is above or below a lin