A Fan-type result on k-ordered graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart.     

Acyclic and k-distance coloring of the grid

In this paper, we give a relatively simple though very efficient way to color the d-dimensional grid G(n₁,n₂,…,n_d) (with n_i vertices in each dimension 1?i?d), for two different types of vertex colorings: (1) acyclic coloring of graphs, in which we color the vertices such that (i) no two neighbors are assigned the same color and (ii) for any two colors i and j, the subgraph induced by the vertices colored i or j is acyclic; and (2) k-distance coloring of graphs, in which every vertex must be colored in such a way that two vertices lying at distance less than or equal to k must be assigned different colors. The minimum number of colors needed to acyclically color (respectively k-distance color) a graph G is called acyclic chromatic number of G (respectively k-distance chromatic number), and denoted a(G) (respectively χ_k(G)).The method we propose for coloring the d-dimensional grid in those two variants relies on the representation of the vertices of G_d(n₁,…,n_d) thanks to its coordinates in each dimension; this gives us upper bounds on a(G_d(n₁,…,n_d)) and χ_k(G_d(n₁,…,n_d)).We also give lower bounds on a(G_d(n₁,…,n_d)) and χ_k(G_d(n₁,…,n_d)). In particular, we give a lower bound on a(G) for any graph G; surprisingly, as far as we know this result was never mentioned before. Applied to the d-dimensional grid G_d(n₁,…,n_d), the lower and upper bounds for a(G_d(n₁,…,n_d)) match (and thus give an optimal result) when the lengths in each dimension are “sufficiently large” (more precisely, if ). If this is not the case, then these bounds differ by an additive constant at most equal to . Concerning χ_k(G_d(n₁,…,n_d)), we give exact results on its value for (1) k=2 and any d?1, and (2) d=2 and any k?1.In the case of acyclic coloring, we also apply our results to hypercubes of dimension d, H_d, which are a particular case of G_d(n₁,…,n_d) in which there are only 2 vertices in each dimension. In that case, the bounds we obtain differ by a multiplicative constant equal to 2.     

On the k-tuple domination of generalized de Brujin and Kautz digraphs

A vertex u in a digraph G = (V, A) is said to dominate itself and vertices v such that (u, v) ∈ A. For a positive integer k, a k-tuple dominating set of G is a subset D of vertices such that every vertex in G is dominated by at least k vertices in D. The k-tuple domination number of G is the minimum cardinality of a k-tuple dominating set of G. This paper deals with the k-tuple domination problem on generalized de Bruijn and Kautz digraphs. We establish bounds on the k-tuple domination number for the generalized de Bruijn and Kautz digraphs and we obtain some conditions for the k-tuple domination number attaining the bounds.     

Upper signed k-domination in a general graph

Let k be a positive integer, and let G=(V,E) be a graph with minimum degree at least k−1. A function f:V→{−1,1} is said to be a signed k-dominating function (SkDF) if _∑u∈N[v]f(u)?k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)?f(v) for every v∈V. The maximum of the values of _∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination numberΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.     

Distributed algorithms for finding and maintaining a k-tree core in a dynamic network

A k-core C_k of a tree T is subtree with exactly k leaves for k?n_l, where n_l the number of leaves in T, and minimizes the sum of the distances of all nodes from C_k. In this paper first we propose a distributed algorithm for constructing a rooted spanning tree of a dynamic graph such that root of the tree is located near the center of the graph. Then we provide a distributed algorithm for finding k-core of that spanning tree. The spanning tree is constructed in two stages. In the first stage, a forest of trees is generated. In the next stage these trees are connected to form a single rooted tree. An interesting aspect of the first stage of proposed spanning algorithm is that it implicitly constructs the (convex) hull of those nodes which are not already included in the spanning forest. The process is repeated till all non root nodes of the graph have chosen a unique parent. We implemented the algorithms for finding spanning tree and its k-core. A core can be quite useful for routing messages in a dynamic network consisting of a set of mobile devices.     

Spanners of bounded degree graphs

A k-spanner of a graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most k times the distance in G. We prove that for fixed k,w, the problem of deciding if a given graph has a k-spanner of treewidth w is fixed-parameter tractable on graphs of bounded degree. In particular, this implies that finding a k-spanner that is a tree (a tree k-spanner) is fixed-parameter tractable on graphs of bounded degree. In contrast, we observe that if the graph has only one vertex of unbounded degree, then Treek-Spanner is NP-complete for k?4.     

On the k-tuple domination of de Bruijn and Kautz digraphs

In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs.     

Monochromatic geometric k-factors for bicolored point sets with auxiliary points

Given a bicolored point set S, it is not always possible to construct a monochromatic geometric planar k-factor of S. We consider the problem of finding such a k-factor of S by using auxiliary points. Two types are considered: white points whose position is fixed, and Steiner points which have no fixed position. Our approach provides algorithms for constructing those k-factors, and gives bounds on the number of auxiliary points needed to draw a monochromatic geometric planar k-factor of S.     

Minimizing the sum of the k largest functions in linear time

Given a collection of n functions defined on , and a polyhedral set , we consider the problem of minimizing the sum of the k largest functions of the collection over Q. Specifically we focus on collections of linear functions and several classes of convex, piecewise linear functions which are defined by location models. We present simple linear programming formulations for these optimization models which give rise to linear time algorithms when the dimension d is fixed. Our results improve complexity bounds of several problems reported recently by Tamir [Discrete Appl. Math. 109 (2001) 293-307], Tokuyama [Proc. 33rd Annual ACM Symp. on Theory of Computing, 2001, pp. 75-84] and Kalcsics, Nickel, Puerto and Tamir [Oper. Res. Lett. 31 (1984) 114-127].     

Improved algorithms for the k simple shortest paths and the replacement paths problems

Given a directed, non-negatively weighted graph G=(V,E) and s,t∈V, we consider two problems. In the k simple shortest paths problem, we want to find the k simple paths from s to t with the k smallest weights. In the replacement paths problem, we want the shortest path from s to t that avoids e, for every edge e in the original shortest path from s to t. The best known algorithm for the k simple shortest paths problem has a running of O(k(mn+n²logn)). For the replacement paths problem the best known result is the trivial one running in time O(mn+n²logn).In this paper we present two simple algorithms for the replacement paths problem and the k simple shortest paths problem in weighted directed graphs (using a solution of the All Pairs Shortest Paths problem). The running time of our algorithm for the replacement paths problem is O(mn+n²loglogn). For the k simple shortest paths we will perform O(k) iterations of the second simple shortest path (each in O(mn+n²loglogn) running time) using a useful property of Roditty and Zwick [L. Roditty, U. Zwick, Replacement paths and k simple shortest paths in unweighted directed graphs, in: Proc. of International Conference on Automata, Languages and Programming (ICALP), 2005, pp. 249-260]. These running times immediately improve the best known results for both problems over sparse graphs.Moreover, we prove that both the replacement paths and the k simple shortest paths (for constant k) problems are not harder than APSP (All Pairs Shortest Paths) in weighted directed graphs.     

On a simple randomized algorithm for finding a 2-factor in sparse graphs

We analyze the performance of a simple randomized algorithm for finding 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of degree at most three. For the directed case, the algorithm finds a 2-factor in O(n²) expected time. The analysis of our algorithm is based on random walks on the line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [On selecting a satisfying truth assignment, in: Proc. 32nd Annual IEEE Symp. on the Foundations of Computer Science (FOCS), 1991, p. 163]. For the undirected case, the algorithm finds a 2-factor in O(n³) expected time. We also analyze random versions of these graphs and show that cycles of length Ω(n/logn) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [Finding hidden Hamilton cycles, Random Structures Algorithms 5 (1994) 395] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [On approximating the longest path in a graph, Algorithmica 18 (1997) 82].     

A linear-time algorithm to compute a MAD tree of an interval graph

The average distance of a connected graph G is the average of the distances between all pairs of vertices of G. We present a linear time algorithm that determines, for a given interval graph G, a spanning tree of G with minimum average distance (MAD tree). Such a tree is sometimes referred to as a minimum routing cost spanning tree.     

Spanning trees with minimum weighted degrees

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this problem within a 2−ε factor.     

Computing all efficient solutions of the biobjective minimum spanning tree problem

A common way of computing all efficient (Pareto optimal) solutions for a biobjective combinatorial optimisation problem is to compute first the extreme efficient solutions and then the remaining, non-extreme solutions. The second phase, the computation of non-extreme solutions, can be based on a “k-best” algorithm for the single-objective version of the problem or on the branch-and-bound method. A k-best algorithm computes the k-best solutions in order of their objective values. We compare the performance of these two approaches applied to the biobjective minimum spanning tree problem. Our extensive computational experiments indicate the overwhelming superiority of the k-best approach. We propose heuristic enhancements to this approach which further improve its performance.     

On the tractability of coloring semirandom graphs

As part of the efforts to understand the intricacies of the k-colorability problem, different distributions over k-colorable graphs have been analyzed. While the problem is notoriously hard (not even reasonably approximable) in the worst case, the average case (with respect to such distributions) often turns out to be “easy”. Semi-random models mediate between these two extremes and are more suitable to imitate “real-life” instances than purely random models. In this work we consider semi-random variants of the planted k-colorability distribution. This continues a line of research pursued by Coja-Oghlan, and by Krivelevich and Vilenchik. Our aim is to study a more general semi-random framework than those suggested so far. On the one hand we show that previous algorithmic techniques extend to our more general semi-random setting; on the other hand we give a hardness result, proving that a closely related semi-random model is intractable. Thus we provide some indication about which properties of the input distribution make the k-colorability problem hard.     

Connected sum of digital closed surfaces

During the past 20 years the research of digital surfaces has proceeded to find their properties in the digital space Zⁿ, such as a topological number, a simple k-point, the 3D-Jordan theorem, a k-separating set, a boundary detecting algorithm and so on. Actually, unlike surfaces in a continuous space, the features of digital surfaces have different characteristics. The aim of this paper is to introduce the notion of a digital closed k-surface in Zⁿ, n ? 3, with the general k-adjacency relations as a generalization of Malgouyres’ and Morgenthaler’s k-surfaces in Z³, to establish some minimal simple closed k-surfaces in Z³ and to find their digital topological properties in relation with the k-fundamental group and k-contractibility. Moreover, a connected sum of two digital closed surfaces is introduced and its digital topological properties are investigated.     

The bottleneck k-MST

The k-MST is a well known NP-hard problem and several approximation algorithms exist to solve this problem with a guaranteed performance bound. A closely related problem, called the bottleneck k-MST (BMST(k)) can however be solved in O(mlogn) time on graph with n nodes and m edges. We propose two algorithms to solve BMST(k), one of complexity O(m+nlogn) and the other of O(m) time. We also consider a generalization of BMST(k) which subsumes many bottleneck problems studied in the literature and show that this generalized problem can also be solved in O(m) time.     

On the additive constant of the k-server Work Function Algorithm

We consider the Work Function Algorithm for the k-server problem (Chrobak and Larmore, 1991; Koutsoupias and Papadimitriou, 1995) [2] and [4]. We show that if the Work Function Algorithm is c-competitive, then it is also strictly(2c)-competitive. As a consequence of (Koutsoupias and Papadimitriou, 1995) [4] this also shows that the Work Function Algorithm is strictly (4k−2)-competitive.     

An algorithm for k-anonymous microaggregation and clustering inspired by the design of distortion-optimized quantizers

We present a multidisciplinary solution to the problems of anonymous microaggregation and clustering, illustrated with two applications, namely privacy protection in databases, and private retrieval of location-based information. Our solution is perturbative, is based on the same privacy criterion used in microdata k-anonymization, and provides anonymity through a substantial modification of the Lloyd algorithm, a celebrated quantization design algorithm, endowed with numerical optimization techniques.Our algorithm is particularly suited to the important problem of k-anonymous microaggregation of databases, with a small integer k representing the number of individual respondents indistinguishable from each other in the published database. Our algorithm also exhibits excellent performance in the problem of clustering or macroaggregation, where k may take on arbitrarily large values. We illustrate its applicability in this second, somewhat less common case, by means of an example of location-based services. Specifically, location-aware devices entrust a third party with accurate location information. This party then uses our algorithm to create distortion-optimized, size-constrained clusters, where k nearby devices share a common centroid location, which may be regarded as a distorted version of the original one. The centroid location is sent back to the devices, which use it when contacting untrusted location-based information providers, in lieu of the exact home location, to enforce k-anonymity.We compare the performance of our novel algorithm to the state-of-the-art microaggregation algorithm MDAV, on both synthetic and standardized real data, which encompass the cases of small and large values of k. The most promising aspect of our proposed algorithm is its capability to maintain the same k-anonymity constraint, while outperforming MDAV by a significant reduction in data distortion, in all the cases considered.     

A linear time algorithm for max-min length triangulation of a convex polygon

We consider the following planar max-min length triangulation problem: given a set of n points in the Euclidean plane, find a triangulation such that the length of the shortest edge in the triangulation is maximized. In this paper, a linear time algorithm is proposed for computing the max-min length triangulation of a set of points in convex position. In addition, an O(nlogn) time algorithm is proposed for computing the max-min length k-set triangulation of a set of points in convex position, where we are to compute a set of k vertices such that the max-min length triangulation on them is minimized over all possible k-set. We further show that the graph version of max-min length triangulation is NP-complete, and some common heuristics such as greedy algorithm are in general not able to give a bounded-ratio approximation to the max-min length triangulation.