On two geometric problems related to the travelling salesman problem

The degree-K Minimum Spanning Tree (MST) problem asks for the minimum length spanning tree that has no vertex of degree greater than K. The Euclidean degree-K MST problem is known to be tractable for K ? 5; the degree-2 MST is simply the Euclidean path-TSP, which is NP-complete. It is proved that the Euclidean degree-3 MST problem is also NP-complete, thus leaving open only the case for K = 4. Among the most illustrious approximation algorithms is the heuristic for the Euclidean TSP due to Christofides. It is proved that implementing the “shortcutting phase” of Christofides' algorithm optimally is NP-hard (even so, Christofides' algorithm guarantees a tour which is no more than 50% longer than the optimal one).     

Approximating k-hop minimum-spanning trees

Given a complete graph on~n nodes with metric edge costs, the minimum-costk-hop spanning tree (kHMST) problem asks for a spanning tree of minimum total cost such that the longest root-leaf-path in the tree has at most k edges. We present an algorithm that computes such a tree of total expected cost times that of a minimum-cost k-hop spanning-tree.     

Connecting colored point sets

We study the following Ramsey-type problem. Let S=B∪R be a two-colored set of n points in the plane. We show how to construct, in time, a crossing-free spanning tree T(B) for B, and a crossing-free spanning tree T(R) for R, such that both the number of crossings between T(B) and T(R) and the diameters of T(B) and T(R) are kept small. The algorithm is conceptually simple and is implementable without using any non-trivial data structure. This improves over a previous method in Tokunaga [Intersection number of two connected geometric graphs, Inform. Process. Lett. 59 (1996) 331-333] that is less efficient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in the reference above on the minimum number of crossings between T(B) and T(R).     

Bounds on isoperimetric values of trees

Let G=(V,E) be a finite, simple and undirected graph. For S⊆V, let δ(S,G)={(u,v)∈E:u∈S and v∈V−S} be the edge boundary of S. Given an integer i, 1≤i≤|V|, let the edge isoperimetric value of G at i be defined as b_e(i,G)=min_S⊆V;|S|=i|δ(S,G)|. The edge isoperimetric peak of G is defined as b_e(G)=max_1≤j≤|V|b_e(j,G). Let b_v(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees.The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as ), and where c₁, c₂ are constants. For a complete t-ary tree of depth d (denoted as ) and d≥clogt where c is a constant, we show that and where c₁, c₂ are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T=(V,E,r) be a finite, connected and rooted tree — the root being the vertex r. Define a weight function w:V→N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index η(T) be defined as the number of distinct weights in the tree, i.e η(T)=|{w(u):u∈V}|. For a positive integer k, let ?(k)=|{i∈N:1≤i≤|V|,b_e(i,G)≤k}|. We show that .     

A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ^*/logΔ^*) of the optimum in O(mnlogn)-time, where Δ^* is the minimum k such that there is a spanning tree of the graph with maximum degree k.     

Further analysis of the number of spanning trees in circulant graphs

Let 1?s₁<s₂<?<s_k?⌊n/2⌋ be given integers. An undirected even-valent circulant graph, has n vertices 0,1,2,…, n-1, and for each and j(0?j?n-1) there is an edge between j and . Let stand for the number of spanning trees of . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]], is that where a_n satisfies a linear recurrence relation of order 2^s_k-1. And, most recently, for odd-valent circulant graphs, a nice investigation on the number a_n is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69-79].In this paper, we explore further properties of the numbers a_n from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for a_n, we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]. As examples, we describe our technique and the asymptotics of the numbers.     

A fast parallel algorithm for finding Hamiltonian cycles in dense graphs

Suppose that 0<η<1 is given. We call a graph, G, on n vertices an η-Chvátal graph if its degree sequence d₁≤d₂≤?≤d_n satisfies: for k<n/2, d_k≤min{k+ηn,n/2} implies d_n−k−ηn≥n−k. (Thus for η=0 we get the well-known Chvátal graphs.) An -algorithm is presented which accepts as input an η-Chvátal graph and produces a Hamiltonian cycle in G as an output. This is a significant improvement on the previous best -algorithm for the problem, which finds a Hamiltonian cycle only in Dirac graphs (δ(G)≥n/2 where δ(G) is the minimum degree in G).     

A note on the edge cover chromatic index of multigraphs

Let G be a multigraph with vertex set V(G). An edge coloring C of G is called an edge-cover-coloring if each color appears at least once at each vertex v∈V(G). The maximum positive integer k such that G has a k-edge-cover-coloring is called the edge cover chromatic index of G and is denoted by . It is well known that , where μ(v) is the multiplicity of v and δ(G) is the minimum degree of G. We improve this lower bound to δ(G)−1 when 2≤δ(G)≤5. Furthermore we show that this lower bound is best possible.     

Polynomial area bounds for MST embeddings of trees

In their seminal paper on geometric minimum spanning trees, Monma and Suri (1992) [31] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane. The embeddings provided by their algorithm require area O(n²²)×O(n²²) and the authors conjectured that an improvement below cn×cn is not possible, for some constant c>0. In this paper, we show how to construct MST embeddings of arbitrary trees of maximum degree 3 and 4 within polynomial area.     

Finding 2-edge connected spanning subgraphs

This paper studies the NP-hard problem of finding a minimum size 2-edge connected spanning subgraph (2-ECSS). An algorithm is given that on an r-edge connected input graph G=(V,E) finds a 2-ECSS of size at most |V|+(|E|−|V|)/(r−1). For r-regular, r-edge connected input graphs for r=3, 4, 5 and 6, this gives approximation guarantees of and , respectively.     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

A greedy approximation algorithm for the group Steiner problem

In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices . Each subset g_i is called a group and the vertices in ?_ig_i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265-285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73-91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59-63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(max_i|g_i|)·logm) provided by the LP based approaches.     

A subexponential algorithm for the coloured tree partition problem

Given a tree of n vertices and a list of feasible colours for each vertex, the coloured tree partition problem (CTPP) consists in partitioning the tree into p vertex-disjoint subtrees of minimum total cost, and assigning to each subtree a different colour, which must be feasible for all of its vertices. The problem is strongly NP-hard on general graphs, as well as on grid and bipartite graphs. This paper deals with the previously open case of tree graphs, showing that it is strongly NP-complete to determine whether a feasible solution exists. It presents reduction, decomposition and bounding procedures to simplify the problem and an exact algorithm of complexity (with ) for the special case in which a vertex of each subtree is given.     

On better heuristics for Steiner minimum trees

Finding a shortest network interconnecting a given set of points in a metric space is called the Steiner minimum tree problem. The Steiner ratio is the largest lower bound for the ratio between lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points. In this paper, we show that in a metric space, if the Steiner ratio is less than one and finding a Steiner minimum tree for a set of size bounded by a fixed number can be performed in polynomial time, then there exists a polynomialtime heuristic for the Steiner minimum tree problem with performance ratio bigger than the Steiner ratio. It follows that in the Euclidean plane, there exists a polynomial-time heuristic for Steiner minimum trees with performance ratio bigger than . This solves a long-standing open problem.Part of this work was done while this author visited the Department of Computer Science, Princeton University, supported in part by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648, supported in part by NSF grant No. CCR-8920505, and also supported in part by the National Natural Science Foundation of China.     

On spanning tree congestion

We prove that every connected graph G of order n has a spanning tree T such that for every edge e of T the edge cut defined in G by the vertex sets of the two components of T−e contains at most edges. This result solves a problem posed by Ostrovskii (M.I. Ostrovskii, Minimal congestion trees, Discrete Math. 285 (2004) 219-226).     

On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means

For independently distributed observables: X_i∼N(θ_i,σ²),i=1,…,p, we consider estimating the vector θ=(θ₁,…,θ_p)^′ with loss ‖d−θ‖² under the constraint , with known τ₁,…,τ_p,σ²,m. In comparing the risk performance of Bayesian estimators δ_α associated with uniform priors on spheres of radius α centered at (τ₁,…,τ_p) with that of the maximum likelihood estimator , we make use of Stein’s unbiased estimate of risk technique, Karlin’s sign change arguments, and a conditional risk analysis to obtain for a fixed (m,p) necessary and sufficient conditions on α for δ_α to dominate . Large sample determinations of these conditions are provided. Both cases where all such δ_α’s and cases where no such δ_α’s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δ_m dominates if and only if m≤k(p) with , improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators δ_π with π spherically symmetric and supported on the parameter space dominate whenever m≤c₁(p) with .