A tight bound on the min-ratio edge-partitioning problem of a tree

In this paper, we study how to partition a tree into edge-disjoint subtrees of approximately the same size. Given a tree T with n edges and a positive integer k≤n, we design an algorithm to partition T into k edge-disjoint subtrees such that the ratio of the maximum number to the minimum number of edges of the subtrees is at most two. The best previous upper bound of the ratio is three, given by Wu et al. [B.Y. Wu, H.-L. Wang, S.-T. Kuan, K.-M. Chao, On the uniform edge-partition of a tree, Discrete Applied Mathematics 155 (10) (2007) 1213-1223]. Wu et al. also showed that for some instances, it is impossible to achieve a ratio better than two. Therefore, there is a lower bound of two on the ratio. It follows that the ratio upper bound attained in this paper is already tight.     

Revisiting dynamic programming for finding optimal subtrees in trees

In this paper we revisit an existing dynamic programming algorithm for finding optimal subtrees in edge weighted trees. This algorithm was sketched by Maffioli in a technical report in 1991. First, we adapt this algorithm for the application to trees that can have both node and edge weights. Second, we extend the algorithm such that it does not only deliver the values of optimal trees, but also the trees themselves. Finally, we use our extended algorithm for developing heuristics for the k-cardinality tree problem in undirected graphs G with node and edge weights. This NP-hard problem consists of finding in the given graph a tree with exactly k edges such that the sum of the node and the edge weights is minimal. In order to show the usefulness of our heuristics we conduct an extensive computational analysis that concerns most of the existing problem instances. Our results show that with growing problem size the proposed heuristics reach the performance of state-of-the-art metaheuristics. Therefore, this study can be seen as a cautious note on the scaling of metaheuristics.     

Dynamic Enumeration of All Mixed Cells

The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family of homotopy functions. The mixed cells are reformulated in terms of a linear inequality system with an additional combinatorial condition. An enumeration tree is constructed among a family of linear inequality systems induced from it such that every mixed cell corresponds to a unique feasible leaf node, and the depth-first search is applied to the enumeration tree for finding all the feasible leaf nodes. How to construct such an enumeration tree is crucial in computational efficiency. This paper proposes a dynamic construction of an enumeration tree, which branches each parent node into its child nodes so that the number of feasible child nodes is expected to be small; hence we can prune many subtrees which do not contain any mixed cell. Numerical results exhibit that the proposed dynamic construction of an enumeration tree works very efficiently for large scale polynomial systems; for example, it generated all mixed cells of the cyclic-15 problem for the first time in less than 16 hours.     

Persistence of centrality in random growing trees

We investigate properties of node centrality in random growing tree models. We focus on a measure of centrality that computes the maximum subtree size of the tree rooted at each node, with the most central node being the tree centroid. For random trees grown according to a preferential attachment model, a uniform attachment model, or a diffusion processes over a regular tree, we prove that a single node persists as the tree centroid after a finite number of steps, with probability 1. Furthermore, this persistence property generalizes to the top K ≥ 1 nodes with respect to the same centrality measure. We also establish necessary and sufficient conditions for the size of an initial seed graph required to ensure persistence of a particular node with probability , as a function of ϵ: In the case of preferential and uniform attachment models, we derive bounds for the size of an initial hub constructed around the special node. In the case of a diffusion process over a regular tree, we derive bounds for the radius of an initial ball centered around the special node. Our necessary and sufficient conditions match up to constant factors for preferential attachment and diffusion tree models.     

Converting suffix trees into factor/suffix oracles

Several methods to compress suffix trees were defined, most of them with the aim of obtaining compact (that is, space economical) index structures. Besides this practical aspect, a compression method can reveal structural properties of the resulting data structure, allowing a better understanding of it and a better estimation of its performances.

In this paper, we propose a simple method to compress suffix trees by merging couples of nodes. This idea was already used in the literature in a context different from ours. The originality of our approach is that the nodes we merge are not chosen with respect to their subtrees (which is difficult to test algorithmically), nor with respect to the words spelled along branches (which usually requires testing several branches before finding the good one) but with respect to their position in the tree (which is easy to compute). Another particularity of our method is it needs to read no edge label: it is exclusively based on the topology of the suffix tree. The compact structure resulting after compression is the factor/suffix oracle introduced by Allauzen, Crochemore and Raffinot whose accepted language includes the accepted language of the corresponding suffix tree.

The interest of our paper is therefore threefold:

1. A topology-based compression method is defined for (compact) suffix trees.

2. A new property of a factor/suffix oracle is established, that is, like a DAG, it results from the corresponding suffix tree after a linear number of appropriate node mergings; unlike a DAG, the merged nodes do not necessarily have isomorphical subtrees.

3. A new algorithm to transform a suffix tree into a factor/suffix oracle is given, which has linear running time and thus improves the quadratic complexity previously known for the same task.

Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node. The CMST problem is an NP-complete problem, and existing exact algorithms can solve only small size problems. Currently, the best available heuristic procedures for the CMST problem are tabu search algorithms due to Amberg et al. and Sharaiha et al. These algorithms use two-exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees. In this paper, we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously; we refer to such neighborhood structures as multi-exchange neighborhood structures. Our first multi-exchange neighborhood structure allows exchanges of single nodes among several subtrees. Our second multi-exchange neighborhood structure allows exchanges that involve multiple subtrees. The size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors. Our approach, which is based on the cyclic transfer neighborhood structure due to Thompson and Psaraftis and Thompson and Orlin transforms a profitable exchange into a negative cost subset-disjoint cycle in a graph, called an improvement graph, and identifies these cycles using variants of shortest path label-correcting algorithms. Our computational results with GRASP and tabu search algorithms based on these neighborhood structures reveal that (i) for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some; and (ii) for the heterogeneous demand case our algorithms improved the best available solutions for most of the benchmark instances with improvements by as much as 18%. The running times our multi-exchange neighborhood search algorithms are comparable to those taken by two-exchange neighborhood search algorithms. Received: September 1998 / Accepted: March 2001?Published online May 18, 2001     

A random tree model associated with random graphs

Grow a tree on n vertices by starting with no edges and successively adding an edge chosen uniformly from the set of possible edges whose addition would not create a cycle. This process is closely related to the classical random graph process. We describe the asymptotic structure of the tree, as seen locally from a given vertex. In particular, we give an explicit expression for the asymptotic degree distribution. Our results an be applied to study the random minimum-weight spanning tree question, when the edge-weight distribution is allowed to vary almost arbitrarily with n.     

On the Local and Global Means of Subtree Orders

The global mean of subtrees of a tree is the average order (i.e., average number of vertices) of its subtrees. Analogously, the local mean of a vertex in a tree is the average order of subtrees containing this vertex. In the comprehensive study of these concepts by Jamison (J Combin Theory Ser B 35 (1983), 207–223 and J Combin Theory Ser B 37 (1984), 70–78), several open questions were proposed. One of them asks if the largest local mean always occurs at a leaf vertex. Another asks if it is true that the local mean of any vertex of any tree is at most twice the global mean. In this note, we answer the first question by showing that the largest local mean always occurs at a leaf or a vertex of degree 2 and that both cases are possible. With this result, a positive answer to the second question is provided. We also show some related results on local mean and global mean of trees.     

A composite very large-scale neighborhood structure for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree in a network where nodes have specified demands, with an additional capacity constraints on the subtrees incident to a given source node s. The capacitated minimum spanning tree problem arises as an important subproblem in many telecommunication network design problems. In a recent paper, Ahuja et al. (Math. Program. 91 (2001) 71) proposed two very large-scale neighborhood search algorithms for the capacitated minimum spanning tree problem. Their first node-based neighborhood structure is obtained by performing multi-exchanges involving several trees where each tree contributes a single node. Their second tree-based neighborhood structure is obtained by performing multi-exchanges where each tree contributes a subtree. The computational investigations found that node-based multi-exchange neighborhood gives the best performance for the homogenous demand case (when all nodes have the same demand), and the tree-based multi-exchange neighborhood gives the best performance for the heterogeneous demand case (when nodes may have different demands). In this paper, we propose a composite neighborhood structure that subsumes both the node-based and tree-based neighborhoods, and outperforms both the previous neighborhood search algorithms for solving the capacitated minimum spanning tree problem on standard benchmark instances. We also develop improved dynamic programming based exact algorithms for searching the composite neighborhood.     

Efficient implementation of a shifting algorithm

An efficient implementation of the shifting algorithm [2] for min-max tree partitioning is given. The complexity is reduced from ORk³ + kn) to O(Rk(k + log d) + n) where a tree of n vertices, radius, of R edges, and maximum degree d is partitioned into k + 1 subtrees. The improvement is mainly due to the new junction tree data structure which suggests a succinct representation for subsets of edges, of a given tree, that preserves the interrelation betqween the edges on the tree.     

Join tree propagation utilizing both arc reversal and variable elimination

In this paper, we put forth the first join tree propagation algorithm that selectively applies either arc reversal (AR) or variable elimination (VE) to build the propagated messages. Our approach utilizes a recent method for identifying the propagated join tree messages à priori. When it is determined that a join tree node will construct a single distribution to be sent to a neighbouring node, VE is utilized as it builds a single distribution in the most direct fashion; otherwise, AR is applied as it maintains a factorization of distributions allowing for barren variables to be exploited during propagation later on in the join tree. Experimental results, involving evidence processing in four benchmark Bayesian networks, empirically demonstrate that selectively applying VE and AR is faster than applying one of these methods exclusively on the entire network.     

Characterization of the Distance between Subtrees of a Tree by the Associated Tight Span

A characterization is given to the distance between subtrees of a tree defined as the shortest path length between subtrees. This is a generalization of the four-point condition for tree metrics. For this, we use the theory of the tight span and obtain an extension of the famous result by Dress that a metric is a tree metric if and only if its tight span is a tree. Received July 13, 2004     

Highway games on weakly cyclic graphs

A highway problem is determined by a connected graph which provides all potential entry and exit vertices and all possible edges that can be constructed between vertices, a cost function on the edges of the graph and a set of players, each in need of constructing a connection between a specific entry and exit vertex. Mosquera (2007) introduce highway problems and the corresponding cooperative cost games called highway games to address the problem of fair allocation of the construction costs in case the underlying graph is a tree. In this paper, we study the concavity and the balancedness of highway games on weakly cyclic graphs. A graph G is called highway-game concave if for each highway problem in which G is the underlying graph the corresponding highway game is concave. We show that a graph is highway-game concave if and only if it is weakly triangular. Moreover, we prove that highway games on weakly cyclic graphs are balanced.     

Intersection graphs of proper subtrees of unicyclic graphs

A graph is fraternally oriented iff for every three vertices u, ν, w the existence of the edges u → w and ν → w implies that u and ν are adjacent. A directed unicyclic graph is obtained from a unicyclic graph by orienting the unique cycle clockwise and by orienting the appended subtrees from the cycle outwardly. Two directed subtrees s, t of a directed unicyclic graph are proper if their union contains no (directed or undirected) cycle and either they are disjoint or one of them s has its root r(s) in t and contains all the successors of r(s) in t. In the present paper we prove that G is an intersection graph of a family of proper directed subtrees of a directed unicyclic graph iff it has a fraternal orientation such that for every vertex ν, G(Γⁱⁿν) is acyclic and G(Γ^outν) is the transitive closure of a tree. We describe efficient algorithms for recognizing when such graphs are perfect and for testing isomorphism of proper circular-arc graphs.     

Continuous bottleneck tree partitioning problems

We study continuous partitioning problems on tree network spaces whose edges and nodes are points in Euclidean spaces. A continuous partition of this space into p connected components is a collection of p subtrees, such that no pair of them intersect at more than one point, and their union is the tree space. An edge-partition is a continuous partition defined by selecting p−1 cut points along the edges of the underlying tree, which is assumed to have n nodes. These cut points induce a partition into p subtrees (connected components). The objective is to minimize (maximize) the maximum (minimum) “size” of the components (the min–max (max–min) problem). When the size is the length of a subtree, the min–max and the max–min partitioning problems are NP-hard. We present O(n² log(min(p,n))) algorithms for the edge-partitioning versions of the problem. When the size is the diameter, the min–max problems coincide with the continuous p-center problem. We describe O(n log³ n) and O(n log² n) algorithms for the max–min partitioning and edge-partitioning problems, respectively, where the size is the diameter of a component.     

Improving multicut in directed trees by upgrading nodes

In this paper, we consider the network improvement problem for multicut by upgrading nodes in a directed tree T = (V, E) with multiple sources and multiple terminals. In a node based upgrading model, a node v can be upgraded at the expense of c(v) and such an upgrade reduces weights on all edges incident to v. The objective is to upgrade a minimum cost subset S ⊆ V of nodes such that the resulting network has a multicut in which no edge has weight larger than a given value D. We first obtain a minimum cardinality node multicut V_c for tree T, then find the minimum cost upgrading set based on the upgrading sets for the subtrees rooted at the nodes in V_c. We show that our algorithm is polynomial when the number of source–terminal pairs is upper bounded by a given value.     

Tight Bounds for Connecting Sites Across Barriers

Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines, it is known that there is a spanning tree such that every barrier is crossed by tree edges, and this bound is asymptotically optimal. Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Lower bound constructions are known with 3 crossings per barrier and 2n total cost. We obtain tight bounds on the minimum cost of spanning trees in the special case where the barriers are interior disjoint line segments that form a convex subdivision of the plane and there is a point in every cell of the subdivision. In particular, we show that there is a spanning tree such that every barrier crosses at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are the best possible. Work by Eynat Rafalin and Diane Souvaine was supported by the National Science Foundation under Grant #CCF-0431027. E. Rafalin’s research conducted while at Tufts University.     

Patching upX-trees

A fundamental problem in many areas of classification, and particularly in biology, is the reconstruction of a leaf-labeled tree from just a subset of its induced subtrees. Without loss of generality, we may assume that these induced subtrees all have precisely four leaves. Of particular interest is determining whether a collection of quartet subtrees uniquely defines a parent tree. Here, we solve this problem in the case where the collection of quartet trees is of minimal size, by studyingencodings of binary trees by such quartet trees. We obtain a characterization of minimal encodings that exploits an underlying patchwork structure. As we will show elsewhere, this allows one to obtain a polynomial time algorithm for certain instances of the problem of reconstructing trees from subtrees.Supported by DFG — Graduiertenkolleg Strukturbildungsprozesse, Forschungsschwerpunkt Mathematisierung, University of Bielefeld, Germany.Supported by the New Zealand Marsden Fund.     

A node-based layered graph approach for the Steiner tree problem with revenues,budget and hop-constraints

The Steiner tree problem with revenues, budget and hop-constraints (STPRBH) is a variant of the classical Steiner tree problem. The goal is to find a tree maximizing the collected revenue, which is associated with nodes, subject to a given budget for the edge cost of the tree and a hop-limit for the distance between the given root node and any other node in that tree. In this work, we introduce a novel generic way to model hop-constrained tree problems as integer linear programs and apply it to the STPRBH. Our approach is based on the concept of layered graphs that gained widespread attention in the recent years, due to their computational advantage when compared to previous formulations for modeling hop-constraints. Contrary to previous MIP formulations based on layered graphs (that are arc-based models), our model is node-based. Thus it contains much less variables and allows to tackle large-scale instances and/or instances with large hop-limits, for which the size of arc-based layered graph models may become prohibitive. The aim of our model is to provide a good compromise between quality of root relaxation bounds and the size of the underlying MIP formulation. We implemented a branch-and-cut algorithm for the STPRBH based on our new model. Most of the instances available for the DIMACS challenge, including 78 (out of 86) previously unsolved ones, can be solved to proven optimality within a time limit of 1000 s, most of them being solved within a few seconds only. These instances contain up to 500 nodes and 12,500 edges, with hop-limit up to 25.     

Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$-norm

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.