An Improved Approximation Algorithm for Multicast <Emphasis Type="Italic">k</Emphasis>-Tree Routing

An improved approximation algorithm is presented in this paper for the multicast k-tree routing problem. The algorithm has a worst case performance ratio of (2.4 + ρ), where ρ is the best approximation ratio for the metric Steiner tree problem (and is about 1.55 so far). The previous best approximation algorithm for the multicast k-tree routing problem has a performance ratio of 4. Two techniques, weight averaging and tree partitioning, are developed to facilitate the algorithm design and analysis.Research supported by AICML, CFI, NSERC, PENCE, a Startup Grant from the University of Alberta, and NNSF Grant 60373012.     

Node-weighted Steiner tree approximation in unit disk graphs

Given a graph G=(V,E) with node weight w:V→R ⁺ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(n ^O(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+ε)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+ε)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.     

An Approximation for Minimum Multicast Route in Optical Networks with Nonsplitting Nodes

Consider the problem of computing the minimum-weight multicast route in an optical network with both nonsplitting and splitting nodes. This problem can be reduced to the minimum Hamiltonian path problem when all nodes are nonsplitting, and the Steiner minimum tree problem when all nodes are splitting. Therefore, the problem is NP-hard. Previously, the best known polynomial-time approximation has the performance ratio 3. In this paper, we present a new polynomial-time approximation with performance ratio of 1+ρ, where ρ is the best known approximation performance ratio for the Steiner minimum tree in graph and it has been known that ρ < 1.55. Support in part by National Science Foundation under grants CCF-0514796 and CNS-0524429     

Approximation algorithms for multicast routing in ad hoc wireless networks

Energy efficient multicast problem is one of important issues in ad hoc networks. In this paper, we address the energy efficient multicast problem for discrete power levels in ad hoc wireless networks. The problem of our concern is: given n nodes deployed over 2-D plane and each node v has l(v) transmission power levels and a multicast request (s,D) (clearly, when D is V∖{s}, the multicast request is a broadcast request), how to find a multicast tree rooted at s and spanning all destinations in D such that the total energy cost of the multicast tree is minimized. We first prove that this problem is NP-hard and it is unlikely to have an approximation algorithm with performance ratio ρlnn(ρ<1). Then, we propose a general algorithm for the multicast/broadcast tree problem. And based on the general algorithm, we propose an approximation algorithm and a heuristics for multicast tree problem. Especially, we also propose an efficient heuristic for broadcast tree problem. Simulations ensure our algorithms are efficient.     

Inapproximability results for the lateral gene transfer problem

This paper concerns the Lateral Gene Transfer Problem. This minimization problem, defined by Hallett and Lagergren (2001), is that of finding the most parsimonious lateral gene transfer scenario for a given pair of gene and species trees. Our main results are the following:

The Polymatroid Steiner Problems

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S⊂eqV in a weighted graph G = (V,E,c), c:E→ R⁺. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P(V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S⊂eqV). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s₁,…,s_k}, each sensor s_i can choose to monitor only a single target from a subset of targets X_i, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X₁ ∪ … ∪ X_k. The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et al. (2002).We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Chekuri et al. (2002).A preliminary version of this paper appeared in ISAAC 2004     

Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem

Given a set N of n terminals in the first quadrant of the Euclidean plane E ², find a minimum length directed tree rooted at the origin o, connecting to all terminals in N, and consisting of only horizontal and vertical arcs oriented from left to right or from bottom to top. This problem is called rectilinear Steiner arborescence problem, which has been proved to be NP-complete recently (Shi and Su, 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2000, to appear). In this paper, we present a polynomial time approximation scheme for this problem.     

Inapproximability and approximability of maximal tree routing and coloring

Let G be a undirected connected graph. Given g groups each being a subset of V(G) and a number of colors, we consider how to find a subgroup of subsets such that there exists a tree interconnecting all vertices in each subset and all trees can be colored properly with given colors (no two trees sharing a common edge receive the same color); the objective is to maximize the number of subsets in the subgroup. This problem arises from the application of multicast communication in all optical networks. In this paper, we first obtain an explicit lower bound on the approximability of this problem and prove Ω(g^1−ε)-inapproximability even when G is a mesh. We then propose a simple greedy algorithm that achieves performance ratio O√|E(G)|, which matches the theoretical bounds. Supported in part by the NSF of China under Grant No. 70221001 and 60373012.     

Minimum entropy coloring

We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that, for every constant ε>0, it is NP-hard to find a coloring whose entropy is within (1−ε)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem. S. Fiorini acknowledges the support from the Fonds National de la Recherche Scientifique and GERAD-HEC Montréal. G. Joret is a F.R.S.-FNRS Research Fellow.     

Tree edge decomposition with an application to minimum ultrametric tree approximation

A k-decomposition of a tree is a process in which the tree is recursively partitioned into k edge-disjoint subtrees until each subtree contains only one edge. We investigated the problem how many levels it is sufficient to decompose the edges of a tree. In this paper, we show that any n-edge tree can be 2-decomposed (and 3-decomposed) within at most ⌈1.44 log n⌉ (and ⌈log n⌉ respectively) levels. Extreme trees are given to show that the bounds are asymptotically tight. Based on the result, we designed an improved approximation algorithm for the minimum ultrametric tree.     

An asymptotic PTAS for batch scheduling with nonidentical job sizes to minimize makespan

Motivated by the existence of an APTAS (Asymptotic PTAS) for bin packing problem, we consider the batch scheduling problem with nonidentical job sizes to minimize makespan. For the proportional special version, i.e., there exists a fixed number α such that p _j =α s _j for every 1≤j≤n, we first present a lower bound of 3/2 for the approximation ratio and then design an APTAS. Supported by NNSF of China (No.10671108).     

A quadratic lower bound for colourful simplicial depth

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in ℝ ^d is contained in at least ⌊(d+2)²/4⌋ simplices with one vertex from each set. Both authors were supported by NSERC Discovery grants. Additionally, T. Stephen was supported by DFG FG-468 and the Dynamical Systems research focus at the University of Magdeburg.     

Hardness and algorithms for rainbow connection

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In the first result of this paper we prove that computing rc(G) is NP-Hard solving an open problem from Caro et al. (Electron. J. Comb. 15, 2008, Paper R57). In fact, we prove that it is already NP-Complete to decide if rc(G)=2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ε>0, a connected graph with minimum degree at least ε n has bounded rainbow connection, where the bound depends only on ε, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.     

New Approximation Algorithms for the Steiner Tree Problems

The Steiner tree problem asks for the shortest tree connecting a given set of terminal points in a metric space. We design new approximation algorithms for the Steiner tree problems using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solutions. We achieve the best up to now approximation ratios of 1.644 in arbitrary metric and 1.267 in rectilinear plane, respectively.     

Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

Given a simple, undirected graph G=(V,E) and a weight function w:E→ℤ⁺, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1,k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1+1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2−1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a ratio of 3/2 for k=2 (note that this matches the inapproximability bound above), and (2−2/(k+1)) for any k≥3, respectively, in polynomial time.     

Minimum diameter cost-constrained Steiner trees

Given an edge-weighted undirected graph $G=(V,E,c,w)$ where each edge $e\in E$ has a cost $c(e)\ge 0$ and another weight $w(e)\ge 0$ , a set $S\subseteq V$ of terminals and a given constant $\mathrm{C}_0\ge 0$ , the aim is to find a minimum diameter Steiner tree whose all terminals appear as leaves and the cost of tree is bounded by $\mathrm{C}_0$ . The diameter of a tree refers to the maximum weight of the path connecting two different leaves in the tree. This problem is called the minimum diameter cost-constrained Steiner tree problem, which is NP-hard even when the topology of the Steiner tree is fixed. In this paper, we deal with the fixed-topology restricted version. We prove the restricted version to be polynomially solvable when the topology is not part of the input and propose a weakly fully polynomial time approximation scheme (weakly FPTAS) when the topology is part of the input, which can find a $(1+\epsilon )$ –approximation of the restricted version problem for any $\epsilon >0$ with a specific characteristic.     

Polynomial time approximation schemes for minimum disk cover problems

The following planar minimum disk cover problem is considered in this paper: given a set D\mathcal{D} of n disks and a set ℘ of m points in the Euclidean plane, where each disk covers a subset of points in ℘, to compute a subset of disks with minimum cardinality covering ℘. This problem is known to be NP-hard and an algorithm which approximates the optimal disk cover within a factor of (1+ε) in O(mn^{O(\frac1e2log2\frac1e)})\mathcal{O}(mn^{\mathcal{O}(\frac{1}{\epsilon^{2}}\log^{2}\frac{1}{\epsilon})}) time is proposed in this paper. This work presents the first polynomial time approximation scheme for the minimum disk cover problem where the best known algorithm can approximate the optimal solution with a large constant factor. Further, several variants of the minimum disk cover problem such as the incongruent disk cover problem and the weighted disk cover problem are considered and approximation schemes are designed.     

Some inverse min-max network problems under weighted l1 and l∞ norms with bound constraints on changes

We consider some inverse min-max (or max-min) network problems. Such an inverse problem is to modify the weights with bound constraints so that a given feasible solution becomes an optimal solution of a min-max (or max-min) network problem, and the deviation of the weights, measured by the weighted l ₁ norm or weighted l _∞ norm, is minimum. In this paper, we present strongly polynomial time algorithms to solve the inverse min-max spanning tree problem and the inverse maximum capacity path problem.     

Flattening topologically spherical surface

The problem of optimal surface flattening in 3-D finds many applications in engineering and manufacturing. However, previous algorithms for this problem are all heuristics without any quality guarantee and the computational complexity of the problem was not well understood. In this paper, we prove that the optimal surface flattening problem is NP-hard. Further, we show that the problem of flattening a topologically spherical surface admits a PTAS and can be solved by a (1+ε)-approximation algorithm in O(nlog n) time for any constant ε>0, where n is the input size of the problem.     

Approximation algorithms for the makespan minimization with positive tails on a single machine with a fixed non-availability interval

In this article, we consider the non-resumable case of the single machine scheduling problem with a fixed non-availability interval. We aim to minimize the makespan when every job has a positive tail. We propose a polynomial approximation algorithm with a worst-case performance ratio of 3/2 for this problem. We show that this bound is tight. We present a dynamic programming algorithm and we show that the problem has an FPTAS (Fully Polynomial Time Approximation Algorithm) by exploiting the well-known approach of Ibarra and Kim (J. ACM 22:463–468, 1975). Such an FPTAS is strongly polynomial. The obtained results outperform the previous polynomial approximation algorithms for this problem.