On Constrained Facility Location Problems

Given m facilities each with an opening cost, n demands, and distance between every demand and facility, the Facility Location problem finds a solution which opens some facilities to connect every demand to an opened facility such that the total cost of the solution is minimized. The k-Facility Location problem further requires that the number of opened facilities is at most k, where k is a parameter given in the instance of the problem. We consider the Facility Location problems satisfying that for every demand the ratio of the longest distance to facilities and the shortest distance to facilities is at most ω, where ω is a predefined constant. Using the local search approach with scaling technique and error control technique, for any arbitrarily small constant > 0, we give a polynomial-time approximation algorithm for the ω-constrained Facility Location problem with approximation ratio 1 + ω + 1 + ε, which significantly improves the previous best known ratio (ω + 1)/α for some 1≤α≤2, and a polynomial-time approximation algorithm for the ω-constrained k- Facility Location problem with approximation ratio ω+1+ε. On the aspect of approximation hardness, we prove that unless NP■DTIME(nO(loglogn)), the ω-constrained Facility Location problem cannot be approximated within 1 + lnω - 1, which slightly improves the previous best known hardness result 1.243 + 0.316ln(ω - 1). The experimental results on the standard test instances of Facility Location problem show that our algorithm also has good performance in practice.     

Approximation Algorithms for Requirement Cut on Graphs

In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to the requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X ₁,…,X _g ⊆V, with each group X _i having a requirement r _i between 0 and |X _i |. The goal is to find a minimum cost set of edges whose removal separates each group X _i into at least r _i disconnected components. We give an O(log n⋅log (gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log n⋅log (gR)). On trees, we obtain an improved guarantee of O(log (gR)). There is an Ω(log g) hardness of approximation for the requirement cut problem, even on trees.     

Chordal Deletion is Fixed-Parameter Tractable

It is known to be NP-hard to decide whether a graph can be made chordal by the deletion of k vertices or by the deletion of k edges. Here we present a uniformly polynomial-time algorithm for both problems: the running time is f(k)⋅n ^α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n ^k ) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices or edges to be deleted is fixed-parameter tractable. This answers an open question of Cai (Discrete Appl. Math. 127:415–429, 2003).     

A series of approximation algorithms for the acyclic directed steiner tree problem

Given an acyclic directed network, a subsetS of nodes (terminals), and a rootr, theacyclic directed Steiner tree problem requires a minimum-cost subnetwork which contains paths fromr to each terminal. It is known that unlessNP⊆DTIME[n ^polylogn ] no polynomial-time algorithm can guarantee better than (lnk)/4-approximation, wherek is the number of terminals. In this paper we give anO(k ^ε)-approximation algorithm for any ε>0. This result improves the previously knownk-approximation. This research was supported in part by Volkswagen-Stiftung and Packard Foundation.     

Exact Algorithms for the Bottleneck Steiner Tree Problem

Given n points, called terminals, in the plane ℝ² and a positive integer k, the bottleneck Steiner tree problem is to find k Steiner points from ℝ² and a spanning tree on the n+k points that minimizes its longest edge length. Edge length is measured by an underlying distance function on ℝ², usually, the Euclidean or the L ₁ metric. This problem is known to be NP-hard. In this paper, we study this problem in the L _p metric for any 1≤p≤∞, and aim to find an exact algorithm which is efficient for small fixed k. We present the first fixed-parameter tractable algorithm running in f(k)⋅nlog ² n time for the L ₁ and the L _∞ metrics, and the first exact algorithm for the L _p metric for any fixed rational p with 1<p<∞ whose time complexity is f(k)⋅(n ^k +nlog n), where f(k) is a function dependent only on k. Note that prior to this paper there was no known exact algorithm even for the L ₂ metric.     

Maximum Dispersion and Geometric Maximum Weight Cliques

We consider a facility location problem, where the objective is to “disperse” a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.     

Erratum: An Approximation Algorithm for Minimum-Cost Vertex-Connectivity Problems

Abstract. There is an error in our paper ``An Approximation Algorithm for Minimum-Cost Vertex- Connectivity Problems' (Algorithmica (1997), 18:21—43). In that paper we considered the following problem: given an undirected graph and values r _ij for each pair of vertices i and j , find a minimum-cost set of edges such that there are r _ij vertex-disjoint paths between vertices i and j . We gave approximation algorithms for two special cases of this problem. Our algorithms rely on a primal—dual approach which has led to approximation algorithms for many edge-connectivity problems. The algorithms work in a series of stages; in each stage an augmentation subroutine augments the connectivity of the current solution. The error is in a lemma for the proof of the performance guarantee of the augmentation subroutine. In the case r _ij = k for all i,j , we described a polynomial-time algorithm that claimed to output a solution of cost no more than 2 H (k) times optimal, where H = 1 + 1/2 + · · · + 1/n . This result is erroneous. We describe an example where our primal—dual augmentation subroutine, when augmenting a k -vertex connected graph to a (k+1) -vertex connected graph, gives solutions that are a factor Ω(k) away from the minimum. In the case r _ij ∈ {0,1,2} for all i,j , we gave a polynomial-time algorithm which outputs a solution of cost no more than three times the optimal. In this case we prove that the statement in the lemma that was erroneous for the k -vertex connected case does hold, and that the algorithm performs as claimed.     

Tight Results on Minimum Entropy Set Cover

In the minimum entropy set cover problem, one is given a collection of k sets which collectively cover an n-element ground set. A feasible solution of the problem is a partition of the ground set into parts such that each part is included in some of the k given sets. Such a partition defines a probability distribution, obtained by dividing each part size by n. The goal is to find a feasible solution minimizing the (binary) entropy of the corresponding distribution. Halperin and Karp have recently proved that the greedy algorithm always returns a solution whose cost is at most the optimum plus a constant. We improve their result by showing that the greedy algorithm approximates the minimum entropy set cover problem within an additive error of 1 nat =log ₂ e bits ≃1.4427 bits. Moreover, inspired by recent work by Feige, Lovász and Tetali on the minimum sum set cover problem, we prove that no polynomial-time algorithm can achieve a better constant, unless P = NP. We also discuss some consequences for the related minimum entropy coloring problem. G. Joret is a Research Fellow of the Fonds National de la Recherche Scientifique (FNRS).     

Approximation Algorithms for <Emphasis Type="Italic">k</Emphasis>-hurdle Problems

The polynomial-time solvable k-hurdle problem is a natural generalization of the classical s-t minimum cut problem where we must select a minimum-cost subset S of the edges of a graph such that |p∩S|≥k for every s-t path p. In this paper, we describe a set of approximation algorithms for “k-hurdle” variants of the NP-hard multiway cut and multicut problems. For the k-hurdle multiway cut problem with r terminals, we give two results, the first being a pseudo-approximation algorithm that outputs a (k−1)-hurdle solution whose cost is at most that of an optimal solution for k hurdles. Secondly, we provide a 2(1-\frac1r)2(1-\frac{1}{r})-approximation algorithm based on rounding the solution of a linear program, for which we give a simple randomized half-integrality proof that works for both edge and vertex k-hurdle multiway cuts that generalizes the half-integrality results of Garg et al. for the vertex multiway cut problem. We also describe an approximation-preserving reduction from vertex cover as evidence that it may be difficult to achieve a better approximation ratio than 2(1-\frac1r)2(1-\frac{1}{r}). For the k-hurdle multicut problem in an n-vertex graph, we provide an algorithm that, for any constant ε>0, outputs a ⌈(1−ε)k⌉-hurdle solution of cost at most O(log n) times that of an optimal k-hurdle solution, and we obtain a 2-approximation algorithm for trees.     

Approximation Algorithms for Bounded Degree Phylogenetic Roots

The Degree- Δ Closest Phylogenetic k th Root Problem (ΔCPR _k ) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) the degree of each internal node in T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, i.e., |E ⊕{{u,v} : u,v are leaves of T and d _T (u,v)≤k}|, is minimized, where d _T (u,v) denotes the distance between u and v in tree T. This problem arises from theoretical studies in evolutionary biology and generalizes several important combinatorial optimization problems such as the maximum matching problem. Unfortunately, it is known to be NP-hard for all fixed constants Δ,k such that either both Δ≥3 and k≥3, or Δ>3 and k=2. This paper presents a polynomial-time 8-approximation algorithm for Δ CPR ₂ for any fixed Δ>3, a quadratic-time 12-approximation algorithm for 3CPR ₃, and a polynomial-time approximation scheme for the maximization version of Δ CPR _k for any fixed Δ and k.     

Approximating Buy-at-Bulk and Shallow-Light <Emphasis Type="Italic">k</Emphasis>-Steiner Trees

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T⊆V including a particular vertex s called the root, and an integer k≤|T|. There are two cost functions on the edges of G, a buy cost b:E→ℝ⁺ and a distance cost r:E→ℝ⁺. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑ _e∈H b(e)+∑ _t∈T−s dist(t,s) is minimized, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log ⁴ n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n),O(log ³ n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least terminals. Using this we obtain an (O(log ² n),O(log ⁴ n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log ⁴ n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 677–686, 2006). A preliminary version of this paper appeared in the Proceedings of 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) 2006, LNCS 4110, pp. 153–163, 2006. M.T. Hajiaghayi supported in part by IPM under grant number CS1383-2-02. M.R. Salavatipour supported by NSERC grant No. G121210990, and a faculty start-up grant from University of Alberta.     

<Emphasis Type="Italic">f</Emphasis>-Sensitivity Distance Oracles and Routing Schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V,E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s,t,F), where s and t are vertices and F is a set of forbidden edges such that |F|≤f, returns an estimate of the distance between s and t in G(V,E∖F). For an integer parameter k≥1, the size of the data structure is O(fkn ^1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k−2)(f+1), and the query time is O(|F|⋅log ² n⋅log log n⋅log log d), where d is the distance between s and t in G(V,E∖F).     

A Fast Algorithm for the Path 2-Packing Problem

Let G be an undirected graph and $\mathcal{T}=\{T_{1},\ldots,T_{k}\}Let G be an undirected graph and T={T₁,?,T_k}\mathcal{T}=\{T_{1},\ldots,T_{k}\} be a collection of disjoint subsets of nodes. Nodes in T ₁∪⋅⋅⋅∪T _k are called terminals, other nodes are called inner. By a T\mathcal{T} -path we mean a path P such that P connects terminals from distinct sets in T\mathcal{T} and all internal nodes of P are inner. We study the problem of finding a maximum cardinality collection ℘ of T\mathcal{T} -paths such that at most two paths in ℘ pass through any node. Our algorithm is purely combinatorial and has the time complexity O(mn ²), where n and m denote the numbers of nodes and edges in G, respectively.     

Local Search Algorithms for the Red-Blue Median Problem

In this paper, we consider the following red-blue median problem which is a generalization of the well-studied k-median problem. The input consists of a set of red facilities, a set of blue facilities, and a set of clients in a metric space and two integers k _r ,k _b ≥0. The problem is to open at most k _r red facilities and at most k _b blue facilities and minimize the sum of distances of clients to their respective closest open facilities.     

Competitive Analysis of Scheduling Algorithms for Aggregated Links

We study an online job scheduling problem arising in networks with aggregated links. The goal is to schedule n jobs, divided into k disjoint chains, on m identical machines, without preemption, so that the jobs within each chain complete in the order of release times and the maximum flow time is minimized. We present a deterministic online algorithm with competitive ratio , and show a matching lower bound, even for randomized algorithms. The performance bound for we derive in the paper is, in fact, more subtle than a standard competitive ratio bound, and it shows that in overload conditions (when many jobs are released in a short amount of time), ’s performance is close to the optimum. We also show how to compute an offline solution efficiently for k=1, and that minimizing the maximum flow time for k,m≥2 is -hard. As by-products of our method, we obtain two offline polynomial-time algorithms for minimizing makespan: an optimal algorithm for k=1, and a 2-approximation algorithm for any k. W. Jawor and M. Chrobak supported by NSF grants OISE-0340752 and CCR-0208856. Work of C. Dürr conducted while being affiliated with the Laboratoire de Recherche en Informatique, Université Paris-Sud, 91405 Orsay. Supported by the CNRS/NSF grant 17171 and ANR Alpage.     

The Expressibility of First Order Dynamic Logic

This paper resolved an open problem proposed by A .P. Stolboushkin and M .A. Taitslin.We studied the expressibility of first order dynamic logic, and constructed infinite recursive programclasses K_1 , K_2, …, RG K_1 K_2 … RF, such that L (RG)相似文献   

An Approximation Algorithm for the Fault Tolerant Metric Facility Location Problem

We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to r _j facilities. We give the first non-trivial approximation algorithm for this problem, having an approximation guarantee of 3 · H _k , where k is the maximum requirement and H _k is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primal–dual algorithm of [5] for the metric facility location problem, reduces the maximum residual requirement by one.     

On Metric Clustering to Minimize the Sum of Radii

Given an n-point metric (P,d) and an integer k>0, we consider the problem of covering P by k balls so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in n ^{O(log n⋅log Δ)} time and returns with high probability the optimal solution. Here, Δ is the ratio between the maximum and minimum interpoint distances in the metric space. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and in metrics of constant doubling dimension.