In a column-restricted covering integer program (CCIP), all the non-zero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the integrality gaps of the underlying 0,1-CIP. If the underlying 0,1-CIP has an integrality gap O(γ), and assuming that the integrality gap of the priority version of the 0,1-CIP is O(ω), we give a factor O(γ+ω) approximation algorithm for the CCIP. Priority versions of 0,1-CIPs (PCIPs) naturally capture quality of service type constraints in a covering problem. We investigate priority versions of the line (PLC) and the (rooted) tree cover (PTC) problems. Apart from being natural objects to study, these problems fall in a class of fundamental geometric covering problems. We bound the integrality of certain classes of this PCIP by a constant. Algorithmically, we give a polytime exact algorithm for PLC, show that the PTC problem is APX-hard, and give a factor 2-approximation algorithm for it.
The only work on priority versions of covering problems that we are aware of is due to Charikar, Naor and Schieber @cite_8 who studied the priority Steiner tree and forest problems in the context of QoS management in a network multicasting application. present a @math -approximation algorithm for the problem, and @cite_2 later show that no efficient @math approximation algorithm can exist unless @math ( @math is the number of vertices).
@cite_2 @cite_8
2009976792 2167240010 1528210381 2949832116
We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)-approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i?1)k1 i in time O(nik2i) for any fixed i1, where k is the number of terminals. Thus, an O(k?) approximation ratio can be achieved in polynomial time for any fixed ?0. Setting i=logk, we obtain an O(log2k) approximation ratio in quasi-polynomial time. For the directed generalized Steiner network problem we give an algorithm that achieves an approximation ratio of O(k2 3log1 3k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner tree problem, the set TSP problem, and several others in both directed and undirected graphs can be reduced in an approximation preserving fashion to the directed Steiner tree problem. Thus, we obtain the first nontrivial approximations to those as well. All these problems are known to be as hard as the Set cover problem to approximate. We study two-stage robustvariants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by [1], [6], and [4], are two-stage planning problems in which the requirements are revealed after some decisions are taken in stage one. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust Steiner tree problem, for example, one buys some edges in stage one after which some terminals are revealed. In the second stage, one has to buy more edges, at a higher cost, to complete the stage one solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially manyscenarios given implicitly. A scenario consists of any subset of kterminals (for Steiner tree), or any subset of kterminal-pairs (for Steiner forest), or any subset of kclients (for facility location). We present the first constant-factor approximation algorithms for the robust Steiner tree and robust uncapacitated facility location problems. For the robust Steiner forest problem with uniform inflation, we present an O(logn)-approximation and show that the problem with two inflation factors is impossible to approximate within O(log1 2 i¾? i¾?n) factor, for any constant i¾?> 0, unless NP has randomized quasi-polynomial time algorithms. Finally, we show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question by [1] and [6]. This paper considers the Steiner tree problem in the model of two-stage stochastic optimization with recourse. This model, the focus of much recent research [11, 16, 8, 18], tries to capture the fact that many infrastructure planning problems have to be solved in the presence of uncertainty, and that we have make decisions knowing merely market forecasts (and not the precise set of demands); by the time the actual demands arrive, the costs may be higher due to inflation. In the context of the Stochastic Steiner Tree problem on a graph G = (V,E), the model can be paraphrased thus: on Monday, we are given a probability distribution π on subsets of vertices, and can build some subset EM of edges. On Tuesday, a set of terminals D materializes (drawn from the same distribution π). We now have to buy edges ET so that the set EM ∪ ET forms a Steiner tree on D. The goal is to minimize the expected cost of the solution. We give the first constant-factor approximation algorithm for this problem. To the best of our knowledge, this is the first O(1)-approximation for the stochastic version of a non sub-additive problem. In fact, algorithms for the unrooted stochastic Steiner tree problem we consider are powerful enough to solve the Multicommodity Rent-or-Buy problem, itself a topic of recent interest [3, 7, 15]. In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an @math -vertex graph @math with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set @math , which changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). Our task here is twofold. We want to support updates in sublinear @math time, and keep the approximation factor of the algorithm as small as possible. We show that we can maintain a @math -approximate Steiner tree of a general graph in @math time per terminal addition or removal. Here, @math denotes the stretch of the metric induced by @math . For planar graphs we achieve the same running time and the approximation ratio of @math . Moreover, we show faster algorithms for incremental and decremental scenarios. Finally, we show that if we allow higher approximation ratio, even more efficient algorithms are possible. In particular we show a polylogarithmic time @math -approximate algorithm for planar graphs. One of the main building blocks of our algorithms are dynamic distance oracles for vertex-labeled graphs, which are of independent interest. We also improve and use the online algorithms for the Steiner tree problem.