Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling @math maximizing the number of forward edges, i.e. edges (u,v) such that @math (u) 0 @math @math n^ (1 k ) @math S_v Z ^+ @math 4 2 ( 2 +1) 2.344 @math 2 2 2.828 @math k [n]$.
For the vertex deletion version of , @cite_10 gave linear time and quadratic time algorithms for rooted trees and series-parallel graphs respectively. The problem reduces to vertex cover on @math -uniform hypergraphs for any constant @math thereby admitting a @math -approximation, and a matching @math -inapproximability assuming the UGC was obtained by Svensson @cite_3 .
@cite_10 @cite_3
2949832116 2950822364 2079297844 2091147489
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. Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer @math and arbitrary small @math , the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor @math even on graphs where the vertices can be almost partitioned into @math solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations. Examines the vertex deletion problem for weighted directed acyclic graphs (WDAGs). The objective is to delete the fewest number of vertices so that the resulting WDAG has no path of length > spl delta . Several simplified versions of this problem are shown to be NP-hard. However, the problem is solved in linear time when the WDAG is a rooted tree, and in quadratic time when the WDAG is a series-parallel graph. > For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the cyclic cover time is @math , where @math is the number of vertices in the graph, @math is the average degree of its vertices, and @math is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most @math .