1,040,723 research outputs found
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, are FTP when
parameterized by the size of the edge set modification. However polynomial
kernel was known for none of these three problems. For each of them, we provide
cubic kernels that can be computed in linear time for each of these problems.
We thereby answer an open problem first mentioned by Dom, Guo, Huffner and
Niedermeier (2005).Comment: Submitte
Approximating minimum power covers of intersecting families and directed edge-connectivity problems
AbstractGiven a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Let G=(V,E) be a graph with edge costs {c(e):e∈E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r∈V, and require that G contains k pairwise edge-disjoint rv-paths for all v∈V−r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k=1, these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln|V|) approximation threshold. For k=Ω(nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O(kln|V|). Our algorithms are based on a more general O(ln|V|)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X∩Y,X∪Y∈F for any intersecting X,Y∈F, and an edge set I covers F if for every X∈F there is an edge in I entering X
A Primal-Dual Based Power Control Approach for Capacitated Edge Servers
The intensity of radio waves decays rapidly with increasing propagation
distance, and an edge server's antenna needs more power to form a larger signal
coverage area. Therefore, the power of the edge server should be controlled to
reduce energy consumption. In addition, edge servers with capacitated resources
provide services for only a limited number of users to ensure the quality of
service (QoS). We set the signal transmission power for the antenna of each
edge server and formed a signal disk, ensuring that all users were covered by
the edge server signal and minimizing the total power of the system. This
scenario is a typical geometric set covering problem, and even simple cases
without capacity limits are NP-hard problems. In this paper, we propose a
primal-dual-based algorithm and obtain an -approximation result. We compare
our algorithm with two other algorithms through simulation experiments. The
results show that our algorithm obtains a result close to the optimal value in
polynomial time
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