9 research outputs found
Integrality Gap of Time-Indexed Linear Programming Relaxation for Coflow Scheduling
Coflow is a set of related parallel data flows in a network. The goal of the coflow scheduling is to process all the demands of the given coflows while minimizing the weighted completion time. It is known that the coflow scheduling problem admits several polynomial-time 5-approximation algorithms that compute solutions by rounding linear programming (LP) relaxations of the problem. In this paper, we investigate the time-indexed LP relaxation for coflow scheduling. We show that the integrality gap of the time-indexed LP relaxation is at most 4. We also show that yet another polynomial-time 5-approximation algorithm can be obtained by rounding the solutions to the time-indexed LP relaxation
Finding Independent Transversals Efficiently
Let G be a graph and (V_1,...,V_m) be a vertex partition of G. An independent transversal (IT) of G with respect to (V_1,...,V_m) is an independent set {v_1,...,v_m} in G such that v_i is in V_i for each i in {1,...,m}.
There exist various theorems that give sufficient conditions for the existence of ITs. These theorems have been used to solve problems in graph theory (e.g. list colouring, strong colouring, delay edge colouring, circular colouring, various graph partitioning and special independent set problems), hypergraphs (e.g. hypergraph matching), group theory (e.g. generators in linear groups), and theoretical computer science (e.g. job scheduling and other resource allocation problems). However, the proofs of the existence theorems that give the best possible bounds do not provide efficient algorithms for finding an IT. In this
thesis, we give poly-time algorithms for finding an IT under certain conditions and some applications, while weakening the original theorems only slightly. We also give e fficient poly-time algorithms for finding partial ITs and ITs of large weight in vertex-weighted graphs, as well as an application of these weighted results
Placements of virtual network functions for effective network functions virtualization
In the future wireless networks, network function virtualization will lay the foun-
dation for establishing a new resource management framework to e ciently utilize
network resources. The rst part of this thesis deals in the minimization of the to-
tal latency for a network and how to solve it e ciently. A model of users, Virtual
Network Functions (vNFs) and hosting devices have been considered and was used
to nd the minimum latency using Integer Linear Programming (ILP). The problem
is NP-hard and takes exponential time to solve in the worst case. A Stable Matching
based heuristic has been proposed to solve the problem in polynomial time and then
the local search is utilized to improve the e ciency of the result.
The second part of this thesis proposes the problem of fair allocation of the vNFs to
hosting devices. A mathematical programming based model (ILP) has been designed
to solve the problem which takes exponential time to solve in the worst case, due
to its NP-hard nature. Thus an heuristic approach has been provided to solve the
problem in polynomial time
Finding Perfect Matchings in Bipartite Hypergraphs
Haxell's condition [14] is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell's condition is satisfied.We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of Haxell's condition holds. Our algorithm can be seen as a generalization of the classical Hungarian algorithm for finding perfect matchings in bipartite graphs. The techniques we use to achieve this result could be of use more generally in other combinatorial problems on hypergraphs where disjointness structure is crucial, e.g., Set Packin