16 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Randomized rounding algorithms for large scale unsplittable flow problems
Unsplittable flow problems cover a wide range of telecommunication and
transportation problems and their efficient resolution is key to a number of
applications. In this work, we study algorithms that can scale up to large
graphs and important numbers of commodities. We present and analyze in detail a
heuristic based on the linear relaxation of the problem and randomized
rounding. We provide empirical evidence that this approach is competitive with
state-of-the-art resolution methods either by its scaling performance or by the
quality of its solutions. We provide a variation of the heuristic which has the
same approximation factor as the state-of-the-art approximation algorithm. We
also derive a tighter analysis for the approximation factor of both the
variation and the state-of-the-art algorithm. We introduce a new objective
function for the unsplittable flow problem and discuss its differences with the
classical congestion objective function. Finally, we discuss the gap in
practical performance and theoretical guarantees between all the aforementioned
algorithms
The Parameterised Complexity of Integer Multicommodity Flow
The Integer Multicommodity Flow problem has been studied extensively in the
literature. However, from a parameterised perspective, mostly special cases,
such as the Disjoint Paths problem, have been considered. Therefore, we
investigate the parameterised complexity of the general Integer Multicommodity
Flow problem. We show that the decision version of this problem on directed
graphs for a constant number of commodities, when the capacities are given in
unary, is XNLP-complete with pathwidth as parameter and XALP-complete with
treewidth as parameter. When the capacities are given in binary, the problem is
NP-complete even for graphs of pathwidth at most 13. We give related results
for undirected graphs. These results imply that the problem is unlikely to be
fixed-parameter tractable by these parameters.
In contrast, we show that the problem does become fixed-parameter tractable
when weighted tree partition width (a variant of tree partition width for edge
weighted graphs) is used as parameter
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
On the capacity provisioning on dynamic networks
In this thesis, we consider the development of algorithms suitable for designing evacuation
procedures in sparse or remote communities. The works are extensions of sink location
problems on dynamic networks, which are motivated by real-life disaster events such as
the Tohoku Japanese Tsunami, the Australian wildfire and many more. The available algorithms in this context consider the location of the sinks (safe-havens) with the assumptions
that the evacuation by foot is possible, which is reasonable when immediate evacuation
is needed in urban settings. However, for remote communities, emergency vehicles may
need to be dispatched or situated strategically for an efficient evacuation process. With
the assumption removed, our problems transform to the task of allocating capacities on
the edges of dynamic networks given a budget capacity c. We first of all consider this
problem on a dynamic path network of n vertices with the objective of minimizing the
completion time (minmax criterion) given that the position of the sink is known. This leads
to an O(nlogn + nlog(c/ξ)) time, where ξ is a refinement or precision parameter for an
additional binary search in the worst case scenario. Next, we extend the problem to star
topologies. The case where the sink is located at the middle of the star network follows
the same approach for the path network. However, when the sink is located on a leaf node,
the problem becomes more complicated when the number of links (edges) exceeds three.
The second phase of this thesis focuses on allocating capacities on the edges of dynamic
path networks with the objective of minimizing the total evacuation time (minsum criterion)
given the position of the sink and the budget (fixed) capacity. In general, minsum problems
are more difficult than minmax problems in the context of sink location problems. Due to
few combinatorial properties discovered together with the possibility of changing objective.
function configuration in the course of the optimization process, we consider the development of numerical procedure which involves the use of sequential quadratic programming
(SQP). The sequential quadratic programming employed allows the specification of an arbitrary initial capacities and also helps in monitoring the changing configuration of the
objective function. We propose to consider these problems on more complex topolgies
such as trees and general graph in future.NSERC Discovery Grants program.
University of Lethbridge Graduate Research Award.
Alberta Innovates Awar
Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems
Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard.
One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing.
We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include:
a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an O(log n) approximation factor, unless P=NP.
b) Polylogarithmic Bicriteria Approximations: Polynomial time (O(log^8 n), O(log^2 kappa)) bicritera approximation algorithms for the Confluent Quickest Flow problem where kappa is the number of sinks, in both directed and undirected graphs.
Corresponding results are also developed for the Confluent Maximum Flow over time problem. The techniques developed also improve recent approximation algorithms for static confluent flows