6 research outputs found
Improved Bi-criteria Approximation for the All-or-Nothing Multicommodity Flow Problem in Arbitrary Networks
This paper addresses the following fundamental maximum throughput routing
problem: Given an arbitrary edge-capacitated -node directed network and a
set of commodities, with source-destination pairs and demands
, admit and route the largest possible number of commodities -- i.e.,
the maximum {\em throughput} -- to satisfy their demands. The main
contributions of this paper are two-fold: First, we present a bi-criteria
approximation algorithm for this all-or-nothing multicommodity flow (ANF)
problem. Our algorithm is the first to achieve a {\em constant approximation of
the maximum throughput} with an {\em edge capacity violation ratio that is at
most logarithmic in }, with high probability. Our approach is based on a
version of randomized rounding that keeps splittable flows, rather than
approximating those via a non-splittable path for each commodity: This allows
our approach to work for {\em arbitrary directed edge-capacitated graphs},
unlike most of the prior work on the ANF problem. Our algorithm also works if
we consider the weighted throughput, where the benefit gained by fully
satisfying the demand for commodity is determined by a given weight
. Second, we present a derandomization of our algorithm that maintains
the same approximation bounds, using novel pessimistic estimators for
Bernstein's inequality. In addition, we show how our framework can be adapted
to achieve a polylogarithmic fraction of the maximum throughput while
maintaining a constant edge capacity violation, if the network capacity is
large enough. One important aspect of our randomized and derandomized
algorithms is their {\em simplicity}, which lends to efficient implementations
in practice
Packing Directed Cycles Quarter- and Half-Integrally
The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph
that does not admit a family of vertex-disjoint cycles contains a feedback
vertex set (a set of vertices hitting all cycles in the graph) of size . After being known for long as Younger's conjecture, a similar
statement for directed graphs has been proven in 1996 by Reed, Robertson,
Seymour, and Thomas. However, in their proof, the dependency of the size of the
feedback vertex set on the size of vertex-disjoint cycle packing is not
elementary.
We show that if we compare the size of a minimum feedback vertex set in a
directed graph with the quarter-integral cycle packing number, we obtain a
polynomial bound. More precisely, we show that if in a directed graph there
is no family of cycles such that every vertex of is in at most four of
the cycles, then there exists a feedback vertex set in of size .
Furthermore, a variant of our proof shows that if in a directed graph there
is no family of cycles such that every vertex of is in at most two of
the cycles, then there exists a feedback vertex set in of size .
On the way there we prove a more general result about quarter-integral
packing of subgraphs of high directed treewidth: for every pair of positive
integers and , if a directed graph has directed treewidth
, then one can find in a family of
subgraphs, each of directed treewidth at least , such that every vertex of
is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19
Maximizing Routing Throughput with Applications to Delay Tolerant Networks
abstract: Many applications require efficient data routing and dissemination in Delay Tolerant Networks (DTNs) in order to maximize the throughput of data in the network, such as providing healthcare to remote communities, and spreading related information in Mobile Social Networks (MSNs). In this thesis, the feasibility of using boats in the Amazon Delta Riverine region as data mule nodes is investigated and a robust data routing algorithm based on a fountain code approach is designed to ensure fast and timely data delivery considering unpredictable boat delays, break-downs, and high transmission failures. Then, the scenario of providing healthcare in Amazon Delta Region is extended to a general All-or-Nothing (Splittable) Multicommodity Flow (ANF) problem and a polynomial time constant approximation algorithm is designed for the maximum throughput routing problem based on a randomized rounding scheme with applications to DTNs. In an MSN, message content is closely related to users’ preferences, and can be used to significantly impact the performance of data dissemination. An interest- and content-based algorithm is developed where the contents of the messages, along with the network structural information are taken into consideration when making message relay decisions in order to maximize data throughput in an MSN. Extensive experiments show the effectiveness of the above proposed data dissemination algorithm by comparing it with state-of-the-art techniques.Dissertation/ThesisDoctoral Dissertation Computer Science 201
Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing
Let be a weighted, digraph subject to a sequence of adversarial
edge deletions. In the decremental single-source reachability problem (SSR), we
are given a fixed source and the goal is to maintain a data structure that
can answer path-queries for any . In the more
general single-source shortest paths (SSSP) problem the goal is to return an
approximate shortest path to , and in the SCC problem the goal is to
maintain strongly connected components of and to answer path queries within
each component. All of these problems have been very actively studied over the
past two decades, but all the fast algorithms are randomized and, more
significantly, they can only answer path queries if they assume a weaker model:
they assume an oblivious adversary which is not adaptive and must fix the
update sequence in advance. This assumption significantly limits the use of
these data structures, most notably preventing them from being used as
subroutines in static algorithms. All the above problems are notoriously
difficult in the adaptive setting. In fact, the state-of-the-art is still the
Even and Shiloach tree, which dates back all the way to 1981 and achieves total
update time . We present the first algorithms to break through this
barrier:
1) deterministic decremental SSR/SCC with total update time
2) deterministic decremental SSSP with total update time .
To achieve these results, we develop two general techniques of broader
interest for working with dynamic graphs: 1) a generalization of expander-based
tools to dynamic directed graphs, and 2) a technique that we call congestion
balancing and which provides a new method for maintaining flow under
adversarial deletions. Using the second technique, we provide the first
near-optimal algorithm for decremental bipartite matching.Comment: Reuploaded with some generalizations of previous theorem