92 research outputs found
Multi-commodity Flows and Cuts in Polymatroidal Networks
IMA, Google Corp., Microsoft Corp., Yandex Corp
Heuristics for Sparsest Cut Approximations in Network Flow Applications
The Maximum Concurrent Flow Problem (MCFP) is a polynomially bounded problem that has been used over the years in a variety of applications. Sometimes it is used to attempt to find the Sparsest Cut, an NP-hard problem, and other times to find communities in Social Network Analysis (SNA) in its hierarchical formulation, the HMCFP. Though it is polynomially bounded, the MCFP quickly grows in space utilization, rendering it useful on only small problems. When it was defined, only a few hundred nodes could be solved, where a few decades later, graphs of one to two thousand nodes can still be too much for modern commodity hardware to handle.
This dissertation covers three approaches to heuristics to the MCFP that run significantly faster in practice than the LP formulation with far less memory utilization. The first two approaches are based on the Maximum Adjacency Search (MAS) and apply to both the MCFP and the HMCFP used for community detection. We compare the three approaches to the LP performance in terms of accuracy, runtime, and memory utilization on several classes of synthetic graphs representing potential real-world applications. We find that the heuristics are often correct, and run using orders of magnitude less memory and time
On Routing Disjoint Paths in Bounded Treewidth Graphs
We study the problem of routing on disjoint paths in bounded treewidth graphs
with both edge and node capacities. The input consists of a capacitated graph
and a collection of source-destination pairs . The goal is to maximize the number of pairs that
can be routed subject to the capacities in the graph. A routing of a subset
of the pairs is a collection of paths such that,
for each pair , there is a path in
connecting to . In the Maximum Edge Disjoint Paths (MaxEDP) problem,
the graph has capacities on the edges and a routing
is feasible if each edge is in at most of
the paths of . The Maximum Node Disjoint Paths (MaxNDP) problem is
the node-capacitated counterpart of MaxEDP.
In this paper we obtain an approximation for MaxEDP on graphs of
treewidth at most and a matching approximation for MaxNDP on graphs of
pathwidth at most . Our results build on and significantly improve the work
by Chekuri et al. [ICALP 2013] who obtained an approximation
for MaxEDP
Maximum Edge-Disjoint Paths in -sums of Graphs
We consider the approximability of the maximum edge-disjoint paths problem
(MEDP) in undirected graphs, and in particular, the integrality gap of the
natural multicommodity flow based relaxation for it. The integrality gap is
known to be even for planar graphs due to a simple
topological obstruction and a major focus, following earlier work, has been
understanding the gap if some constant congestion is allowed.
In this context, it is natural to ask for which classes of graphs does a
constant-factor constant-congestion property hold. It is easy to deduce that
for given constant bounds on the approximation and congestion, the class of
"nice" graphs is nor-closed. Is the converse true? Does every proper
minor-closed family of graphs exhibit a constant factor, constant congestion
bound relative to the LP relaxation? We conjecture that the answer is yes.
One stumbling block has been that such bounds were not known for bounded
treewidth graphs (or even treewidth 3). In this paper we give a polytime
algorithm which takes a fractional routing solution in a graph of bounded
treewidth and is able to integrally route a constant fraction of the LP
solution's value. Note that we do not incur any edge congestion. Previously
this was not known even for series parallel graphs which have treewidth 2. The
algorithm is based on a more general argument that applies to -sums of
graphs in some graph family, as long as the graph family has a constant factor,
constant congestion bound. We then use this to show that such bounds hold for
the class of -sums of bounded genus graphs
Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing
An elementary h-route flow, for an integer h ≥ 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h ≤ 3: The size of a minimum h-route cut is at least f/h and at most O(log 3 k·f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h = 3 that has an approximation ratio of O(log 3 k). Previously, polylogarithmic approximation was known only for h-route cuts for h ≤ 2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem
- …