224 research outputs found
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
Probabilistic Spectral Sparsification In Sublinear Time
In this paper, we introduce a variant of spectral sparsification, called
probabilistic -spectral sparsification. Roughly speaking,
it preserves the cut value of any cut with an
multiplicative error and a additive error. We show how
to produce a probabilistic -spectral sparsifier with
edges in time
time for unweighted undirected graph. This gives fastest known sub-linear time
algorithms for different cut problems on unweighted undirected graph such as
- An time -approximation
algorithm for the sparsest cut problem and the balanced separator problem.
- A time approximation minimum s-t cut algorithm
with an additive error
Extensions and limits to vertex sparsification
Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, E[subscript H]) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there exist flow sparsifiers that simultaneously preserve the congestion of all multicommodity flows within an O(log k / log log k)-factor where |K| = k. This bound improves to O(1) if G excludes any fixed minor. This is a strengthening of previous results, which consider the problem of finding a graph H = (K, E[subscript H]) (a cut sparsifier) that approximately preserves the value of minimum cuts separating any partition of the terminals. Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers). Thereby, we immediately improve all approximation ratios derived using vertex sparsification in [14].
We also prove an Ω(log log k) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. The proof of this theorem relies on a technique (which we refer to as oblivious dual certifcates) for proving super-constant congestion lower bounds against many multicommodity flows at once. Our result implies that approximation algorithms for multicommodity flow-type problems designed by a black box reduction to a "uniform" case on k nodes (see [14] for examples) must incur a super-constant cost in the approximation ratio
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
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