6,065 research outputs found
Integral Symmetric 2-Commodity Flows
Let be a symmetric digraph, and be a symmetric capacity. Let and . An integral symmetric 2-commodity flow in from to of value is an integral 4-commodity flow from to of value . The Integral Symmetric 2-Commodity Flow Problem consists in finding a symmetric 2-commodity flow from to of value such that . It is known that the Integral 2-Commodity Flow Problem is NP-complete for both directed and undirected graphs (\cite{FHW80} and \cite{EIS76}). We prove that the cut criterion is a necessary and sufficient condition for the existence of a solution to the Integral Symmetric 2-Commodity Flow Problem, and give a polynomial-time algorithm in that provides a solution to this problem. The time complexity of our algorithm is \textbf{}, where is the time complexity of your favorite flow algorithm (usually in )
New and simple algorithms for stable flow problems
Stable flows generalize the well-known concept of stable matchings to markets
in which transactions may involve several agents, forwarding flow from one to
another. An instance of the problem consists of a capacitated directed network,
in which vertices express their preferences over their incident edges. A
network flow is stable if there is no group of vertices that all could benefit
from rerouting the flow along a walk.
Fleiner established that a stable flow always exists by reducing it to the
stable allocation problem. We present an augmenting-path algorithm for
computing a stable flow, the first algorithm that achieves polynomial running
time for this problem without using stable allocation as a black-box
subroutine. We further consider the problem of finding a stable flow such that
the flow value on every edge is within a given interval. For this problem, we
present an elegant graph transformation and based on this, we devise a simple
and fast algorithm, which also can be used to find a solution to the stable
marriage problem with forced and forbidden edges.
Finally, we study the stable multicommodity flow model introduced by
Kir\'{a}ly and Pap. The original model is highly involved and allows for
commodity-dependent preference lists at the vertices and commodity-specific
edge capacities. We present several graph-based reductions that show
equivalence to a significantly simpler model. We further show that it is
NP-complete to decide whether an integral solution exists
Cut-Matching Games on Directed Graphs
We give O(log^2 n)-approximation algorithm based on the cut-matching
framework of [10, 13, 14] for computing the sparsest cut on directed graphs.
Our algorithm uses only O(log^2 n) single commodity max-flow computations and
thus breaks the multicommodity-flow barrier for computing the sparsest cut on
directed graph
Hardness Results for Structured Linear Systems
We show that if the nearly-linear time solvers for Laplacian matrices and
their generalizations can be extended to solve just slightly larger families of
linear systems, then they can be used to quickly solve all systems of linear
equations over the reals. This result can be viewed either positively or
negatively: either we will develop nearly-linear time algorithms for solving
all systems of linear equations over the reals, or progress on the families we
can solve in nearly-linear time will soon halt
Complexity and Approximation of the Continuous Network Design Problem
We revisit a classical problem in transportation, known as the continuous
(bilevel) network design problem, CNDP for short. We are given a graph for
which the latency of each edge depends on the ratio of the edge flow and the
capacity installed. The goal is to find an optimal investment in edge
capacities so as to minimize the sum of the routing cost of the induced Wardrop
equilibrium and the investment cost. While this problem is considered as
challenging in the literature, its complexity status was still unknown. We
close this gap showing that CNDP is strongly NP-complete and APX-hard, both on
directed and undirected networks and even for instances with affine latencies.
As for the approximation of the problem, we first provide a detailed analysis
for a heuristic studied by Marcotte for the special case of monomial latency
functions (Mathematical Programming, Vol.~34, 1986). Specifically, we derive a
closed form expression of its approximation guarantee for arbitrary sets S of
allowed latency functions. Second, we propose a different approximation
algorithm and show that it has the same approximation guarantee. As our final
-- and arguably most interesting -- result regarding approximation, we show
that using the better of the two approximation algorithms results in a strictly
improved approximation guarantee for which we give a closed form expression.
For affine latencies, e.g., this algorithm achieves a 1.195-approximation which
improves on the 5/4 that has been shown before by Marcotte. We finally discuss
the case of hard budget constraints on the capacity investment.Comment: 27 page
Routing Games with Progressive Filling
Max-min fairness (MMF) is a widely known approach to a fair allocation of
bandwidth to each of the users in a network. This allocation can be computed by
uniformly raising the bandwidths of all users without violating capacity
constraints. We consider an extension of these allocations by raising the
bandwidth with arbitrary and not necessarily uniform time-depending velocities
(allocation rates). These allocations are used in a game-theoretic context for
routing choices, which we formalize in progressive filling games (PFGs).
We present a variety of results for equilibria in PFGs. We show that these
games possess pure Nash and strong equilibria. While computation in general is
NP-hard, there are polynomial-time algorithms for prominent classes of
Max-Min-Fair Games (MMFG), including the case when all users have the same
source-destination pair. We characterize prices of anarchy and stability for
pure Nash and strong equilibria in PFGs and MMFGs when players have different
or the same source-destination pairs. In addition, we show that when a designer
can adjust allocation rates, it is possible to design games with optimal strong
equilibria. Some initial results on polynomial-time algorithms in this
direction are also derived
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}))
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