292,731 research outputs found
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
Truthful Assignment without Money
We study the design of truthful mechanisms that do not use payments for the
generalized assignment problem (GAP) and its variants. An instance of the GAP
consists of a bipartite graph with jobs on one side and machines on the other.
Machines have capacities and edges have values and sizes; the goal is to
construct a welfare maximizing feasible assignment. In our model of private
valuations, motivated by impossibility results, the value and sizes on all
job-machine pairs are public information; however, whether an edge exists or
not in the bipartite graph is a job's private information.
We study several variants of the GAP starting with matching. For the
unweighted version, we give an optimal strategyproof mechanism; for maximum
weight bipartite matching, however, we show give a 2-approximate strategyproof
mechanism and show by a matching lowerbound that this is optimal. Next we study
knapsack-like problems, which are APX-hard. For these problems, we develop a
general LP-based technique that extends the ideas of Lavi and Swamy to reduce
designing a truthful mechanism without money to designing such a mechanism for
the fractional version of the problem, at a loss of a factor equal to the
integrality gap in the approximation ratio. We use this technique to obtain
strategyproof mechanisms with constant approximation ratios for these problems.
We then design an O(log n)-approximate strategyproof mechanism for the GAP by
reducing, with logarithmic loss in the approximation, to our solution for the
value-invariant GAP. Our technique may be of independent interest for designing
truthful mechanisms without money for other LP-based problems.Comment: Extended abstract appears in the 11th ACM Conference on Electronic
Commerce (EC), 201
On robust network coding subgraph construction under uncertainty
We consider the problem of network coding subgraph
construction in networks where there is uncertainty about
link loss rates. For a given set of scenarios specified by an uncertainty
set of link loss rates, we provide a robust optimization-based
formulation to construct a single subgraph that would work
relatively well across all scenarios. We show that this problem
is coNP-hard in general for both objectives: minimizing cost
of subgraph construction and maximizing throughput given a
cost constraint. To solve the problem tractably, we approximate
the problem by introducing path constraints, which results
in polynomial time-solvable solution in terms of the problem
size. The simulation results show that the robust optimization
solution is better and more stable than the deterministic solution
in terms of worst-case performance. From these results, we
compare the tractability of robust network design problems with
different uncertain network components and different problem
formulations
Path deviations outperform approximate stability in heterogeneous congestion games
We consider non-atomic network congestion games with heterogeneous players
where the latencies of the paths are subject to some bounded deviations. This
model encompasses several well-studied extensions of the classical Wardrop
model which incorporate, for example, risk-aversion, altruism or travel time
delays. Our main goal is to analyze the worst-case deterioration in social cost
of a perturbed Nash flow (i.e., for the perturbed latencies) with respect to an
original Nash flow. We show that for homogeneous players perturbed Nash flows
coincide with approximate Nash flows and derive tight bounds on their
inefficiency. In contrast, we show that for heterogeneous populations this
equivalence does not hold. We derive tight bounds on the inefficiency of both
perturbed and approximate Nash flows for arbitrary player sensitivity
distributions. Intuitively, our results suggest that the negative impact of
path deviations (e.g., caused by risk-averse behavior or latency perturbations)
is less severe than approximate stability (e.g., caused by limited
responsiveness or bounded rationality). We also obtain a tight bound on the
inefficiency of perturbed Nash flows for matroid congestion games and
homogeneous populations if the path deviations can be decomposed into edge
deviations. In particular, this provides a tight bound on the Price of
Risk-Aversion for matroid congestion games
Randomized parallel approximations to max flow
The final publication is available at link.springer.comPeer ReviewedPostprint (author's final draft
Graph Orientation and Flows Over Time
Flows over time are used to model many real-world logistic and routing
problems. The networks underlying such problems -- streets, tracks, etc. -- are
inherently undirected and directions are only imposed on them to reduce the
danger of colliding vehicles and similar problems. Thus the question arises,
what influence the orientation of the network has on the network flow over time
problem that is being solved on the oriented network. In the literature, this
is also referred to as the contraflow or lane reversal problem.
We introduce and analyze the price of orientation: How much flow is lost in
any orientation of the network if the time horizon remains fixed? We prove that
there is always an orientation where we can still send of the
flow and this bound is tight. For the special case of networks with a single
source or sink, this fraction is which is again tight. We present
more results of similar flavor and also show non-approximability results for
finding the best orientation for single and multicommodity maximum flows over
time
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