7,815 research outputs found
A strongly polynomial algorithm for generalized flow maximization
A strongly polynomial algorithm is given for the generalized flow
maximization problem. It uses a new variant of the scaling technique, called
continuous scaling. The main measure of progress is that within a strongly
polynomial number of steps, an arc can be identified that must be tight in
every dual optimal solution, and thus can be contracted. As a consequence of
the result, we also obtain a strongly polynomial algorithm for the linear
feasibility problem with at most two nonzero entries per column in the
constraint matrix.Comment: minor correction
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
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine
learning: approximating a submodular function everywhere, learning a submodular
function (in a PAC-like setting [53]), and constrained minimization of
submodular functions. We show that the complexity of all three problems depends
on the 'curvature' of the submodular function, and provide lower and upper
bounds that refine and improve previous results [3, 16, 18, 52]. Our proof
techniques are fairly generic. We either use a black-box transformation of the
function (for approximation and learning), or a transformation of algorithms to
use an appropriate surrogate function (for minimization). Curiously, curvature
has been known to influence approximations for submodular maximization [7, 55],
but its effect on minimization, approximation and learning has hitherto been
open. We complete this picture, and also support our theoretical claims by
empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201
Shortest path and maximum flow problems in planar flow networks with additive gains and losses
In contrast to traditional flow networks, in additive flow networks, to every
edge e is assigned a gain factor g(e) which represents the loss or gain of the
flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is
less than the designated capacity of e, then f(e) + g(e) = 0 units of flow
reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this
report we study the maximum flow problem in additive flow networks, which we
prove to be NP-hard even when the underlying graphs of additive flow networks
are planar. We also investigate the shortest path problem, when to every edge e
is assigned a cost value for every unit flow entering edge e, which we show to
be NP-hard in the strong sense even when the additive flow networks are planar
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