104 research outputs found
Strategy-Proof Facility Location for Concave Cost Functions
We consider k-Facility Location games, where n strategic agents report their
locations on the real line, and a mechanism maps them to k facilities. Each
agent seeks to minimize his connection cost, given by a nonnegative increasing
function of his distance to the nearest facility. Departing from previous work,
that mostly considers the identity cost function, we are interested in
mechanisms without payments that are (group) strategyproof for any given cost
function, and achieve a good approximation ratio for the social cost and/or the
maximum cost of the agents.
We present a randomized mechanism, called Equal Cost, which is group
strategyproof and achieves a bounded approximation ratio for all k and n, for
any given concave cost function. The approximation ratio is at most 2 for Max
Cost and at most n for Social Cost. To the best of our knowledge, this is the
first mechanism with a bounded approximation ratio for instances with k > 2
facilities and any number of agents. Our result implies an interesting
separation between deterministic mechanisms, whose approximation ratio for Max
Cost jumps from 2 to unbounded when k increases from 2 to 3, and randomized
mechanisms, whose approximation ratio remains at most 2 for all k. On the
negative side, we exclude the possibility of a mechanism with the properties of
Equal Cost for strictly convex cost functions. We also present a randomized
mechanism, called Pick the Loser, which applies to instances with k facilities
and n = k+1 agents, and for any given concave cost function, is strongly group
strategyproof and achieves an approximation ratio of 2 for Social Cost
On the Size and the Approximability of Minimum Temporally Connected Subgraphs
We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with
vertices and edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least and at most , for
any constant , where is the number of temporal edges and
is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and -approximable in cycles
Malleable Scheduling Beyond Identical Machines
In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e/(e-1), unless P = NP. On the positive side, we present polynomial-time algorithms with approximation ratios 2e/(e-1) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1+phi for unrelated speeds (phi is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the L_p norm of their speeds
Strategyproof Facility Location in Perturbation Stable Instances
We consider -Facility Location games, where strategic agents report
their locations on the real line, and a mechanism maps them to
facilities. Each agent seeks to minimize her distance to the nearest facility.
We are interested in (deterministic or randomized) strategyproof mechanisms
without payments that achieve a reasonable approximation ratio to the optimal
social cost of the agents. To circumvent the inapproximability of -Facility
Location by deterministic strategyproof mechanisms, we restrict our attention
to perturbation stable instances. An instance of -Facility Location on the
line is -perturbation stable (or simply, -stable), for some
, if the optimal agent clustering is not affected by moving any
subset of consecutive agent locations closer to each other by a factor at most
. We show that the optimal solution is strategyproof in
-stable instances whose optimal solution does not include any
singleton clusters, and that allocating the facility to the agent next to the
rightmost one in each optimal cluster (or to the unique agent, for singleton
clusters) is strategyproof and -approximate for -stable instances
(even if their optimal solution includes singleton clusters). On the negative
side, we show that for any and any , there is no
deterministic anonymous mechanism that achieves a bounded approximation ratio
and is strategyproof in -stable instances. We also prove
that allocating the facility to a random agent of each optimal cluster is
strategyproof and -approximate in -stable instances. To the best of our
knowledge, this is the first time that the existence of deterministic (resp.
randomized) strategyproof mechanisms with a bounded (resp. constant)
approximation ratio is shown for a large and natural class of -Facility
Location instances
Memoryless Algorithms for the Generalized -server Problem on Uniform Metrics
We consider the generalized -server problem on uniform metrics. We study
the power of memoryless algorithms and show tight bounds of on
their competitive ratio. In particular we show that the \textit{Harmonic
Algorithm} achieves this competitive ratio and provide matching lower bounds.
This improves the doubly-exponential bound of Chiplunkar and
Vishwanathan for the more general setting of uniform metrics with different
weights
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