31,604 research outputs found
The Maximum Traveling Salesman Problem with Submodular Rewards
In this paper, we look at the problem of finding the tour of maximum reward
on an undirected graph where the reward is a submodular function, that has a
curvature of , of the edges in the tour. This problem is known to be
NP-hard. We analyze two simple algorithms for finding an approximate solution.
Both algorithms require oracle calls to the submodular function. The
approximation factors are shown to be and
, respectively; so the second
method has better bounds for low values of . We also look at how these
algorithms perform for a directed graph and investigate a method to consider
edge costs in addition to rewards. The problem has direct applications in
monitoring an environment using autonomous mobile sensors where the sensing
reward depends on the path taken. We provide simulation results to empirically
evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy
bound with curvature
Distance Oracles for Time-Dependent Networks
We present the first approximate distance oracle for sparse directed networks
with time-dependent arc-travel-times determined by continuous, piecewise
linear, positive functions possessing the FIFO property.
Our approach precomputes approximate distance summaries from
selected landmark vertices to all other vertices in the network. Our oracle
uses subquadratic space and time preprocessing, and provides two sublinear-time
query algorithms that deliver constant and approximate
shortest-travel-times, respectively, for arbitrary origin-destination pairs in
the network, for any constant . Our oracle is based only on
the sparsity of the network, along with two quite natural assumptions about
travel-time functions which allow the smooth transition towards asymmetric and
time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of
EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An
extended abstract also appeared in the 41st International Colloquium on
Automata, Languages, and Programming (ICALP 2014, track-A
Integrals of motion in the Many-Body localized phase
We construct a complete set of quasi-local integrals of motion for the
many-body localized phase of interacting fermions in a disordered potential.
The integrals of motion can be chosen to have binary spectrum , thus
constituting exact quasiparticle occupation number operators for the Fermi
insulator. We map the problem onto a non-Hermitian hopping problem on a lattice
in operator space. We show how the integrals of motion can be built, under
certain approximations, as a convergent series in the interaction strength. An
estimate of its radius of convergence is given, which also provides an estimate
for the many-body localization-delocalization transition. Finally, we discuss
how the properties of the operator expansion for the integrals of motion imply
the presence or absence of a finite temperature transition.Comment: 65 pages, 12 figures. Corrected typos, added reference
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