24 research outputs found
Near-Optimal UGC-hardness of Approximating Max k-CSP_R
In this paper, we prove an almost-optimal hardness for Max -CSP based
on Khot's Unique Games Conjecture (UGC). In Max -CSP, we are given a set
of predicates each of which depends on exactly variables. Each variable can
take any value from . The goal is to find an assignment to
variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to
approximate Max -CSP to within factor for any . To the best of our knowledge, this result
improves on all the known hardness of approximation results when . In this case, the previous best hardness result was
NP-hardness of approximating within a factor by Chan. When , our result matches the best known UGC-hardness result of Khot, Kindler,
Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP by Kindler, Kolla
and Trevisan, we provide an -approximation algorithm
for Max -CSP. This algorithm implies that our inapproximability result
is tight up to a factor of . In comparison,
when is a constant, the previously known gap was , which is
significantly larger than our gap of .
Finally, we show that we can replace the Unique Games Conjecture assumption
with Khot's -to-1 Conjecture and still get asymptotically the same hardness
of approximation
Fundamental Limits on Communication for Oblivious Updates in Storage Networks
In distributed storage systems, storage nodes intermittently go offline for
numerous reasons. On coming back online, nodes need to update their contents to
reflect any modifications to the data in the interim. In this paper, we
consider a setting where no information regarding modified data needs to be
logged in the system. In such a setting, a 'stale' node needs to update its
contents by downloading data from already updated nodes, while neither the
stale node nor the updated nodes have any knowledge as to which data symbols
are modified and what their value is. We investigate the fundamental limits on
the amount of communication necessary for such an "oblivious" update process.
We first present a generic lower bound on the amount of communication that is
necessary under any storage code with a linear encoding (while allowing
non-linear update protocols). This lower bound is derived under a set of
extremely weak conditions, giving all updated nodes access to the entire
modified data and the stale node access to the entire stale data as side
information. We then present codes and update algorithms that are optimal in
that they meet this lower bound. Next, we present a lower bound for an
important subclass of codes, that of linear Maximum-Distance-Separable (MDS)
codes. We then present an MDS code construction and an associated update
algorithm that meets this lower bound. These results thus establish the
capacity of oblivious updates in terms of the communication requirements under
these settings.Comment: IEEE Global Communications Conference (GLOBECOM) 201
Tracking the l_2 Norm with Constant Update Time
The l_2 tracking problem is the task of obtaining a streaming algorithm that, given access to a stream of items a_1,a_2,a_3,... from a universe [n], outputs at each time t an estimate to the l_2 norm of the frequency vector f^{(t)}in R^n (where f^{(t)}_i is the number of occurrences of item i in the stream up to time t). The previous work [Braverman-Chestnut-Ivkin-Nelson-Wang-Woodruff, PODS 2017] gave a streaming algorithm with (the optimal) space using O(epsilon^{-2}log(1/delta)) words and O(epsilon^{-2}log(1/delta)) update time to obtain an epsilon-accurate estimate with probability at least 1-delta. We give the first algorithm that achieves update time of O(log 1/delta) which is independent of the accuracy parameter epsilon, together with the nearly optimal space using O(epsilon^{-2}log(1/delta)) words. Our algorithm is obtained using the Count Sketch of [Charilkar-Chen-Farach-Colton, ICALP 2002]