Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Complexity of Partial Function Extension for Coverage Functions   Coverage functions are an important subclass of submodular functions, finding
applications in machine learning, game theory, social networks, and facility
location. We study the complexity of partial function extension to coverage
functions. That is, given a partial function consisting of a family of subsets
of $[m]$ and a value at each point, does there exist a coverage function
defined on all subsets of $[m]$ that extends this partial function? Partial
function extension is previously studied for other function classes, including
boolean functions and convex functions, and is useful in many fields, such as
obtaining bounds on learning these function classes.
  We show that determining extendibility of a partial function to a coverage
function is NP-complete, establishing in the process that there is a
polynomial-sized certificate of extendibility. The hardness also gives us a
lower bound for learning coverage functions. We then study two natural notions
of approximate extension, to account for errors in the data set. The two
notions correspond roughly to multiplicative point-wise approximation and
additive $L_1$ approximation. We show upper and lower bounds for both notions
of approximation. In the second case we obtain nearly tight bounds.

23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc