Abstract. We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.
Given a function f in a finite field IF q of q elements, we define the functional graph of f as a directed graph on q nodes labelled by the elements of IF q where there is an edge from u to v if and only if f (u) = v. We obtain some theoretic estimates on the number of non-isomorphic graphs generated by all polynomials of a given degree. We then develop a simple and practical algorithm to test the isomorphism of quadratic polynomials that has linear memory and time complexities. Furthermore, we extend this isomorphism testing algorithm to the general case of functional graphs, and prove that, while its time complexity increases only slightly, its memory complexity remains linear. We exploit this algorithm to provide an upper bound on the number of functional graphs corresponding to polynomials of degree d over IF q . Finally, we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.2010 Mathematics Subject Classification. 05C20, 05C85, 11T06, 11T24.
Therapies consisting of a combination of agents are an attractive proposition,
especially in the context of diseases such as cancer, which can manifest with a
variety of tumor types in a single case. However uncovering usable drug
combinations is expensive both financially and temporally. By employing
computational methods to identify candidate combinations with a greater
likelihood of success we can avoid these problems, even when the amount of data
is prohibitively large. Hitting Set is a combinatorial problem
that has useful application across many fields, however as it is
NP-complete it is traditionally considered hard to solve
exactly. We introduce a more general version of the problem
(α,β,d)-Hitting Set,
which allows more precise control over how and what the hitting set targets.
Employing the framework of Parameterized Complexity we show that despite being
NP-complete, the
(α,β,d)-Hitting Set
problem is fixed-parameter tractable with a kernel of size O(αdkd) when we parameterize by the size k of the
hitting set and the maximum number α of the minimum number of hits,
and taking the maximum degree d of the target sets as a
constant. We demonstrate the application of this problem to multiple drug
selection for cancer therapy, showing the flexibility of the problem in
tailoring such drug sets. The fixed-parameter tractability result indicates that
for low values of the parameters the problem can be solved quickly using exact
methods. We also demonstrate that the problem is indeed practical, with
computation times on the order of 5 seconds, as compared to previous Hitting Set
applications using the same dataset which exhibited times on the order of 1 day,
even with relatively relaxed notions for what constitutes a low value for the
parameters. Furthermore the existence of a kernelization for
(α,β,d)-Hitting Set
indicates that the problem is readily scalable to large datasets.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.