7 research outputs found
Fixed-Parameter Algorithms for Fair Hitting Set Problems
Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a fair version of Hitting Set. In the classical Hitting Set problem, the input is a universe ?, a family ? of subsets of ?, and a non-negative integer k. The goal is to determine whether there exists a subset S ? ? of size k that hits (i.e., intersects) every set in ?. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family ? of subsets of ?, where each subset in ? can be thought of as the group of elements of the same type. We want to find a set S ? ? of size k that (i) hits all sets of ?, and (ii) does not contain too many elements of each type. We call this problem Fair Hitting Set, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernels for Hitting Set
Fixed-Parameter Algorithms for Fair Hitting Set Problems
Selection of a group of representatives satisfying certain fairness
constraints, is a commonly occurring scenario. Motivated by this, we initiate a
systematic algorithmic study of a \emph{fair} version of \textsc{Hitting Set}.
In the classical \textsc{Hitting Set} problem, the input is a universe
, a family of subsets of , and a
non-negative integer . The goal is to determine whether there exists a
subset of size that \emph{hits} (i.e.,
intersects) every set in . Inspired by several recent works, we
formulate a fair version of this problem, as follows. The input additionally
contains a family of subsets of , where each subset
in can be thought of as the group of elements of the same
\emph{type}. We want to find a set of size that
(i) hits all sets of , and (ii) does not contain \emph{too many}
elements of each type. We call this problem \textsc{Fair Hitting Set}, and
chart out its tractability boundary from both classical as well as multivariate
perspective. Our results use a multitude of techniques from parameterized
complexity including classical to advanced tools, such as, methods of
representative sets for matroids, FO model checking, and a generalization of
best known kernels for \textsc{Hitting Set}
Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity
This paper settles the computational complexity of model checking of several
extensions of the monadic second order (MSO) logic on two classes of graphs:
graphs of bounded treewidth and graphs of bounded neighborhood diversity.
A classical theorem of Courcelle states that any graph property definable in
MSO is decidable in linear time on graphs of bounded treewidth. Algorithmic
metatheorems like Courcelle's serve to generalize known positive results on
various graph classes. We explore and extend three previously studied MSO
extensions: global and local cardinality constraints (CardMSO and MSO-LCC) and
optimizing the fair objective function (fairMSO).
First, we show how these extensions of MSO relate to each other in their
expressive power. Furthermore, we highlight a certain "linearity" of some of
the newly introduced extensions which turns out to play an important role.
Second, we provide parameterized algorithm for the aforementioned structural
parameters. On the side of neighborhood diversity, we show that combining the
linear variants of local and global cardinality constraints is possible while
keeping the linear (FPT) runtime but removing linearity of either makes this
impossible. Moreover, we provide a polynomial time (XP) algorithm for the most
powerful of studied extensions, i.e. the combination of global and local
constraints. Furthermore, we show a polynomial time (XP) algorithm on graphs of
bounded treewidth for the same extension. In addition, we propose a general
procedure of deriving XP algorithms on graphs on bounded treewidth via
formulation as Constraint Satisfaction Problems (CSP). This shows an alternate
approach as compared to standard dynamic programming formulations
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum