24 research outputs found
Approximating the MaxCover Problem with Bounded Frequencies in FPT Time
We study approximation algorithms for several variants of the MaxCover
problem, with the focus on algorithms that run in FPT time. In the MaxCover
problem we are given a set N of elements, a family S of subsets of N, and an
integer K. The goal is to find up to K sets from S that jointly cover (i.e.,
include) as many elements as possible. This problem is well-known to be NP-hard
and, under standard complexity-theoretic assumptions, the best possible
polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We
first consider a variant of MaxCover with bounded element frequencies, i.e., a
variant where there is a constant p such that each element belongs to at most p
sets in S. For this case we show that there is an FPT approximation scheme
(i.e., for each B there is a B-approximation algorithm running in FPT time) for
the problem of maximizing the number of covered elements, and a randomized FPT
approximation scheme for the problem of minimizing the number of elements left
uncovered (we take K to be the parameter). Then, for the case where there is a
constant p such that each element belongs to at least p sets from S, we show
that the standard greedy approximation algorithm achieves approximation ratio
exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted
variant of MaxCover, and show approximation algorithms that run in exponential
time and combine an exact algorithm with a greedy approximation. Some of our
results improve currently known results for MaxVertexCover
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Parameterized Matroid-Constrained Maximum Coverage
In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended.
We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid ? = (V, ?) of rank k on a ground set V and a coverage function f on V, the goal is to find an independent set S ? ? maximizing f(S). This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum k-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency ? (i.e., any element of the underlying universe of the coverage function appears in at most ? sets), we design a procedure, parameterized by some integer ?, to extract in polynomial time an approximate kernel of size ? ? k that is guaranteed to contain a 1 - (? - 1)/? approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a 1 - ? approximation in time (?/?)^O(k) ? |V|^O(1). This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, as the kernel has a very simple characterization, it can be constructed in the streaming setting
Multiwinner Analogues of Plurality Rule: Axiomatic and Algorithmic Perspectives
We characterize the class of committee scoring rules that satisfy the
fixed-majority criterion. In some sense, the committee scoring rules in this
class are multiwinner analogues of the single-winner Plurality rule, which is
uniquely characterized as the only single-winner scoring rule that satisfies
the simple majority criterion. We define top--counting committee scoring
rules and show that the fixed majority consistent rules are a subclass of the
top--counting rules. We give necessary and sufficient conditions for a
top--counting rule to satisfy the fixed-majority criterion. We find that,
for most of the rules in our new class, the complexity of winner determination
is high (that is, the problem of computing the winners is NP-hard), but we also
show examples of rules with polynomial-time winner determination procedures.
For some of the computationally hard rules, we provide either exact FPT
algorithms or approximate polynomial-time algorithms
Parameterized Exact and Approximation Algorithms for Maximum -Set Cover and Related Satisfiability Problems
Given a family of subsets over a set of elements~ and two
integers~ and~, Max k-Set Cover consists of finding a subfamily~ of cardinality at most~, covering at least~
elements of~. This problem is W[2]-hard when parameterized by~, and FPT
when parameterized by . We investigate the parameterized approximability of
the problem with respect to parameters~ and~. Then, we show that Max
Sat-k, a satisfiability problem generalizing Max k-Set Cover, is also FPT with
respect to parameter~.Comment: Accepted in RAIRO - Theoretical Informatics and Application
Baby PIH: Parameterized Inapproximability of Min CSP
The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP
theorem in the world of parameterized complexity. It asserts that no FPT
algorithm can distinguish a satisfiable 2CSP instance from one which is only
-satisfiable (where the parameter is the number of variables)
for some constant .
We consider a minimization version of CSPs (Min-CSP), where one may assign
values to each variable, and the goal is to ensure that every constraint is
satisfied by some choice among the pairs of values assigned to its
variables (call such a CSP instance -list-satisfiable). We prove the
following strong parameterized inapproximability for Min CSP: For every , it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even
-list-satisfiable. We refer to this statement as "Baby PIH", following the
recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the
combinatorial arguments underlying the Baby PCP theorem, overcoming some basic
obstacles that arise in the parameterized setting. Furthermore, our reduction
runs in time polynomially bounded in both the number of variables and the
alphabet size, and thus implies the Baby PCP theorem as well
Parameterized Matroid-Constrained Maximum Coverage
In this paper, we introduce the concept of Density-Balanced Subset in a
matroid, in which independent sets can be sampled so as to guarantee that (i)
each element has the same probability to be sampled, and (ii) those events are
negatively correlated. These Density-Balanced Subsets are subsets in the ground
set of a matroid in which the traditional notion of uniform random sampling can
be extended. We then provide an application of this concept to the
Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid
of rank on a ground set and a coverage
function on , the goal is to find an independent set
maximizing . This problem is an important special case of the
much-studied submodular function maximization problem subject to a matroid
constraint; this is also a generalization of the maximum -cover problem in a
graph. In this paper, assuming that the coverage function has a bounded
frequency (i.e., any element of the underlying universe of the coverage
function appears in at most sets), we design a procedure, parameterized
by some integer , to extract in polynomial time an approximate kernel of
size that is guaranteed to contain a
approximation of the optimal solution. This procedure can then be used to get a
Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a approximation in time .
This generalizes and improves the results of [Manurangsi, 2019] and [Huang and
Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid.
Moreover, because of its simplicity, the kernel construction can be performed
in the streaming setting
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation
We consider the following problem: There is a set of items (e.g., movies) and
a group of agents (e.g., passengers on a plane); each agent has some intrinsic
utility for each of the items. Our goal is to pick a set of items that
maximize the total derived utility of all the agents (i.e., in our example we
are to pick movies that we put on the plane's entertainment system).
However, the actual utility that an agent derives from a given item is only a
fraction of its intrinsic one, and this fraction depends on how the agent ranks
the item among the chosen, available, ones. We provide a formal specification
of the model and provide concrete examples and settings where it is applicable.
We show that the problem is hard in general, but we show a number of
tractability results for its natural special cases