28 research outputs found
FPT-Algorithms for the l-Matchoid Problem with Linear and Submodular Objectives
We design a fixed-parameter deterministic algorithm for computing a maximum
weight feasible set under a -matchoid of rank , parameterized by
and . Unlike previous work that presumes linear representativity of
matroids, we consider the general oracle model. Our result, combined with the
lower bounds of Lovasz, and Jensen and Korte, demonstrates a separation between
the -matchoid and the matroid -parity problems in the setting of
fixed-parameter tractability.
Our algorithms are obtained by means of kernelization: we construct a small
representative set which contains an optimal solution. Such a set gives us much
flexibility in adapting to other settings, allowing us to optimize not only a
linear function, but also several important submodular functions. It also helps
to transform our algorithms into streaming algorithms.
In the streaming setting, we show that we can find a feasible solution of
value and the number of elements to be stored in memory depends only on
and but totally independent of . This shows that it is possible to
circumvent the recent space lower bound of Feldman et al., by parameterizing
the solution value. This result, combined with existing lower bounds, also
provides a new separation between the space and time complexity of maximizing
an arbitrary submodular function and a coverage function in the value oracle
model
Parameterized Complexity Analysis of Randomized Search Heuristics
This chapter compiles a number of results that apply the theory of
parameterized algorithmics to the running-time analysis of randomized search
heuristics such as evolutionary algorithms. The parameterized approach
articulates the running time of algorithms solving combinatorial problems in
finer detail than traditional approaches from classical complexity theory. We
outline the main results and proof techniques for a collection of randomized
search heuristics tasked to solve NP-hard combinatorial optimization problems
such as finding a minimum vertex cover in a graph, finding a maximum leaf
spanning tree in a graph, and the traveling salesperson problem.Comment: This is a preliminary version of a chapter in the book "Theory of
Evolutionary Computation: Recent Developments in Discrete Optimization",
edited by Benjamin Doerr and Frank Neumann, published by Springe
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
A join-based hybrid parameter for constraint satisfaction
We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP.
Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances
Maximum Volume Subset Selection for Anchored Boxes
Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that:
* The problem is NP-hard already in 3 dimensions.
* In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm.
* For any constant dimension d, we give an efficient polynomial-time approximation scheme
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum