7 research outputs found
Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges
Computational Social Choice is an interdisciplinary research area involving
Economics, Political Science, and Social Science on the one side, and
Mathematics and Computer Science (including Artificial Intelligence and
Multiagent Systems) on the other side. Typical computational problems studied
in this field include the vulnerability of voting procedures against attacks,
or preference aggregation in multi-agent systems. Parameterized Algorithmics is
a subfield of Theoretical Computer Science seeking to exploit meaningful
problem-specific parameters in order to identify tractable special cases of in
general computationally hard problems. In this paper, we propose nine of our
favorite research challenges concerning the parameterized complexity of
problems appearing in this context
On Covering Points with Conics and Strips in the Plane
Geometric covering problems have always been of focus in computer scientific research. The generic geometric covering problem asks to cover a set S of n objects with another set of objects whose cardinality is minimum, in a geometric setting. Many versions of geometric cover have been studied in detail, one of which is line cover: Given a set of points in the plane, find the minimum number of lines to cover them. In Euclidean space Rm, this problem is known as Hyperplane Cover, where lines are replaced by affine hyperplanes bounded by dimension d. Line cover is NP-hard, so is its hyperplane analogue. Our thesis focuses on few extensions of hyperplane cover and line cover.
One of the techniques used to study NP-hard problems is Fixed Parameter Tractability (FPT), where, in addition to input size, a parameter k is provided for input instance. We ask to solve the problem with respect to k, such that the running time is a function in both n and k, strictly polynomial in n, while the exponential component is limited to k. In this thesis, we study FPT and parameterized complexity theory, the theory of classifying hard problems involving a parameter k.
We focus on two new geometric covering problems: covering a set of points in the plane with conics (conic cover) and covering a set of points with strips or fat lines of given width in the plane (fat line cover). A conic is a non-degenerate curve of degree two in the plane. A fat line is defined as a strip of finite width w. In this dissertation, we focus on the parameterized versions of these two problems, where, we are asked to cover the set of points with k conics or k fat lines. We use the existing techniques of FPT algorithms, kernelization and approximation algorithms to study these problems. We do a comprehensive study of these problems, starting with NP-hardness results to studying their parameterized hardness in terms of parameter k.
We show that conic cover is fixed parameter tractable, and give an algorithm of running time O∗ ((k/1.38)^4k), where, O∗ implies that the running time is some polynomial in input size. Utilizing special properties of a parabola, we are able to achieve a faster algorithm and show a running time of O∗ ((k/1.15)^3k).
For fat line cover, first we establish its NP-hardness, then we explore algorithmic possibilities with respect to parameterized complexity theory. We show W [1]-hardness of fat line cover with respect to the number of fat lines, by showing a parameterized reduction from the problem of stabbing axis-parallel squares in the plane. A parameterized reduction is an algorithm which transforms an instance of one parameterized problem into an instance of another parameterized problem using a FPT-algorithm. In addition, we show that some restricted versions of fat line cover are also W [1]-hard. Further, in this thesis, we explore a restricted version of fat line cover, where the set of points are integer coordinates and allow only axis-parallel lines to cover them. We show that the problem is still NP-hard. We also show that this version is fixed parameter tractable having a kernel size of O (k^2) and give a FPT-algorithm with a running time of O∗ (3^k). Finally, we conclude our study on this problem by giving an approximation algorithm for this version having a constant approximation ratio 2
Parameterized Enumeration of Neighbour Strings and Kemeny Aggregations
In this thesis, we consider approaches to enumeration problems in the parameterized
complexity setting. We obtain competitive parameterized algorithms to enumerate all, as well as several of, the solutions for two related problems Neighbour String and Kemeny Rank Aggregation. In both problems, the goal is to find a solution that is as close as possible to a set of inputs (strings and total orders, respectively) according to some distance measure.
We also introduce a notion of enumerative kernels for which there is a bijection between solutions to the original instance and solutions to the kernel, and provide such a kernel for Kemeny Rank Aggregation, improving a previous kernel for the problem.
We demonstrate how several of the algorithms and notions discussed in this thesis are
extensible to a group of parameterized problems, improving published results for some other problems
Instance compression of parametric problems and related hierarchies
We define instance compressibility ([13, 17]) for parametric problems in the classes
PH and PSPACE.We observe that the problem ƩiCIRCUITSAT of deciding satisfiability
of a quantified Boolean circuit with i-1 alternations of quantifiers starting with an
existential quantifier is complete for parametric problems in the class Ʃp/i with respect
to w-reductions, and that analogously the problem QBCSAT (Quantified Boolean Circuit
Satisfiability) is complete for parametric problems in PSPACE with respect to
w-reductions. We show the following results about these problems:
1. If CIRCUITSAT is non-uniformly compressible within NP, then ƩiCIRCUITSAT
is non-uniformly compressible within NP, for any i≥1.
2. If QBCSAT is non-uniformly compressible (or even if satisfiability of quantified
Boolean CNF formulae is non-uniformly compressible), then PSPACE ⊆ NP/poly and PH collapses to the third level.
Next, we define Succinct Interactive Proof (Succinct IP) and by adapting the proof
of IP = PSPACE ([11, 6]) , we show that QBCNFSAT (Quantified Boolean Formula
(in CNF) Satisfiability) is in Succinct IP. On the contrary if QBCNFSAT has Succinct
PCPs ([32]) , Polynomial Hierarchy (PH) collapses.
After extending the notion of instance compression to higher classes, we study the
hierarchical structure of the parametric problems with respect to compressibility. For
that purpose, we extend the existing definition of VC-hierarchy ([13]) to parametric
problems. After that, we have considered a long list of natural NP problems and tried
to classify them into some level of VC-hierarchy. We have shown some of the new w-reductions
in this context and pointed out a few interesting results including the ones
as follows.
1. CLIQUE is VC1-complete (using the results in [14]).
2. SET SPLITTING and NAE-SAT are VC2-complete.
We have also introduced a new complexity class VCE in this context and showed
some hardness and completeness results for this class. We have done a comparison
of VC-hierarchy with other related hierarchies in parameterized complexity domain as
well.
Next, we define the compression of counting problems and the analogous classification
of them with respect to the notion of instance compression. We define #VC-hierarchy
for this purpose and similarly classify a large number of natural counting
problems with respect to this hierarchy, by showing some interesting hardness and
completeness results.
We have considered some of the interesting practical problems as well other than
popular NP problems (e.g., #MULTICOLOURED CLIQUE, #SELECTED DOMINATING
SET etc.) and studied their complexity for both decision and counting version. We have
also considered a large variety of circuit satisfiability problems (e.g., #MONOTONE
WEIGHTED-CNFSAT, #EXACT DNF-SAT etc.) and proved some interesting results
about them with respect to the theory of instance compressibility
Confronting intractability via parameters
One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practitioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques, both practical and theoretical, and a fine-grained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts of Downey and Fellows (1999) [2], Flum and Grohe (2006) [59], Niedermeier (2006) [28], and the upcoming undergraduate text (Downey and Fellows 2012) [278]. © 2011 Elsevier Inc