4,619 research outputs found
Tight lower bounds for certain parameterized NP-hard problems
Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time no(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)-st level W [t − 1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time no(k) poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-sat (for any fixed q ≥ 2), clique, and independent set, cannot be solved in time no(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k poly(m) or O(n k).
Parameterized algorithms and computational lower bounds: a structural approach
Many problems of practical significance are known to be NP-hard, and hence, are unlikely
to be solved by polynomial-time algorithms. There are several ways to cope with
the NP-hardness of a certain problem. The most popular approaches include heuristic
algorithms, approximation algorithms, and randomized algorithms. Recently, parameterized
computation and complexity have been receiving a lot of attention. By
taking advantage of small or moderate parameter values, parameterized algorithms
provide new venues for practically solving problems that are theoretically intractable.
In this dissertation, we design efficient parameterized algorithms for several wellknown
NP-hard problems and prove strong lower bounds for some others. In doing
so, we place emphasis on the development of new techniques that take advantage of
the structural properties of the problems.
We present a simple parameterized algorithm for Vertex Cover that uses polynomial
space and runs in time O(1.2738k + kn). It improves both the previous
O(1.286k + kn)-time polynomial-space algorithm by Chen, Kanj, and Jia, and the
very recent O(1.2745kk4 + kn)-time exponential-space algorithm, by Chandran and
Grandoni. This algorithm stands out for both its performance and its simplicity. Essential
to the design of this algorithm are several new techniques that use structural
information of the underlying graph to bound the search space.
For Vertex Cover on graphs with degree bounded by three, we present a still better algorithm that runs in time O(1.194k + kn), based on an âÂÂalmost-globalâÂÂ
analysis of the search tree.
We also show that an important structural property of the underlying graphs âÂÂ
the graph genus â largely dictates the computational complexity of some important
graph problems including Vertex Cover, Independent Set and Dominating Set.
We present a set of new techniques that allows us to prove almost tight computational
lower bounds for some NP-hard problems, such as Clique, Dominating Set,
Hitting Set, Set Cover, and Independent Set. The techniques are further extended
to derive computational lower bounds on polynomial time approximation schemes for
certain NP-hard problems. Our results illustrate a new approach to proving strong
computational lower bounds for some NP-hard problems under reasonable conditions
Point Line Cover: The Easy Kernel is Essentially Tight
The input to the NP-hard Point Line Cover problem (PLC) consists of a set
of points on the plane and a positive integer , and the question is
whether there exists a set of at most lines which pass through all points
in . A simple polynomial-time reduction reduces any input to one with at
most points. We show that this is essentially tight under standard
assumptions. More precisely, unless the polynomial hierarchy collapses to its
third level, there is no polynomial-time algorithm that reduces every instance
of PLC to an equivalent instance with points, for
any . This answers, in the negative, an open problem posed by
Lokshtanov (PhD Thesis, 2009).
Our proof uses the machinery for deriving lower bounds on the size of kernels
developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients:
We first show, by reduction from Vertex Cover, that PLC---conditionally---has
no kernel of total size bits. This does not directly imply
the claimed lower bound on the number of points, since the best known
polynomial-time encoding of a PLC instance with points requires
bits. To get around this we build on work of Goodman et al.
(STOC 1989) and devise an oracle communication protocol of cost
for PLC; its main building block is a bound of for the order
types of points that are not necessarily in general position, and an
explicit algorithm that enumerates all possible order types of n points. This
protocol and the lower bound on total size together yield the stated lower
bound on the number of points.
While a number of essentially tight polynomial lower bounds on total sizes of
kernels are known, our result is---to the best of our knowledge---the first to
show a nontrivial lower bound for structural/secondary parameters
Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems
In the {\sc Hitting Set} problem, we are given a collection of
subsets of a ground set and an integer , and asked whether has a
-element subset that intersects each set in . We consider two
parameterizations of {\sc Hitting Set} below tight upper bounds: and
. In both cases is the parameter. We prove that the first
parameterization is fixed-parameter tractable, but has no polynomial kernel
unless coNPNP/poly. The second parameterization is W[1]-complete,
but the introduction of an additional parameter, the degeneracy of the
hypergraph , makes the problem not only fixed-parameter
tractable, but also one with a linear kernel. Here the degeneracy of
is the minimum integer such that for each the
hypergraph with vertex set and edge set containing all edges of
without vertices in , has a vertex of degree at most
In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected
graph (a directed graph) on vertices and an integer , and asked
whether has a set of vertices such that for each vertex there is an edge (arc) from a vertex in to . {\sc Nonblocker} can be
viewed as a special case of {\sc Directed Nonblocker} (replace an undirected
graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that
{\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for
{\sc Directed Nonblocker}
Counting Complexity for Reasoning in Abstract Argumentation
In this paper, we consider counting and projected model counting of
extensions in abstract argumentation for various semantics. When asking for
projected counts we are interested in counting the number of extensions of a
given argumentation framework while multiple extensions that are identical when
restricted to the projected arguments count as only one projected extension. We
establish classical complexity results and parameterized complexity results
when the problems are parameterized by treewidth of the undirected
argumentation graph. To obtain upper bounds for counting projected extensions,
we introduce novel algorithms that exploit small treewidth of the undirected
argumentation graph of the input instance by dynamic programming (DP). Our
algorithms run in time double or triple exponential in the treewidth depending
on the considered semantics. Finally, we take the exponential time hypothesis
(ETH) into account and establish lower bounds of bounded treewidth algorithms
for counting extensions and projected extension.Comment: Extended version of a paper published at AAAI-1
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