88 research outputs found
Complexity and Approximability of Parameterized MAX-CSPs
International audienceWe study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance.We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a (1−ϵ)-approximation in time f(k,ϵ)⋅poly(n). 3. There is no FPTAS unless FPT=W[1].For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results
The Complexity of Finding S-Factors in Regular Graphs
A graph G has an S-factor if there exists a spanning subgraph F of G such that for all v in V: deg_F(v) in S. The simplest example of such factor is a 1-factor, which corresponds to a perfect matching in a graph. In this paper we study the computational complexity of finding S-factors in regular graphs. Our techniques combine some classical as well as recent tools from graph theory
Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets
We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, such as Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for an improved running time analysis. We illustrate the method with improved algorithms for Max (r,2) -CSP and #Dominating Set. For Max (r,2) -CSP instances with domain size r and m constraints, the running time improves from r m/6 to r m/7.5 for cubic instances and from r 0.19⋅m to r 0.18⋅m for general instances, omitting subexponential factors. For #Dominating Set instances with n vertices, the running time improves from 1.4143 n to 1.2458 n for cubic instances and from 1.5673 n to 1.5183 n for general instances. It is likely that other algorithms relying on local transformations can be improved using our method, which exploits a non-local property of graphs
Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree
"In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1,2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97^k n^4 log W) using O(n^3 log nW log k) space. This is the first single-exponential time, polynomial-space FPT algorithm for the weighted Steiner tree problem."
PLEASE NOTE:This is an author-created version that the author has self-archived to the "Aaltodoc" (aaltodoc.aalto.fi) faculty-level repository at Aalto University. The final publication is available at link.springer.com via the link http://dx.doi.org/10.1007/978-3-662-47672-7_40Peer reviewe
Online Linear Extractors for Independent Sources
In this work, we characterize online linear extractors. In other words, given a matrix , we study the convergence of the iterated process , where is repeatedly sampled independently from some fixed (but unknown) distribution with (min)-entropy at least . Here, we think of as the state of an online extractor, and as its input.
As our main result, we show that the state converges to the uniform distribution for all input distributions with entropy if and only if the matrix has no non-trivial invariant subspace (i.e., a non-zero subspace such that ). In other words, a matrix yields an online linear extractor if and only if has no non-trivial invariant subspace. For example, the linear transformation corresponding to multiplication by a generator of the field yields a good online linear extractor. Furthermore, for any such matrix convergence takes at most steps.
We also study the more general notion of condensing---that is, we ask when this process converges to a distribution with entropy at least , when the input distribution has entropy greater than . (Extractors corresponding to the special case when .) We show that a matrix gives a good condenser if there are relatively few vectors such that are linearly dependent. As an application, we show that the very simple cyclic rotation transformation condenses to bits for any if is a prime satisfying a certain simple number-theoretic condition.
Our proofs are Fourier-analytic and rely on a novel lemma, which gives a tight bound on the product of certain Fourier coefficients of any entropic distribution
Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain
We study the computation of the diameter and radius under the rectilinear
link distance within a rectilinear polygonal domain of vertices and
holes. We introduce a \emph{graph of oriented distances} to encode the distance
between pairs of points of the domain. This helps us transform the problem so
that we can search through the candidates more efficiently. Our algorithm
computes both the diameter and the radius in time, where denotes the matrix
multiplication exponent and is the number of
edges of the graph of oriented distances. We also provide a faster algorithm
for computing the diameter that runs in time
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