65 research outputs found
Certifying the restricted isometry property is hard
This paper is concerned with an important matrix condition in compressed
sensing known as the restricted isometry property (RIP). We demonstrate that
testing whether a matrix satisfies RIP is NP-hard. As a consequence of our
result, it is impossible to efficiently test for RIP provided P \neq NP
Parameterized Complexity of Conflict-Free Matchings and Paths
An input to a conflict-free variant of a classical problem Gamma, called Conflict-Free Gamma, consists of an instance I of Gamma coupled with a graph H, called the conflict graph. A solution to Conflict-Free Gamma in (I,H) is a solution to I in Gamma, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions
Dynamic programming on tree decompositions is a frequently used approach to
solve otherwise intractable problems on instances of small treewidth. In recent
work by Bodlaender et al., it was shown that for many connectivity problems,
there exist algorithms that use time, linear in the number of vertices, and
single exponential in the width of the tree decomposition that is used. The
central idea is that it suffices to compute representative sets, and these can
be computed efficiently with help of Gaussian elimination.
In this paper, we give an experimental evaluation of this technique for the
Steiner Tree problem. A comparison of the classic dynamic programming algorithm
and the improved dynamic programming algorithm that employs the table reduction
shows that the new approach gives significant improvements on the running time
of the algorithm and the size of the tables computed by the dynamic programming
algorithm, and thus that the rank based approach from Bodlaender et al. does
not only give significant theoretical improvements but also is a viable
approach in a practical setting, and showcases the potential of exploiting the
idea of representative sets for speeding up dynamic programming algorithms
Finding Even Subgraphs Even Faster
Problems of the following kind have been the focus of much recent research in
the realm of parameterized complexity: Given an input graph (digraph) on
vertices and a positive integer parameter , find if there exist edges
(arcs) whose deletion results in a graph that satisfies some specified parity
constraints. In particular, when the objective is to obtain a connected graph
in which all the vertices have even degrees---where the resulting graph is
\emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The
corresponding problem in digraphs where the resulting graph should be strongly
connected and every vertex should have the same in-degree as its out-degree is
called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica,
2014}] showed that these problems are fixed parameter tractable (FPT), and gave
algorithms with the running time . They also asked, as
an open problem, whether there exist FPT algorithms which solve these problems
in time . In this paper we answer their question in the
affirmative: using the technique of computing \emph{representative families of
co-graphic matroids} we design algorithms which solve these problems in time
. The crucial insight we bring to these problems is to view
the solution as an independent set of a co-graphic matroid. We believe that
this view-point/approach will be useful in other problems where one of the
constraints that need to be satisfied is that of connectivity
Intertwining connectivities in representable matroids
Let be a representable matroid and subsets of the ground set such that the smallest separation that separates from has order , and the smallest separation that separates from has order . We prove that if is sufficiently large, then there is an element such that in one of and both connectivities are preserved. For matroids representable over a finite field we prove a stronger result: we show that we can remove such that both a connectivity and a minor of are preserved
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