5,576 research outputs found
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
Pareto-Path Multi-Task Multiple Kernel Learning
A traditional and intuitively appealing Multi-Task Multiple Kernel Learning
(MT-MKL) method is to optimize the sum (thus, the average) of objective
functions with (partially) shared kernel function, which allows information
sharing amongst tasks. We point out that the obtained solution corresponds to a
single point on the Pareto Front (PF) of a Multi-Objective Optimization (MOO)
problem, which considers the concurrent optimization of all task objectives
involved in the Multi-Task Learning (MTL) problem. Motivated by this last
observation and arguing that the former approach is heuristic, we propose a
novel Support Vector Machine (SVM) MT-MKL framework, that considers an
implicitly-defined set of conic combinations of task objectives. We show that
solving our framework produces solutions along a path on the aforementioned PF
and that it subsumes the optimization of the average of objective functions as
a special case. Using algorithms we derived, we demonstrate through a series of
experimental results that the framework is capable of achieving better
classification performance, when compared to other similar MTL approaches.Comment: Accepted by IEEE Transactions on Neural Networks and Learning System
Iterative Delegations in Liquid Democracy with Restricted Preferences
In this paper, we study liquid democracy, a collective decision making
paradigm which lies between direct and representative democracy. One main
feature of liquid democracy is that voters can delegate their votes in a
transitive manner so that: A delegates to B and B delegates to C leads to A
delegates to C. Unfortunately, this process may not converge as there may not
even exist a stable state (also called equilibrium). In this paper, we
investigate the stability of the delegation process in liquid democracy when
voters have restricted types of preference on the agent representing them
(e.g., single-peaked preferences). We show that various natural structures of
preferences guarantee the existence of an equilibrium and we obtain both
tractability and hardness results for the problem of computing several
equilibria with some desirable properties
Limits of Preprocessing
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a
complexity theoretic assumption, none of the considered problems can be reduced
by polynomial-time preprocessing to a problem kernel whose size is polynomial
in a structural problem parameter of the input, such as induced width or
backdoor size. Our results provide a firm theoretical boundary for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: This is a slightly longer version of a paper that appeared in the
proceedings of AAAI 201
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
On the Threshold of Intractability
We study the computational complexity of the graph modification problems
Threshold Editing and Chain Editing, adding and deleting as few edges as
possible to transform the input into a threshold (or chain) graph. In this
article, we show that both problems are NP-complete, resolving a conjecture by
Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128,
2001). On the positive side, we show the problem admits a quadratic vertex
kernel. Furthermore, we give a subexponential time parameterized algorithm
solving Threshold Editing in time,
making it one of relatively few natural problems in this complexity class on
general graphs. These results are of broader interest to the field of social
network analysis, where recent work of Brandes (ISAAC, 2014) posits that the
minimum edit distance to a threshold graph gives a good measure of consistency
for node centralities. Finally, we show that all our positive results extend to
the related problem of Chain Editing, as well as the completion and deletion
variants of both problems
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