117 research outputs found
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
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
Randomized Algorithms for the Loop Cutset Problem
We show how to find a minimum weight loop cutset in a Bayesian network with
high probability. Finding such a loop cutset is the first step in the method of
conditioning for inference. Our randomized algorithm for finding a loop cutset
outputs a minimum loop cutset after O(c 6^k kn) steps with probability at least
1 - (1 - 1/(6^k))^c6^k, where c > 1 is a constant specified by the user, k is
the minimal size of a minimum weight loop cutset, and n is the number of
vertices. We also show empirically that a variant of this algorithm often finds
a loop cutset that is closer to the minimum weight loop cutset than the ones
found by the best deterministic algorithms known
On Weighted Graph Separation Problems and Flow-Augmentation
One of the first application of the recently introduced technique of\emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithmfor the weighted version of \textsc{Directed Feedback Vertex Set}, a landmarkproblem in parameterized complexity. In this note we explore applicability offlow-augmentation to other weighted graph separation problems parameterized bythe size of the cutset. We show the following. -- In weighted undirected graphs\textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- Theweighted version of \textsc{Group Feedback Vertex Set} is FPT, even with anoracle access to group operations. -- The weighted version of \textsc{DirectedSubset Feedback Vertex Set} is FPT. Our study reveals \textsc{DirectedSymmetric Multicut} as the next important graph separation problem whoseparameterized complexity remains unknown, even in the unweighted setting.<br
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Given an undirected graph , a collection of
pairs of vertices, and an integer , the Edge Multicut problem ask if there
is a set of at most edges such that the removal of disconnects
every from the corresponding . Vertex Multicut is the analogous
problem where is a set of at most vertices. Our main result is that
both problems can be solved in time , i.e.,
fixed-parameter tractable parameterized by the size of the cutset in the
solution. By contrast, it is unlikely that an algorithm with running time of
the form exists for the directed version of the problem, as
we show it to be W[1]-hard parameterized by the size of the cutset
Fixed-Parameter Tractability of Multicut in Directed Acyclic Graphs
The Multicut problem, given a graph G, a set of terminal pairs , and an integer , asks whether one can find a cutset consisting of at most nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, is not reachable from for each . The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355--388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459--468], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains -hard
Designing FPT algorithms for cut problems using randomized contractions
We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. More precisely, we show the following: • We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2O(k 2 log |Σ|)n4 log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |Σ | is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by O( log n) to optimality, which improves over the trivial O(1) upper bound
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