269,545 research outputs found
On the complexity of dominating set problems related to the minimum all-ones problem
The minimum all-ones problem and the connected odd dominating set problem were shown to be NP-complete in different papers for general graphs, while they are solvable in linear time (or trivial) for trees, unicyclic graphs, and series-parallel graphs. The complexity of both problems when restricted to bipartite graphs was raised as an open question. Here we solve both problems. For this purpose, we introduce the related decision problem of the existence of an odd dominating set without isolated vertices, and study its complexity. Our main result shows that this new problem is NP-complete, even when restricted to bipartite graphs. We use this result to deduce that the minimum all-ones problem and the connected odd dominating set problem are also NP-complete for bipartite graphs. We show that all three problems are solvable in linear time for graphs with bounded treewidth. We also show that the new problem remains NP-complete when restricted to other graph classes, e.g., planar graphs, graphs with girth at least five, and graphs with a small maximum degree, in particular 3-regular graphs. \ud
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Simplifying Random Satisfiability Problem by Removing Frustrating Interactions
How can we remove some interactions in a constraint satisfaction problem
(CSP) such that it still remains satisfiable? In this paper we study a modified
survey propagation algorithm that enables us to address this question for a
prototypical CSP, i.e. random K-satisfiability problem. The average number of
removed interactions is controlled by a tuning parameter in the algorithm. If
the original problem is satisfiable then we are able to construct satisfiable
subproblems ranging from the original one to a minimal one with minimum
possible number of interactions. The minimal satisfiable subproblems will
provide directly the solutions of the original problem.Comment: 21 pages, 16 figure
Solving Multiclass Learning Problems via Error-Correcting Output Codes
Multiclass learning problems involve finding a definition for an unknown
function f(x) whose range is a discrete set containing k > 2 values (i.e., k
``classes''). The definition is acquired by studying collections of training
examples of the form [x_i, f (x_i)]. Existing approaches to multiclass learning
problems include direct application of multiclass algorithms such as the
decision-tree algorithms C4.5 and CART, application of binary concept learning
algorithms to learn individual binary functions for each of the k classes, and
application of binary concept learning algorithms with distributed output
representations. This paper compares these three approaches to a new technique
in which error-correcting codes are employed as a distributed output
representation. We show that these output representations improve the
generalization performance of both C4.5 and backpropagation on a wide range of
multiclass learning tasks. We also demonstrate that this approach is robust
with respect to changes in the size of the training sample, the assignment of
distributed representations to particular classes, and the application of
overfitting avoidance techniques such as decision-tree pruning. Finally, we
show that---like the other methods---the error-correcting code technique can
provide reliable class probability estimates. Taken together, these results
demonstrate that error-correcting output codes provide a general-purpose method
for improving the performance of inductive learning programs on multiclass
problems.Comment: See http://www.jair.org/ for any accompanying file
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
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