1,307 research outputs found
On treewidth approximations
We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a approximation algorithm for the treewidth of arbitrary graphs, where is the treewidth of the input graph
On Low Treewidth Approximations of Conjunctive Queries
We recently initiated the study of approximations of conjunctive queries within classes that admit tractable query evaluation (with respect to combined complexity). Those include classes of acyclic, bounded treewidth, or bounded hypertreewidth queries. Such approximations are always guaranteed to exist. However, while for acyclic and bounded hypertreewidth queries we have shown a number of examples of interesting approximations, for queries of bounded treewidth the study had been restricted to queries over graphs, where such approximations usually trivialize. In this note we show that for relations of arity greater than two, the notion of low treewidth approximations is a rich one, as many queries possess them. In fact we look at approximations of queries of maximum possible treewidth by queries of minimum possible treewidth (i.e., one), and show that even in this case the structure of approximations remain rather rich as long as input relations are not binary
Approximation Algorithms for the Capacitated Domination Problem
We consider the {\em Capacitated Domination} problem, which models a
service-requirement assignment scenario and is also a generalization of the
well-known {\em Dominating Set} problem. In this problem, given a graph with
three parameters defined on each vertex, namely cost, capacity, and demand, we
want to find an assignment of demands to vertices of least cost such that the
demand of each vertex is satisfied subject to the capacity constraint of each
vertex providing the service. In terms of polynomial time approximations, we
present logarithmic approximation algorithms with respect to different demand
assignment models for this problem on general graphs, which also establishes
the corresponding approximation results to the well-known approximations of the
traditional {\em Dominating Set} problem. Together with our previous work, this
closes the problem of generally approximating the optimal solution. On the
other hand, from the perspective of parameterization, we prove that this
problem is {\it W[1]}-hard when parameterized by a structure of the graph
called treewidth. Based on this hardness result, we present exact
fixed-parameter tractable algorithms when parameterized by treewidth and
maximum capacity of the vertices. This algorithm is further extended to obtain
pseudo-polynomial time approximation schemes for planar graphs
On the Complexity and Approximation of Binary Evidence in Lifted Inference
Lifted inference algorithms exploit symmetries in probabilistic models to
speed up inference. They show impressive performance when calculating
unconditional probabilities in relational models, but often resort to
non-lifted inference when computing conditional probabilities. The reason is
that conditioning on evidence breaks many of the model's symmetries, which can
preempt standard lifting techniques. Recent theoretical results show, for
example, that conditioning on evidence which corresponds to binary relations is
#P-hard, suggesting that no lifting is to be expected in the worst case. In
this paper, we balance this negative result by identifying the Boolean rank of
the evidence as a key parameter for characterizing the complexity of
conditioning in lifted inference. In particular, we show that conditioning on
binary evidence with bounded Boolean rank is efficient. This opens up the
possibility of approximating evidence by a low-rank Boolean matrix
factorization, which we investigate both theoretically and empirically.Comment: To appear in Advances in Neural Information Processing Systems 26
(NIPS), Lake Tahoe, USA, December 201
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