Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

Abstract

We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class Π. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most k from a graph with maximum degree 1 can be computed in polynomial time when k is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree 2 or connected components of size at most c, c ≥ 3, is NP-hard. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably has no f(k) · |I|O(1)-time algorithm and that, in contrast, an optimal network with at most k arcs can be computed in 2O(k) · |I|O(1) time where |I| is the total input size.

Bayesian network structure learning with improved genetic algorithm

“…In the past 11 years, many researchers have been working on BN structure learning and some progress has achieved. Grüttemeier et al 1 proposed a BN structure learning algorithm with structural constraints. Gu et al 2 learnt large-scale BNs (>50 nodes) by partitioning nodes into clusters to learn subgraphs separately and combining all these subgraph to obtain the final complete BN.…”

“…RKGA and BRKGA contain mutation operator and cross operator the same as GA, but RKGA and BRKGA add a block called decoder to convert random keys into initial solutions to optimization problem. Random keys are consisting of real numbers in the interval [0,1] and its length is the same as the final solution. And the decoder can be designed according to different optimization problems, which can be very simple or act as a local optimizer.…”

“…Parameterized uniform crossover randomly generates a vector consisting of random numbers in the interval [0,1]. Each random number in the vector is compared to a pre-defined parameter, ρ e .…”

“…The whole steps for BRKGA solving BNSL can be found in Algorithm 1. The first step is to generate p vectors of random keys of size n. Each random key is in the real interval [0,1] and random keys are the input of the decoder. The decoder interprets random keys as initial solutions to the optimization problem and the fitness values are then calculated using loss function defined in the above section.…”

Efficient Bayesian Network Structure Learning via Parameterized Local Search on Topological Orderings

“…Moreover, even very restricted special cases of BNSL remain NPhard, for example the case where every possible parent set has constant size (Ordyniak and Szeider 2013). Finally, restricting the topology of the DAG to sparse classes such as graphs of bounded treewidth or bounded degree leads to NPhard learning problems as well (Korhonen andParviainen 2013, 2015;Grüttemeier and Komusiewicz 2020).…”