Boolean dimension and tree-width

The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the $d$ linear extensions of a witnessing realizer. Focusing on the encoding aspect Ne\v{s}et\v{r}il and Pudl\'{a}k defined a more expressive version of dimension. A poset $P$ has boolean dimension at most $d$ if it is possible to decide whether $x \leq y$ in $P$ by looking at the relative position of $x$ and $y$ in only $d$ permutations of the elements of $P$. We prove that posets with cover graphs of bounded tree-width have bounded boolean dimension. This stays in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. This result might be a step towards a resolution of the long-standing open problem: Do planar posets have bounded boolean dimension?Comment: one more reference added; paper revised along the suggestion of three reviewer

Balancing Bounded Treewidth Circuits

Algorithmic tools for graphs of small treewidth are used to address questions in complexity theory. For both arithmetic and Boolean circuits, it is shown that any circuit of size $n^{O(1)}$ and treewidth $O(\log^i n)$ can be simulated by a circuit of width $O(\log^{i+1} n)$ and size $n^c$, where $c = O(1)$, if $i=0$, and $c=O(\log \log n)$ otherwise. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in arithmetic formulas of depth $O(k^2\log n)$. From this we derive the analogous statement for syntactically multilinear arithmetic circuits, which strengthens a theorem of Mahajan and Rao. As another application, we derive that constant width arithmetic circuits of size $n^{O(1)}$ can be balanced to depth $O(\log n)$, provided certain restrictions are made on the use of iterated multiplication. Also from our main construction, we derive that Boolean bounded fan-in circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in formulas of depth $O(k^2\log n)$. This strengthens in the non-uniform setting the known inclusion that $SC^0 \subseteq NC^1$. Finally, we apply our construction to show that {\sc reachability} for directed graphs of bounded treewidth is in $LogDCFL$

Efficiently Learning Monotone Decision Trees with ID3

Since the Probably Approximately Correct learning model was introduced in 1984, there has been much effort in designing computationally efficient algorithms for learning Boolean functions from random examples drawn from a uniform distribution. In this paper, I take the ID3 information-gain-first classification algorithm and apply it to the task of learning monotone Boolean functions from examples that are uniformly distributed over {0,1}^n. I limited my scope to the class of monotone Boolean functions that can be represented as read-2 width-2 disjunctive normal form expressions. I modeled these functions as graphs and examined each type of connected component contained in these models, i.e. path graphs and cycle graphs. I determined the influence of the variables in the pieces of these graph models in order to understand how ID3 behaves when learning these functions. My findings show that ID3 will produce an optimal decision tree for this class of Boolean functions