| Popis: |
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 $LogDCFL$. |
