Balancing Bounded Treewidth Circuits [chapter]

Maurice Jansen, Jayalal Sarma M.N.
2010 Lecture Notes in Computer Science  
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(^i n) can be simulated by a circuit of width O(^i+1 n) and size n^c, where c = O(1), if i=0, and c=O( 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
more » ... O(k^2 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( 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 n). This strengthens in the non-uniform setting the known inclusion that SC^0 ⊆ NC^1. Finally, we apply our construction to show that reachability for directed graphs of bounded treewidth is in LogDCFL.
doi:10.1007/978-3-642-13182-0_21 fatcat:zbid7xnlpzaerac2zromqibccq