Balancing Bounded Treewidth Circuits

Abstract

We use algorithmic tools for graphs of small treewidth to address questions in complexity theory. For both arithmetic and Boolean circuits, we show that any circuit of size n ^O(1) and treewidth O(logⁱ 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(loglogn) 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 ²logn). From this we derive an analogous statement for syntactically multilinear arithmetic circuits, which strengthens the central theorem of [14]. As another application, we derive that constant width arithmetic circuits of size n ^O(1) can be balanced to depth O(logn), 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 ²logn). This strengthens in the non-uniform setting the known inclusion that SC⁰ ⊆ NC¹. Finally, we apply our construction to show that Reachability and Circuit Value Problem for some treewidth restricted cases can be solved in LogDCFL.

This work was supported in part by the National Natural Science Foundation of China Grant 60553001, and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.