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Depth Lower Bounds against Circuits with Sparse Orientation

Koroth S., Sarma J.

Fundamenta Informaticae

2017

Vol. 152, nr 2

123--144

We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function f is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan circuit (circuits where negations appear only at the leaves) computing f. We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit C computing the CLIQUE function on an n vertex graph. We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d×ω must be Ω(n). In particular, if the weights are o(n/logk n), then C must be of depth ω(logk n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in Amano Maruoka (2005)) and the Karchmer-Wigderson framework (KarchmerWigderson (1988)), we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent. We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting) separates NP from NC. As our main tool, we generalize Karchmer-Wigderson games (Karchmer Wigderson (1988)) for monotone functions to work for non-monotone circuits parametrized by the weight/structure of the orientation. We also prove structural results about orientation and prove connections between number of negations and weight of orientations required to compute a function.

On the Complexity of L-reachability

Komarath B., Sarma J., Sunil K. S.

Fundamenta Informaticae

2016

Vol. 145, nr 4

471--483

We initiate a complexity theoretic study of the language based graph reachability problem (L-REACH) : Fix a language L. Given a graph whose edges are labelled with alphabet symbols of the language L and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterizations for all of them. Our main results are the following: • Restricting the language using formal language theory we show that the complexity of L- REACH increases with the power of the formal language class. We show that there is a regular language for which the L-REACH is NL-complete even for undirected graphs. In the case of linear languages, the complexity of L-REACH does not go beyond the complexity of L itself. Further, there is a deterministic context-free language L for which L-DAGREACH is LogCFL- complete. • We use L-REACH as a lens to study structural complexity. In this direction we show that there is a language A in TC0 for which A-DAGREACH is NP-complete. Using this we show that P vs NP question is equivalent to P vs DAGREACH-1(P)1 question. This leads to the intriguing possibility that by proving DAGREACH-1(P) is contained in some subclass of P, we can prove an upward translation of separation of complexity classes. Note that we do not know a way to upward translate the separation of complexity classes.