SAT, short for satisfiability, is probably the most decorated
and
applicable NP-hard problem. Empirical studies show that:
(i) state-of-the-art SAT solvers abort mainly due to lack of memory, and
(ii) real-life SAT instances come with structure.
We systematically study time-space tradeoffs for solving SAT,
parameterized by path-width & tree-width; two popular ways to quantify
how much structured is the input instance.

In particular, we (conditionally) resolve the main open question
of Alekhnovitch and Razborov (2002) which has as follows:

"Is it possible to solve SAT of treewidth w(n) simultaneously in
time-space (poly(n) 2^O(w), poly(n))?"

We show that in the incidence graph there is a simple
algorithm running in time-space (poly(n) 2^O(w*logn), poly(n)).
We furthermore, show that removing this logn factor from the time
exponent incurs an exponential blow up in the space
unless NC subseteq SC (or its scalled analogs).
That is, ignoring constants in the exponent,
if you pay a logn in the exponent then
you can bring the space down to polynomial, but other than this
there is nothing else you can do.

Finally, we devise algorithms showing that it is possible to trade
constants in the exponent between time and space.

If time permits, I'll outline some new connections and directions
of our work to propositional proof complexity (the first step has
already been taken by Beame-Beck-Impagliazzo 2012).