1 research outputs found
On the Hardness of SAT with Community Structure
Recent attempts to explain the effectiveness of Boolean satisfiability (SAT)
solvers based on conflict-driven clause learning (CDCL) on large industrial
benchmarks have focused on the concept of community structure. Specifically,
industrial benchmarks have been empirically found to have good community
structure, and experiments seem to show a correlation between such structure
and the efficiency of CDCL. However, in this paper we establish hardness
results suggesting that community structure is not sufficient to explain the
success of CDCL in practice. First, we formally characterize a property shared
by a wide class of metrics capturing community structure, including
"modularity". Next, we show that the SAT instances with good community
structure according to any metric with this property are still NP-hard.
Finally, we study a class of random instances generated from the
"pseudo-industrial" community attachment model of Gir\'aldez-Cru and Levy. We
prove that, with high probability, instances from this model that have
relatively few communities but are still highly modular require exponentially
long resolution proofs and so are hard for CDCL. We also present experimental
evidence that our result continues to hold for instances with many more
communities. This indicates that actual industrial instances easily solved by
CDCL may have some other relevant structure not captured by the community
attachment model.Comment: 23 pages. Full version of a SAT 2016 pape