4 research outputs found
On the Subexponential Time Complexity of CSP
A CSP with n variables ranging over a domain of d values can be solved by
brute-force in d^n steps (omitting a polynomial factor). With a more careful
approach, this trivial upper bound can be improved for certain natural
restrictions of the CSP. In this paper we establish theoretical limits to such
improvements, and draw a detailed landscape of the subexponential-time
complexity of CSP.
We first establish relations between the subexponential-time complexity of
CSP and that of other problems, including CNF-Sat. We exploit this connection
to provide tight characterizations of the subexponential-time complexity of CSP
under common assumptions in complexity theory. For several natural CSP
parameters, we obtain threshold functions that precisely dictate the
subexponential-time complexity of CSP with respect to the parameters under
consideration.
Our analysis provides fundamental results indicating whether and when one can
significantly improve on the brute-force search approach for solving CSP
Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives
Proximity graphs have been studied for several decades, motivated by
applications in computational geometry, geography, data mining, and many other
fields. However, the computational complexity of classic graph problems on
proximity graphs mostly remained open. We now study 3-Colorability, Dominating
Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the
proximity graph classes relative neighborhood graphs, Gabriel graphs, and
relatively closest graphs. We prove that all of the problems remain NP-hard on
these graphs, except for 3-Colorability and Hamiltonian Cycle on relatively
closest graphs, where the former is trivial and the latter is left open.
Moreover, for every NP-hard case we additionally show that no
-time algorithm exists unless the ETH fails, where n denotes
the number of vertices