13,154 research outputs found
Improved Exact Algorithms for Mildly Sparse Instances of Max SAT
We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2).
Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively
Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers
A line of work initiated by Fortnow in 1997 has proven model-independent
time-space lower bounds for the problem and related problems
within the polynomial-time hierarchy. For example, for the
problem, the state-of-the-art is that the problem cannot be solved by
random-access machines in time and space simultaneously for .
We extend this lower bound approach to the quantum and randomized domains.
Combining Grover's algorithm with components from time-space
lower bounds, we show that there are problems verifiable in time with
quantum Merlin-Arthur protocols that cannot be solved in time and
space simultaneously for , a
super-quadratic time lower bound. This result and the prior work on
can both be viewed as consequences of a more general formula for
time lower bounds against small-space algorithms, whose asymptotics we study in
full.
We also show lower bounds against randomized algorithms: there are problems
verifiable in time with (classical) Merlin-Arthur protocols that cannot
be solved in randomized time and space simultaneously for , improving a result of Diehl. For quantum Merlin-Arthur protocols, the
lower bound in this setting can be improved to .Comment: 38 pages, 5 figures. To appear in ITCS 202
Breaking the PPSZ Barrier for Unique 3-SAT
The PPSZ algorithm by Paturi, Pudl\'ak, Saks, and Zane (FOCS 1998) is the
fastest known algorithm for (Promise) Unique k-SAT. We give an improved
algorithm with exponentially faster bounds for Unique 3-SAT.
For uniquely satisfiable 3-CNF formulas, we do the following case
distinction: We call a clause critical if exactly one literal is satisfied by
the unique satisfying assignment. If a formula has many critical clauses, we
observe that PPSZ by itself is already faster. If there are only few clauses
allover, we use an algorithm by Wahlstr\"om (ESA 2005) that is faster than PPSZ
in this case. Otherwise we have a formula with few critical and many
non-critical clauses. Non-critical clauses have at least two literals
satisfied; we show how to exploit this to improve PPSZ.Comment: 13 pages; major revision with simplified algorithm but slightly worse
constant
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