843 research outputs found
Satisfiability of Almost Disjoint CNF Formulas
We call a CNF formula linear if any two clauses have at most one variable in
common. Let m(k) be the largest integer m such that any linear k-CNF formula
with <= m clauses is satisfiable. We show that 4^k / (4e^2k^3) <= m(k) < ln(2)
k^4 4^k. More generally, a (k,d)-CSP is a constraint satisfaction problem in
conjunctive normal form where each variable can take on one of d values, and
each constraint contains k variables and forbids exacty one of the d^k possible
assignments to these variables. Call a (k,d)-CSP l-disjoint if no two distinct
constraints have l or more variables in common. Let m_l(k,d) denote the largest
integer m such that any l-disjoint (k,d)-CSP with at most m constraints is
satisfiable. We show that 1/k (d^k/(ed^(l-1)k))^(1+1/(l-1))<= m_l(k,d) < c
(k^2/l ln(d) d^k)^(1+1/(l-1)). for some constant c. This means for constant l,
upper and lower bound differ only in a polynomial factor in d and k
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
Unsatisfiable Linear CNF Formulas Are Large and Complex
We call a CNF formula linear if any two clauses have at most one variable in
common. We show that there exist unsatisfiable linear k-CNF formulas with at
most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at
most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic
means, and we have no explicit construction coming even close to it. One reason
for this is that unsatisfiable linear formulas exhibit a more complex structure
than general (non-linear) formulas: First, any treelike resolution refutation
of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$.
This implies that small unsatisfiable linear k-CNF formulas are hard instances
for Davis-Putnam style splitting algorithms. Second, if we require that the
formula F have a strict resolution tree, i.e. every clause of F is used only
once in the resolution tree, then we need at least a^a^...^a clauses, where a
is approximately 2 and the height of this tower is roughly k.Comment: 12 pages plus a two-page appendix; corrected an inconsistency between
title of the paper and title of the arxiv submissio
Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability
A matched formula is a CNF formula whose incidence graph admits a matching
which matches a distinct variable to every clause. We study phase transition in
a context of matched formulas and their generalization of biclique satisfiable
formulas. We have performed experiments to find a phase transition of property
"being matched" with respect to the ratio where is the number of
clauses and is the number of variables of the input formula . We
compare the results of experiments to a theoretical lower bound which was shown
by Franco and Gelder (2003). Any matched formula is satisfiable, moreover, it
remains satisfiable even if we change polarities of any literal occurrences.
Szeider (2005) generalized matched formulas into two classes having the same
property -- var-satisfiable and biclique satisfiable formulas. A formula is
biclique satisfiable if its incidence graph admits covering by pairwise
disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is
NP-complete. In this paper we describe a heuristic algorithm for recognizing
whether a formula is biclique satisfiable and we evaluate it by experiments on
random formulas. We also describe an encoding of the problem of checking
whether a formula is biclique satisfiable into SAT and we use it to evaluate
the performance of our heuristicComment: Conference version submitted to SOFSEM 2018
(https://beda.dcs.fmph.uniba.sk/sofsem2019/) 18 pages(17 without refernces),
3 figures, 8 tables, an algorithm pseudocod
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
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