274 research outputs found
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
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
Trading inference effort versus size in CNF Knowledge Compilation
Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the
separation results from the discontinued arXiv:1302.442
A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and its Algorithmic Applications
A pair of unit clauses is called conflicting if it is of the form ,
. A CNF formula is unit-conflict free (UCF) if it contains no pair
of conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed
that for each UCF CNF formula with clauses we can simultaneously satisfy at
least \pp m clauses, where \pp =(\sqrt{5}-1)/2. We improve the
Lieberherr-Specker bound by showing that for each UCF CNF formula with
clauses we can find, in polynomial time, a subformula with clauses
such that we can simultaneously satisfy at least \pp m+(1-\pp)m'+(2-3\pp)n"/2
clauses (in ), where is the number of variables in which are not in
.
We consider two parameterized versions of MAX-SAT, where the parameter is the
number of satisfied clauses above the bounds and . The
former bound is tight for general formulas, and the later is tight for UCF
formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance
of the first parameterized problem can be transformed, in polynomial time, into
an equivalent one with at most variables and clauses. We improve
this to variables and clauses. Mahajan and Raman
conjectured that the second parameterized problem is fixed-parameter tractable
(FPT). We show that the problem is indeed FPT by describing a polynomial-time
algorithm that transforms any problem instance into an equivalent one with at
most variables. Our results are obtained using our improvement
of the Lieberherr-Specker bound above
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