2,907 research outputs found
On the Equivalence among Problems of Bounded Width
In this paper, we introduce a methodology, called decomposition-based
reductions, for showing the equivalence among various problems of
bounded-width.
First, we show that the following are equivalent for any :
* SAT can be solved in time,
* 3-SAT can be solved in time,
* Max 2-SAT can be solved in time,
* Independent Set can be solved in time, and
* Independent Set can be solved in time, where
tw and cw are the tree-width and clique-width of the instance, respectively.
Then, we introduce a new parameterized complexity class EPNL, which includes
Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and
Independent Set parameterized by path-width are EPNL-complete. This implies
that if one of these EPNL-complete problems can be solved in time,
then any problem in EPNL can be solved in time.Comment: accepted to ESA 201
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width
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