1,489 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
Model counting for CNF formuals of bounded module treewidth.
The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity
On space efficiency of algorithms working on structural decompositions of graphs
Dynamic programming on path and tree decompositions of graphs is a technique
that is ubiquitous in the field of parameterized and exponential-time
algorithms. However, one of its drawbacks is that the space usage is
exponential in the decomposition's width. Following the work of Allender et al.
[Theory of Computing, '14], we investigate whether this space complexity
explosion is unavoidable. Using the idea of reparameterization of Cai and
Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely
related to a conjecture that the Longest Common Subsequence problem
parameterized by the number of input strings does not admit an algorithm that
simultaneously uses XP time and FPT space. Moreover, we complete the complexity
landscape sketched for pathwidth and treewidth by Allender et al. by
considering the parameter tree-depth. We prove that computations on tree-depth
decompositions correspond to a model of non-deterministic machines that work in
polynomial time and logarithmic space, with access to an auxiliary stack of
maximum height equal to the decomposition's depth. Together with the results of
Allender et al., this describes a hierarchy of complexity classes for
polynomial-time non-deterministic machines with different restrictions on the
access to working space, which mirrors the classic relations between treewidth,
pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new
version is augmented with a space-efficient algorithm for Dominating Set
using the Chinese remainder theore
A contraction-recursive algorithm for treewidth
Let tw(G) denote the treewidth of graph G. Given a graph G and a positive
integer k such that tw(G) <= k + 1, we are to decide if tw(G) <= k. We give a
certifying algorithm RTW ("R" for recursive) for this task: it returns one or
more tree-decompositions of G of width <= k if the answer is YES and a minimal
contraction H of G such that tw(H) > k otherwise.
RTW uses a heuristic variant of Tamaki's PID algorithm for treewidth
(ESA2017), which we call HPID. RTW, given G and k, interleaves the execution of
HPID with recursive calls on G /e for edges e of G, where G / e denotes the
graph obtained from G by contracting edge e. If we find that tw(G / e) > k,
then we have tw(G) > k with the same certificate. If we find that tw(G / e) <=
k, we "uncontract" the bags of the certifying tree-decompositions of G / e into
bags of G and feed them to HPID to help progress. If the question is not
resolved after the recursive calls are made for all edges, we finish HPID in an
exhaustive mode. If it turns out that tw(G) > k, then G is a certificate for
tw(G') > k for every G' of which G is a contraction, because we have found tw(G
/ e) <= k for every edge e of G. This final round of HPID guarantees the
correctness of the algorithm, while its practical efficiency derives from our
methods of "uncontracting" bags of tree-decompositions of G / e to useful bags
of G, as well as of exploiting those bags in HPID.
Experiments show that our algorithm drastically extends the scope of
practically solvable instances. In particular, when applied to the 100
instances in the PACE 2017 bonus set, the number of instances solved by our
implementation on a typical laptop, with the timeout of 100, 1000, and 10000
seconds per instance, are 72, 92, and 98 respectively, while these numbers are
11, 38, and 68 for Tamaki's PID solver and 65, 82, and 85 for his new solver
(SEA 2022).Comment: 17 pages, 2 figures, submitted IPEC 202
Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large
We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC\u2708]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT\u2721] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP
Tag-Cloud Drawing: Algorithms for Cloud Visualization
Tag clouds provide an aggregate of tag-usage statistics. They are typically
sent as in-line HTML to browsers. However, display mechanisms suited for
ordinary text are not ideal for tags, because font sizes may vary widely on a
line. As well, the typical layout does not account for relationships that may
be known between tags. This paper presents models and algorithms to improve the
display of tag clouds that consist of in-line HTML, as well as algorithms that
use nested tables to achieve a more general 2-dimensional layout in which tag
relationships are considered. The first algorithms leverage prior work in
typesetting and rectangle packing, whereas the second group of algorithms
leverage prior work in Electronic Design Automation. Experiments show our
algorithms can be efficiently implemented and perform well.Comment: To appear in proceedings of Tagging and Metadata for Social
Information Organization (WWW 2007
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