12 research outputs found
Faster Existential FO Model Checking on Posets
We prove that the model checking problem for the existential fragment of
first-order (FO) logic on partially ordered sets is fixed-parameter tractable
(FPT) with respect to the formula and the width of a poset (the maximum size of
an antichain). While there is a long line of research into FO model checking on
graphs, the study of this problem on posets has been initiated just recently by
Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential
fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon
their result in two ways: (1) the runtime of our algorithm is
O(f(|{\phi}|,w).n^2) on n-element posets of width w, compared to O(g(|{\phi}|).
n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow.
We complement this result by showing that, under a certain
complexity-theoretical assumption, the existential FO model checking problem
does not have a polynomial kernel.Comment: Paper as accepted to the LMCS journal. An extended abstract of an
earlier version of this paper has appeared at ISAAC'14. Main changes to the
previous version are improvements in the Multicoloured Clique part (Section
4
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
We show that the model-checking problem for successor-invariant first-order
logic is fixed-parameter tractable on graphs with excluded topological
subgraphs when parameterised by both the size of the input formula and the size
of the exluded topological subgraph. Furthermore, we show that model-checking
for order-invariant first-order logic is tractable on coloured posets of
bounded width, parameterised by both the size of the input formula and the
width of the poset.
Our result for successor-invariant FO extends previous results for this logic
on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors
(Eickmeyer et al., LICS 2013), further narrowing the gap between what is known
for FO and what is known for successor-invariant FO. The proof uses Grohe and
Marx's structure theorem for graphs with excluded topological subgraphs. For
order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO
carries over to order-invariant FO
Precedence-constrained scheduling problems parameterized by partial order width
Negatively answering a question posed by Mnich and Wiese (Math. Program.
154(1-2):533-562), we show that P2|prec,|, the
problem of finding a non-preemptive minimum-makespan schedule for
precedence-constrained jobs of lengths 1 and 2 on two parallel identical
machines, is W[2]-hard parameterized by the width of the partial order giving
the precedence constraints. To this end, we show that Shuffle Product, the
problem of deciding whether a given word can be obtained by interleaving the
letters of other given words, is W[2]-hard parameterized by , thus
additionally answering a question posed by Rizzi and Vialette (CSR 2013).
Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled.
Oper. 7(1):75-82), we show that the more general Resource-Constrained Project
Scheduling problem is fixed-parameter tractable parameterized by the partial
order width combined with the maximum allowed difference between the earliest
possible and factual starting time of a job.Comment: 14 pages plus appendi
Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU
The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set
intersecting each of the sets in a given input collection a given number of
times. Generalizing a well-known data reduction algorithm due to Weihe, we show
a problem kernel for Multiple Hitting Set parameterized by the Dilworth number,
a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far
unexplored in the context of parameterized complexity theory. Using matrix
multiplication, we speed up the algorithm to quadratic sequential time and
logarithmic parallel time. We experimentally evaluate our algorithms. By
implementing our algorithm on GPUs, we show the feasability of realizing
kernelization algorithms on SIMD (Single Instruction, Multiple Date)
architectures.Comment: Added experiments on one more data se
Refining Automatically Extracted Knowledge Bases Using Crowdsourcing
Machine-constructed knowledge bases often contain noisy and inaccurate facts. There exists significant work in developing automated algorithms for knowledge base refinement. Automated approaches improve the quality of knowledge bases but are far from perfect. In this paper, we leverage crowdsourcing to improve the quality of automatically extracted knowledge bases. As human labelling is costly, an important research challenge is how we can use limited human resources to maximize the quality improvement for a knowledge base. To address this problem, we first introduce a concept of semantic constraints that can be used to detect potential errors and do inference among candidate facts. Then, based on semantic constraints, we propose rank-based and graph-based algorithms for crowdsourced knowledge refining, which judiciously select the most beneficial candidate facts to conduct crowdsourcing and prune unnecessary questions. Our experiments show that our method improves the quality of knowledge bases significantly and outperforms state-of-the-art automatic methods under a reasonable crowdsourcing cost
Model-Checking on Ordered Structures
We study the model-checking problem for first- and monadic second-order logic
on finite relational structures. The problem of verifying whether a formula of
these logics is true on a given structure is considered intractable in general,
but it does become tractable on interesting classes of structures, such as on
classes whose Gaifman graphs have bounded treewidth. In this paper we continue
this line of research and study model-checking for first- and monadic
second-order logic in the presence of an ordering on the input structure. We do
so in two settings: the general ordered case, where the input structures are
equipped with a fixed order or successor relation, and the order invariant
case, where the formulas may resort to an ordering, but their truth must be
independent of the particular choice of order. In the first setting we show
very strong intractability results for most interesting classes of structures.
In contrast, in the order invariant case we obtain tractability results for
order-invariant monadic second-order formulas on the same classes of graphs as
in the unordered case. For first-order logic, we obtain tractability of
successor-invariant formulas on classes whose Gaifman graphs have bounded
expansion. Furthermore, we show that model-checking for order-invariant
first-order formulas is tractable on coloured posets of bounded width.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0851
Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number
Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can be done more eÆciently than by using the well-known algorithms based on bipartite matching and matrix multiplication. In particular, we show that deciding deciding if an order has width k can be done in O(kn²) time and whether a graph has Dilworth number k can be done in O(k²n²) time. For very small k we have even better results. We show that orders of width at most 3 can be recognized in O(n) time and of width at most 4 in O(n log n)