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Inapproximability of Maximum Biclique Problems, Minimum -Cut and Densest At-Least--Subgraph from the Small Set Expansion Hypothesis
The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly
states that it is NP-hard to distinguish between a graph with a small subset of
vertices whose edge expansion is almost zero and one in which all small subsets
of vertices have expansion almost one. In this work, we prove inapproximability
results for the following graph problems based on this hypothesis:
- Maximum Edge Biclique (MEB): given a bipartite graph , find a complete
bipartite subgraph of with maximum number of edges.
- Maximum Balanced Biclique (MBB): given a bipartite graph , find a
balanced complete bipartite subgraph of with maximum number of vertices.
- Minimum -Cut: given a weighted graph , find a set of edges with
minimum total weight whose removal partitions into connected
components.
- Densest At-Least--Subgraph (DALS): given a weighted graph , find a
set of at least vertices such that the induced subgraph on has
maximum density (the ratio between the total weight of edges and the number of
vertices).
We show that, assuming SSEH and NP BPP, no polynomial time
algorithm gives -approximation for MEB or MBB for every
constant . Moreover, assuming SSEH, we show that it is NP-hard
to approximate Minimum -Cut and DALS to within factor
of the optimum for every constant .
The ratios in our results are essentially tight since trivial algorithms give
-approximation to both MEB and MBB and efficient -approximation
algorithms are known for Minimum -Cut [SV95] and DALS [And07, KS09].
Our first result is proved by combining a technique developed by Raghavendra
et al. [RST12] to avoid locality of gadget reductions with a generalization of
Bansal and Khot's long code test [BK09] whereas our second result is shown via
elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a
different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced
Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper we analyze all Boolean fragments
of autoepistemic logic with respect to the computational complexity of the
three most common decision problems expansion existence, brave reasoning and
cautious reasoning. As a second contribution we classify the computational
complexity of counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
Complexity of Non-Monotonic Logics
Over the past few decades, non-monotonic reasoning has developed to be one of
the most important topics in computational logic and artificial intelligence.
Different ways to introduce non-monotonic aspects to classical logic have been
considered, e.g., extension with default rules, extension with modal belief
operators, or modification of the semantics. In this survey we consider a
logical formalism from each of the above possibilities, namely Reiter's default
logic, Moore's autoepistemic logic and McCarthy's circumscription.
Additionally, we consider abduction, where one is not interested in inferences
from a given knowledge base but in computing possible explanations for an
observation with respect to a given knowledge base.
Complexity results for different reasoning tasks for propositional variants
of these logics have been studied already in the nineties. In recent years,
however, a renewed interest in complexity issues can be observed. One current
focal approach is to consider parameterized problems and identify reasonable
parameters that allow for FPT algorithms. In another approach, the emphasis
lies on identifying fragments, i.e., restriction of the logical language, that
allow more efficient algorithms for the most important reasoning tasks. In this
survey we focus on this second aspect. We describe complexity results for
fragments of logical languages obtained by either restricting the allowed set
of operators (e.g., forbidding negations one might consider only monotone
formulae) or by considering only formulae in conjunctive normal form but with
generalized clause types.
The algorithmic problems we consider are suitable variants of satisfiability
and implication in each of the logics, but also counting problems, where one is
not only interested in the existence of certain objects (e.g., models of a
formula) but asks for their number.Comment: To appear in Bulletin of the EATC
Downsizing of acute inpatient beds associated with private finance initiative: Scotland's case study
OBJECTIVES: To evaluate whether the projected 24% reduction in acute bed numbers in Lothian hospitals, which formed part of the private finance initiative (PFI) plans for the replacement Royal Infirmary of Edinburgh, is being compensated for by improvements in efficiency and greater use of community facilities, and to ascertain whether there is an independent PFI effect by comparing clinical activity and performance in acute specialties in Lothian hospitals with other NHS hospitals in Scotland. DESIGN: Comparison of projected and actual trends in acute bed capacity and inpatient and day case admissions in the first five years (1995-6 to 2000-1) of Lothian Health Board's integrated healthcare plan. Population study of trends in bed rate, hospital activity, length of stay, and throughput in Lothian hospitals compared with the rest of Scotland from 1990-1 to 2000-1. MAIN OUTCOME MEASURES: Staffed bed rates, admission rates, mean lengths of stay, occupancy, and throughput in four adult acute specialty groups in 1990-1, 1995-6, and 2000-1. RESULTS: By 2000-1, rates for inpatient admission in all acute, medical, surgical, and intensive therapy specialties in Lothian hospitals were respectively 20%, 6%, 28%, and 38% below those in the rest of Scotland. Day case rates in all acute and acute surgical specialties were 13% and 33% lower. The proportion of delayed discharges in staffed acute and post-acute NHS beds in Lothian hospitals exceeded the Scottish average (15% and 12% respectively; P<0.001). CONCLUSION: The planning targets and increase in clinical activity in acute specialties in Lothian hospitals associated with PFI had not been achieved by 2000-1. The effect on clinical activity has been a steeper decline in the number of acute beds and rates of admission in Lothian hospitals compared with the rest of Scotland between 1995-6 and 2000-1
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