scholarly journals Backdoor DNFs

2021 ◽  
Author(s):  
Sebastian Ordyniak ◽  
Andre Schidler ◽  
Stefan Szeider
Keyword(s):  
Empirical Study ◽  
Fixed Parameter Tractability ◽  
Detection Problem ◽  
Partial Assignment ◽  
Backdoor Sets ◽  
Fixed Parameter ◽  
Cnf Formulas

We introduce backdoor DNFs, as a tool to measure the theoretical hardness of CNF formulas. Like backdoor sets and backdoor trees, backdoor DNFs are defined relative to a tractable class of CNF formulas. Each conjunctive term of a backdoor DNF defines a partial assignment that moves the input CNF formula into the base class. Backdoor DNFs are more expressive and potentially smaller than their predecessors backdoor sets and backdoor trees. We establish the fixed-parameter tractability of the backdoor DNF detection problem. Our results hold for the fundamental base classes Horn and 2CNF, and their combination. We complement our theoretical findings by an empirical study. Our experiments show that backdoor DNFs provide a significant improvement over their predecessors.

Chapter 17. Fixed-Parameter Tractability

2021 ◽  
Author(s):  
Marko Samer ◽  
Stefan Szeider
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Complexity Analysis ◽  
Fixed Parameter Tractability ◽  
Graph Representations ◽  
Fine Grained ◽  
Input Size ◽  
Backdoor Sets ◽  
Fixed Parameter ◽  
Problem Instances

Parameterized complexity is a new theoretical framework that considers, in addition to the overall input size, the effects on computational complexity of a secondary measurement, the parameter. This two-dimensional viewpoint allows a fine-grained complexity analysis that takes structural properties of problem instances into account. The central notion is “fixed-parameter tractability” which refers to solvability in polynomial time for each fixed value of the parameter such that the order of the polynomial time bound is independent of the parameter. This chapter presents main concepts and recent results on the parameterized complexity of the satisfiability problem and it outlines fundamental algorithmic ideas that arise in this context. Among the parameters considered are the size of backdoor sets with respect to various tractable base classes and the treewidth of graph representations of satisfiability instances.

scholarly journals Fixed-parameter tractability and lower bounds for stabbing problems

2013 ◽  
Vol 46 (7) ◽  
pp. 839-860 ◽  
Author(s):  
Panos Giannopoulos ◽  
Christian Knauer ◽  
Günter Rote ◽  
Daniel Werner
Keyword(s):  
Lower Bounds ◽  
Fixed Parameter Tractability ◽  
Fixed Parameter

Fixed-parameter tractability for the tree assembly problem

2021 ◽  
Author(s):  
Feng Shi ◽  
Jie You ◽  
Zhen Zhang ◽  
Jingyi Liu ◽  
Jianxin Wang
Keyword(s):  
Fixed Parameter Tractability ◽  
Fixed Parameter ◽  
Assembly Problem

Fixed-parameter tractability and data reduction for multicut in trees

Networks ◽  
2005 ◽  
Vol 46 (3) ◽  
pp. 124-135 ◽  
Author(s):  
Jiong Guo ◽  
Rolf Niedermeier
Keyword(s):  
Data Reduction ◽  
Fixed Parameter Tractability ◽  
Fixed Parameter

scholarly journals Fixed-Parameter Tractability of Satisfying Beyond the Number of Variables

Algorithmica ◽  
2012 ◽  
Vol 68 (3) ◽  
pp. 739-757 ◽  
Author(s):  
Robert Crowston ◽  
Gregory Gutin ◽  
Mark Jones ◽  
Venkatesh Raman ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Fixed Parameter Tractability ◽  
Fixed Parameter

scholarly journals Finding Points in General Position

2017 ◽  
Vol 27 (04) ◽  
pp. 277-296 ◽  
Author(s):  
Vincent Froese ◽  
Iyad Kanj ◽  
André Nichterlein ◽  
Rolf Niedermeier
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Subset Selection ◽  
Selection Problem ◽  
Fixed Parameter Tractability ◽  
Maximum Cardinality ◽  
Exponential Time ◽  
Running Time ◽  
Fixed Parameter ◽  
Set Of Points

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

scholarly journals Adapting Stable Matchings to Evolving Preferences

2020 ◽  
Vol 34 (02) ◽  
pp. 1830-1837 ◽  
Author(s):  
Robert Bredereck ◽  
Jiehua Chen ◽  
Dušan Knop ◽  
Junjie Luo ◽  
Rolf Niedermeier
Keyword(s):  
Computational Complexity ◽  
Computational Cost ◽  
Modern Society ◽  
Stable Matching ◽  
Stable Marriage ◽  
Fixed Parameter Tractability ◽  
Stable Matchings ◽  
Changing Environments ◽  
Fixed Parameter ◽  
Comprehensive Picture

Adaptivity to changing environments and constraints is key to success in modern society. We address this by proposing “incrementalized versions” of Stable Marriage and Stable Roommates. That is, we try to answer the following question: for both problems, what is the computational cost of adapting an existing stable matching after some of the preferences of the agents have changed. While doing so, we also model the constraint that the new stable matching shall be not too different from the old one. After formalizing these incremental versions, we provide a fairly comprehensive picture of the computational complexity landscape of Incremental Stable Marriage and Incremental Stable Roommates. To this end, we exploit the parameters “degree of change” both in the input (difference between old and new preference profile) and in the output (difference between old and new stable matching). We obtain both hardness and tractability results, in particular showing a fixed-parameter tractability result with respect to the parameter “distance between old and new stable matching”.

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