scholarly journals A Refined View of Causal Graphs and Component Sizes: SP-Closed Graph Classes and Beyond

2013 ◽  
Vol 47 ◽  
pp. 575-611 ◽  
Author(s):  
C. Bäckström ◽  
P. Jonsson
Keyword(s):  
Domain Size ◽  
Additional Restriction ◽  
Connected Components ◽  
Complexity Classes ◽  
Np Hard ◽  
Causal Graph ◽  
Graph Classes ◽  
Causal Graphs ◽  
Np Hardness

The causal graph of a planning instance is an important tool for planning both in practice and in theory. The theoretical studies of causal graphs have largely analysed the computational complexity of planning for instances where the causal graph has a certain structure, often in combination with other parameters like the domain size of the variables. Chen and Giménez ignored even the structure and considered only the size of the weakly connected components. They proved that planning is tractable if the components are bounded by a constant and otherwise intractable. Their intractability result was, however, conditioned by an assumption from parameterised complexity theory that has no known useful relationship with the standard complexity classes. We approach the same problem from the perspective of standard complexity classes, and prove that planning is NP-hard for classes with unbounded components under an additional restriction we refer to as SP-closed. We then argue that most NP-hardness theorems for causal graphs are difficult to apply and, thus, prove a more general result; even if the component sizes grow slowly and the class is not densely populated with graphs, planning still cannot be tractable unless the polynomial hierachy collapses. Both these results still hold when restricted to the class of acyclic causal graphs. We finally give a partial characterization of the borderline between NP-hard and NP-intermediate classes, giving further insight into the problem.

scholarly journals A Refined Understanding of Cost-optimal Planning with Polytree Causal Graphs

2019 ◽  
Author(s):  
Christer Bäckström ◽  
Peter Jonsson ◽  
Sebastian Ordyniak
Keyword(s):  
Complexity Analysis ◽  
Domain Size ◽  
Research Area ◽  
Optimal Planning ◽  
Causal Graph ◽  
New Methods ◽  
Causal Graphs ◽  
Important Research Area ◽  
Novel Method ◽  
Domain Independent

Complexity analysis based on the causal graphs of planning instances is a highly important research area. In particular, tractability results have led to new methods for constructing domain-independent heuristics. Important early examples of such results were presented by, for instance, Brafman & Domshlak and Katz & Keyder. More general results based on polytrees and bounding certain parameters were subsequently derived by Aghighi et al. and Ståhlberg. We continue this line of research by analyzing cost-optimal planning for instances with a polytree causal graph, bounded domain size and bounded depth. We show that no further restrictions are necessary for tractability, thus generalizing the previous results. Our approach is based on a novel method of closely analysing optimal plans: we recursively decompose the causal graph in a way that allows for bounding the number of variable changes as a function of the depth, using a reording argument and a comparison with prefix trees of known size. We then transform the planning instances into tree-structured constraint satisfaction instances.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals Genetic characterization of a new variant within the ET-37 complex of Neisseria meningitidis associated with outbreaks in various parts of the world

2000 ◽  
Vol 125 (2) ◽  
pp. 285-298 ◽  
Author(s):  
J. JELFS ◽  
R. MUNRO ◽  
F. E. ASHTON ◽  
D. A. CAUGANT
Keyword(s):  
Neisseria Meningitidis ◽  
Genetic Markers ◽  
Southern Hybridization ◽  
Genetic Characterization ◽  
Mortality Rates ◽  
Pulsed Field Gel Electrophoresis ◽  
Additional Restriction ◽  
New Variant ◽  
Genetic Rearrangements

A new variant within the electrophoretic type (ET)-37 complex of Neisseria meningitidis, ET-15, first detected in Canada in 1986, has been associated with severe clinical infections and high mortality rates in several European countries, Israel and Australia. To ascertain the genetic and epidemiological relationships of ET-15 strains from different geographical areas, 72 ET-15 isolates from 10 countries were compared to 13 isolates representing other clones of the ET-37 complex. The 85 strains were analysed by pulsed-field gel electrophoresis (PFGE) using 2 restriction endonucleases and Southern hybridization with 10 genetic markers. Four ET-15 strains and 4 other strains of the ET-37 complex were further examined using an additional restriction enzyme and a total of 18 genetic markers. PFGE fingerprints of the ET-15 strains were closely related. Strains within each country were even more closely related, suggesting single introductions of the clone. Physical mapping of genes in ET-15 and other strains of the ET-37 complex demonstrated that large genetic rearrangements of the genome have occurred in association with the appearance of the ET-15 variant.

scholarly journals Computing the Interleaving Distance is NP-Hard

2019 ◽  
Vol 20 (5) ◽  
pp. 1237-1271 ◽  
Author(s):  
Håvard Bakke Bjerkevik ◽  
Magnus Bakke Botnan ◽  
Michael Kerber
Keyword(s):  
Computational Complexity ◽  
Complete Characterization ◽  
Np Hard ◽  
Slight Improvement ◽  
Approximation Factor ◽  
Distance Computation ◽  
Persistence Modules ◽  
Np Complete ◽  
Interleaving Distance

Abstract We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are 1-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement in the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than 3. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2.

COMPLEXITY AND APPROXIMABILITY OF DOUBLE DIGEST

2005 ◽  
Vol 03 (02) ◽  
pp. 207-223
Author(s):  
MARK CIELIEBAK ◽  
STEPHAN EIDENBENZ ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Real Life ◽  
Approximation Ratio ◽  
Additional Restriction ◽  
Np Hard ◽  
Model Errors ◽  
Np Completeness ◽  
The Third ◽  
Np Complete ◽  
Feasible Solutions

We revisit the DOUBLE DIGEST problem, which occurs in sequencing of large DNA strings and consists of reconstructing the relative positions of cut sites from two different enzymes. We first show that DOUBLE DIGEST is strongly NP-complete, improving upon previous results that only showed weak NP-completeness. Even the (experimentally more meaningful) variation in which we disallow coincident cut sites turns out to be strongly NP-complete. In the second part, we model errors in data as they occur in real-life experiments: we propose several optimization variations of DOUBLE DIGEST that model partial cleavage errors. We then show that most of these variations are hard to approximate. In the third part, we investigate variations with the additional restriction that coincident cut sites are disallowed, and we show that it is NP-hard to even find feasible solutions in this case, thus making it impossible to guarantee any approximation ratio at all.

Logical and schematic characterization of complexity classes

1993 ◽  
Vol 30 (1) ◽  
pp. 61-87 ◽  
Author(s):  
Iain A. Stewart
Keyword(s):  
Complexity Classes

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

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