Graph classes with given 3-connected components: asymptotic counting and critical phenomena

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

Journal of Artificial Intelligence Research ◽

10.1613/jair.3968 ◽

2013 ◽

Vol 47 ◽

pp. 575-611 ◽

Cited By ~ 4

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.

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Graph classes with given 3-connected components: Asymptotic enumeration and random graphs

Random Structures and Algorithms ◽

10.1002/rsa.20421 ◽

2012 ◽

Vol 42 (4) ◽

pp. 438-479 ◽

Cited By ~ 15

Author(s):

Omer Giménez ◽

Marc Noy ◽

Juanjo Rué

Keyword(s):

Random Graphs ◽

Connected Components ◽

Asymptotic Enumeration ◽

Graph Classes

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Connectivity for Bridge-Addable Monotone Graph Classes

Combinatorics Probability Computing ◽

10.1017/s0963548312000272 ◽

2012 ◽

Vol 21 (6) ◽

pp. 803-815 ◽

Cited By ~ 10

Author(s):

L. ADDARIO-BERRY ◽

C. MCDIARMID ◽

B. REED

Keyword(s):

Connected Components ◽

Graph Classes ◽

Vertex Set ◽

Special Case ◽

Monotone Graph ◽

Labelled Graphs

A classof labelled graphs isbridge-addableif, for all graphsGinand all verticesuandvin distinct connected components ofG, the graph obtained by adding an edge betweenuandvis also in; the classismonotoneif, for allG∈and all subgraphsHofG, we haveH∈. We show that for any bridge-addable, monotone classwhose elements have vertex set {1,. . .,n}, the probability that a graph chosen uniformly at random fromis connected is at least (1−on(1))e−½, whereon(1) → 0 asn→ ∞. This establishes the special case of the conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This result has also been obtained independently by Kang and Panagiotou.

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