7 research outputs found
Counting small induced subgraphs satisfying monotone properties
Given a graph property , the problem asks, on input a graph and a positive integer , to compute the number of induced subgraphs of size in that satisfy . The search for explicit criteria on ensuring that is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property , the problem cannot be solved in time for any function , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a -completeness result
Counting Small Induced Subgraphs Satisfying Monotone Properties
Given a graph property , the problem asks, on input a graph and a positive integer , to compute the number of induced subgraphs of size in that satisfy . The search for explicit criteria on ensuring that is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property , the problem cannot be solved in time for any function , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a -completeness result
Counting Small Induced Subgraphs Satisfying Monotone Properties
Given a graph property , the problem asks, on
input a graph and a positive integer , to compute the number of induced
subgraphs of size in that satisfy . The search for explicit
criteria on ensuring that is hard was
initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the
major line of research on counting small patterns in graphs. However, apart
from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving
that a full classification into "easy" and "hard" properties is possible and
some partial results on edge-monotone properties due to Meeks [Discret. Appl.
Math. 16] and D\"orfler et al. [MFCS 19], not much is known.
In this work, we fully answer and explicitly classify the case of monotone,
that is subgraph-closed, properties: We show that for any non-trivial monotone
property , the problem cannot be solved in time
for any function , unless the
Exponential Time Hypothesis fails. By this, we establish that any significant
improvement over the brute-force approach is unlikely; in the language of
parameterized complexity, we also obtain a -completeness
result.Comment: 33 pages, 2 figure
Counting small induced subgraphs satisfying monotone properties
Given a graph property , the problem asks, on input a graph and a positive integer , to compute the number of induced subgraphs of size in that satisfy . The search for explicit criteria on ensuring that is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property , the problem cannot be solved in time for any function , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a -completeness result
Counting small induced subgraphs satisfying monotone properties
Given a graph property Φ, the problem #IndSub(Φ) asks, on input a graph G and a positive integer k, to compute
the number #IndSub(Φ, k → G) of induced subgraphs of size k in G that satisfy Φ. The search for explicit criteria
on Φ ensuring that #IndSub(Φ) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part
of the major line of research on counting small patterns in graphs. However, apart from an implicit result due
to Curticapean, Dell and Marx [STOC 17] proving that a full classification into “easy” and “hard” properties is
possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfler
et al. [MFCS 19], not much is known.
In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties:
We show that for any non-trivial monotone property Φ, the problem #IndSub(Φ) cannot be solved in time
f(k) · |V (G)|
o(k/log1/2(k)) for any function f, unless the Exponential Time Hypothesis fails. By this, we establish
that any significant improvement over the brute-force approach is unlikely; in the language of parameterized
complexity, we also obtain a #W[1]-completeness result.
To prove our result, we use that for fixed Φ and k, we can express the function G 7→ #IndSub(Φ, k → G)
as a finite linear-combination of homomorphism counts from graphs Hi to G. The coefficient vectors of these
homomorphism counts in the linear combination are called the homomorphism vectors associated to Φ; by the
Complexity Monotonicity framework of Curticapean, Dell and Marx [STOC 17], the positions of non-zero entries
of these vectors are known to determine the complexity of #IndSub(Φ). Our main technical result lifts the notion
of f-polynomials from simplicial complexes to graph properties and relates the derivatives of the f-polynomial
of Φ to its homomorphism vector. We then apply results from the theory of Hermite-Birkhoff interpolation to the
f-polynomial to establish sufficient conditions on Φ which ensure that certain entries in the homomorphism vector
do not vanish—which in turn implies hardness. For monotone graph properties, non-triviality then turns out to
be a sufficient condition. Using the same method, we also prove a conjecture by Jerrum and Meeks [TOCT 15,
Combinatorica 19]: #IndSub(Φ) is #W[1]-complete if Φ is a non-trivial graph property only depending on the
number of edges of the graph
Counting small induced subgraphs satisfying monotone properties
Given a graph property Φ , we study the problem #INDSUB(Φ) which asks, on input a graph G and a positive integer k , to compute the number #IndSub(Φ,k→G) of induced subgraphs of size k in G that satisfy Φ . The search for explicit criteria on Φ ensuring that #INDSUB(Φ) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into “easy” and “hard” properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Φ , the problem #INDSUB(Φ) cannot be solved in time f(k).|V(G)|o(k/log1/2(k)) for any function f , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1] - completeness result.</p