5 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
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
Randomized word-parallel algorithms for detection of small induced subgraphs
Induced subgraph detection is a widely studied set of problems in theoretical computer science, with applications in e.g. social networks, molecular biology and other domains that use graph representations. Our focus lies on practical comparison of some well-known deterministic algorithms to recent Monte Carlo algorithms for detecting subgraphs on three and four vertices. For algorithms that involve operations with adjacency matrices, we study the gain of applying word parallelism, i.e. exploiting the parallel nature of common processor operations such as bitwise conjunction and disjunction. We present results of empirical running times for our implementations of the algorithms. Our results reveal insights as to when the Monte Carlo algorithms trump their deterministic counterparts and also include statistically significant improvements of several algorithms when applying word parallelism