7 research outputs found
Evasiveness of Graph Properties and Topological Fixed-Point Theorems
Many graph properties (e.g., connectedness, containing a complete subgraph)
are known to be difficult to check. In a decision-tree model, the cost of an
algorithm is measured by the number of edges in the graph that it queries. R.
Karp conjectured in the early 1970s that all monotone graph properties are
evasive -- that is, any algorithm which computes a monotone graph property must
check all edges in the worst case. This conjecture is unproven, but a lot of
progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in
1984, topological methods have been applied to prove partial results on the
Karp conjecture. This text is a tutorial on these topological methods. I give a
fully self-contained account of the central proofs from the paper of Kahn,
Saks, and Sturtevant, with no prior knowledge of topology assumed. I also
briefly survey some of the more recent results on evasiveness.Comment: Book version, 92 page
Counting induced subgraphs: a topological approach to #W[1]-hardness
We investigate the problem of counting all induced subgraphs of size in a graph that satisfy a given property . This continues the work of Jerrum and Meeks who proved the problem to be -hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties , the problem is hard for if the reduced Euler characteristic of the associated simplicial (graph) complex of is non-zero. This observation links to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that is -hard for every monotone property that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not -edge-connected for . Moreover, we show that for those properties can not be solved in time for any computable function unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that is -hard if is any non-trivial modularity constraint on the number of edges with respect to some prime or if enforces the presence of a fixed isolated subgraph
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