19,862 research outputs found
The parameterised complexity of counting even and odd induced subgraphs
We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory
Some hard families of parameterised counting problems
We consider parameterised subgraph-counting problems of the following form:
given a graph G, how many k-tuples of its vertices have a given property? A
number of such problems are known to be #W[1]-complete; here we substantially
generalise some of these existing results by proving hardness for two large
families of such problems. We demonstrate that it is #W[1]-hard to count the
number of k-vertex subgraphs having any property where the number of distinct
edge-densities of labelled subgraphs that satisfy the property is o(k^2). In
the special case that the property in question depends only on the number of
edges in the subgraph, we give a strengthening of this result which leads to
our second family of hard problems.Comment: A few more minor changes. This version to appear in the ACM
Transactions on Computation Theor
On maximal chain subgraphs and covers of bipartite graphs
In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem.
The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time
The parameterised complexity of counting connected subgraphs and graph motifs
We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem
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