Search

The NP-hard maximum rooted resolved triplets consistency problem (MRTC) takes as input a set S of leaf labels and a set R of resolved triplets over S and asks for a rooted phylogenetic tree that is consistent with the maximum number of elements in R . This article studies the approximability of a generalization of the problem called the maximum rooted triplets consistency problem (MTC) where in addition to resolved triplets, the input may contain fan triplets, forbidden resolved triplets, and forbidden fan triplets. To begin with, we observe that MTC admits a 1 ∕ 4 -approximation in polynomial time. Next, we generalize Wu’s exact exponential-time algorithm for MRTC (Wu, 2004) to MTC. Forcing the algorithm to always output a rooted k -ary phylogenetic tree for any specified k ≥ 2 subsequently leads to an exponential-time approximation scheme (ETAS) for MTC. We then present a polynomial-time approximation scheme (PTAS) for complete instances of MTC (meaning that for every S ′ ⊆ S with | S ′ | = 3 , R contains at least one rooted triplet involving the leaf labels in S ′ ), based on the techniques introduced by Jiang et al. (2001) for a related problem. We also study the computational complexity of MTC restricted to fan triplets and forbidden fan triplets. Finally, extensions to weighted instances are considered.

The complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph. Here, we present two results which, in contrast, provide evidence that no topology of an induced subgraph of fixed size can be substantially easier to detect or count than an independent set of related size.We show that any fixed pattern graph having a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k. It follows in particular that an induced path on 2 k - 1 vertices is not easier to detect than an independent set on k vertices, and that an induced cycle on 2k vertices is not easier to detect than an independent set on k vertices. In view of linear time upper bounds on the detection of induced path of length two and three, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and an induced paw and its generalizations.We show also that for an arbitrary pattern graph H on k vertices with no isolated vertices, there is a simple subdivision of H, resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one, that is not easier to detect or count than an independent set on k vertices, respectively.Next, we show that the so-called diamond and its generalizations on k vertices are not easier to detect as induced subgraphs than an independent set on three vertices or an independent set on k vertices, respectively. For C 4 , we give a weaker evidence of its hardness in terms of an independent set on three vertices.Finally, we derive several results relating the complexity of the edge-colored variant of induced subgraph isomorphism to that of the standard variant.

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k+2 for odd k, in time O(n^3^2logn). Thus, in general, it yields a 223 approximation. For a weighted, undirected graph, with non-negative edge weights in the range {1,2,...,M}, we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O(n^2logn(logn+logM)).