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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.

We study the induced subgraph isomorphism problem and the general subgraph isomorphism problem for small pattern graphs. We present a new general method for detecting induced subgraphs of a host graph isomorphic to a fixed pattern graph by reduction to polynomial testing for nonidentity with zero over a field of finite characteristic. It yields new upper time bounds for several pattern graphs on five vertices and provides an alternative combinatorial method for the majority of pattern graphs on four and three vertices. Since our method avoids the large overhead of fast matrix multiplication, it can be of practical interest even for larger pattern graphs. Next, we derive new upper time bounds on counting the number of isomorphisms between a fixed pattern graph with an independent set of size $s$ and a subgraph of the host graph. We also consider a weighted version of the counting problem, when one counts the number of isomorphisms between the pattern graph and lightest subgraphs, providing a slightly slowe...

Given a pattern graph H with l edges, and a host graph G guaranteed to contain at most one occurrence of a subgraph isomorphic to H , we show that the time complexity of the problem of finding such an occurrence if any in G as well as that of the decision version of the problem are within a multiplicative factor O l of the time complexity for the corresponding problem in the general case, when G may contain several occurrences of H . It follows that for pattern graphs of constant size, the aforementioned uniqueness guarantee cannot yield any asymptotic speed up. We also derive analogous results with the analogous multiplicative factor linear in the number of vertices of H in the induced case when occurrences of induced subgraphs of G isomorphic to H are sought.

We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let $H$ be a fixed graph with $k$ vertices and an independent set of size $s.$ 1. Detecting if an $n$-vertex graph contains a (not necessarily induced) subgraph isomorphic to $H$ can be done in time $O(n^{\omega(\lceil (k-s)/2 \rceil, 1, \lfloor (k-s)/2 \rfloor )})$, where $\omega (p,q,r)$ is the exponent of fast arithmetic matrix multiplication of an $n^p\times n^q$ matrix by an $n^q\times n^r$ matrix. 2. When $s=2,$ counting the number of (not necessarily induced) subgraphs isomorphic to $H$ can be done in the same time, i.e., in time $O(n^{\omega(\lceil (k-2)/2 \rceil, 1, \lfloor (k-2)/2 \rfloor )}).$ It follows in particular that we can count the nu...

For a set of rooted, unordered, distinctly leaf-labeled trees, the NP-hard maximum agreement subtree problem (MAST) asks for a tree contained (up to isomorphism or homeomorphism) in all of the input trees with as many labeled leaves as possible. We study the ordered variants of MAST where the trees are uniformly or non-uniformly ordered. We provide the first known polynomial-time algorithms for the uniformly and non-uniformly ordered homeomorphic variants as well as the uniformly and non-uniformly ordered isomorphic variants of MAST. Our algorithms run in time $O(kn^3)$, $O(n^3 \min \{kn,\, n+\log^{k-1}n\})$, $O(kn^3)$, and $O(n^3 \min \{kn,\, n+\log^{k-1}n\})$, respectively, where n is the number of leaf labels and k is the number of input trees.

We consider the following clustering problems: given an undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

The maximum rooted resolved triplets consistency problem takes as input a set \(\mathcal {R}\) of resolved triplets and asks for a rooted phylogenetic tree that is consistent with the maximum number of elements in \(\mathcal {R}\). This paper studies the polynomial-time approximability of a generalization of the problem where in addition to resolved triplets, the input may contain fan triplets and forbidden triplets. To begin with, we observe that the generalized problem admits a 1/4-approximation in polynomial time. Next, we present a polynomial-time approximation scheme (PTAS) for dense instances based on smooth polynomial integer programming. Finally, we generalize Wu’s exact exponential-time algorithm in [19] for the original problem to also allow fan triplets, forbidden resolved triplets, and forbidden fan triplets. Forcing the algorithm to always output a \(k\)-ary phylogenetic tree for any specified \(k \ge 2\) then leads to an exponential-time approximation scheme (ETAS) for the generalized, unrestricted problem.

We study the induced subgraph isomorphism problem and the general subgraph isomorphism problem for small pattern graphs.

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. For instance, the larger the maximum independent set of the pattern graph is the more efficient algorithms are known. The situation seems to be substantially different in the case of induced subgraph isomorphism for pattern graphs of fixed size. We present two results which provide evidence that no topology of an induced subgraph of fixed size can be easier to detect or count than an independent set of related size. We show that: Any fixed pattern graph that has 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 k vertices is not easier to detect than an independent set on ⌈k/2 ⌉ vertices, and that an induced even cycle on k vertices is not easier to detect than an independent set on k/2 vertices. In view of linear time upper bounds on induced paths of length three and four, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and induced complete split graphs. 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. Finally, we show that the so called diamond, paw and C 4 are not easier to detect as induced subgraphs than an independent set on three vertices.

We present a general framework for computing various optimal embeddings of undirected and directed connected graphs in two and multi-dimensional integer lattices in time sub-exponential either in the minimum number n of lattice points used by such optimal embeddings or in the budget upper bound b on the number of lattice points that may be used in an embedding. The sub-exponential upper bounds in the two dimensional case and d-dimensional case are respectively of the form 2\(^{O(\sqrt{ln}log n)}\), 2\(^{O(\sqrt{lb}log b)}\) and 2\(^{O(dl^{1/d_n (d-1)/d}{\rm log} n)}\), 2\(^{O(dl^{1/d_b (d-1)/d}{\rm log} b)}\), where l stands for the degree of the allowed overlap. For the problem of minimum total edge length planar or multi-dimensional embedding or layout of a graph and the problem of an optimal protein folding in the so called HP model we obtain the upper bounds in terms of n. Note that in case of protein folding n is also the size of the input. The list of problems for which we can derive the upper bounds in terms of b includes among other things