We study classical polynomial-time approximation algorithms for the transverse field Ising model (TFIM), allowing a mixture of ferromagnetic and antiferromagnetic interactions between pairs of qubits, alongside transverse field terms with arbitrary non-negative weights. In this work, we first prove a second-order conic inequality based on the anticommutation property of the two competing terms (Ising Z_i Z_j vs. field X_i terms), and we use this inequality to strengthen the basic SDP relaxation of the problem. By producing two competing rounded product state solutions and taking the better of the two we achieve an approximation ratio γ≈ 0.7860. A further improvement by non-uniform interpolation achieves a ratio γ ≈ 0.82197. Finally, we give an explicit purely antiferromagnetic TFIM instance on three qubits for which every product state achieves at most 169/180≈ 0.9389 of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely antiferromagnetic case.

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To this end we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the “root uncertain” variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees.

We consider the problem of fair allocation of indivisible goods where we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource j in I has a same value vj > 0 for a subset of customers interested in j and it has no value for other customers. The goal is to find a feasible allocation of the resources to the interested customers such that in the Max-Min scenario (also known as Santa Claus problem) the minimum utility (sum of the resources) received by each of the customers is as high as possible and in the Min-Max case (also known as R||C_max problem), the maximum utility is as low as possible. In this paper we are interested in instances of the problem that admit a PTAS. These instances are not only of theoretical interest but also have practical applications. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; there exists an ordering of the resources such that each customer is interested (has positive evaluation) in a set of consecutive resources and we demonstrate a PTAS. For the Min-Max allocation problem, we obtain a PTAS for instances in which there is an ordering of the customers (machines) and each resource (job) is adjacent to a consecutive set of customers (machines). Next we show that our method for the Max-Min scenario, can be extended to a broader class of bipartite graphs where the resources can be viewed as a tree and each customer is interested in a sub-tree of a bounded number of leaves of this tree (e.g. a sub-path).