No abstract

For a binary integer program (IP) max c T x, Ax ≤ b, x ∈ {0, 1} n , where A ∈ R m×n and c ∈ R n have independent Gaussian entries and the right-hand side b ∈ R m satisfies that its negative coordinates have ℓ2 norm at most n/10, we prove that the gap between the value of the linear programming relaxation and the IP is upper bounded by poly(m)(log n) 2 /n with probability at least 1 − 1/n 7 − 1/2 Ω(m) . Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math. of O.R., 1989) in the case of random packing IPs. In constrast to the packing case, our integrality gap depends only polynomially on m instead of exponentially. By recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), the bound on the integrality gap immediately implies that branch and bound requires n poly(m) time on random Gaussian IPs with good probability, which is polynomial when the number of constraints m is fixed.

We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of $$O(5^k\cdot n\cdot m)$$ O ( 5 k · n · m ) . We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in $$O((8k)^d5^ k\cdot k\cdot n\cdot m)$$ O ( ( 8 k ) d 5 k · k · n · m ) time. Lastly, we introduce an $$O(6^kk!\cdot k\cdot n^2)$$ O ( 6 k k ! · k · n 2 ) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of m binary trees with n leaves each with a running time ofwhere k is the minimum temporal hybridization number. We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in O((8k) time algorithm for computing a minimum temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

Explorable heap selection is the problem of selecting the nth smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured by the total distance traveled in the tree (each edge has unit cost). This problem was originally proposed as a model to study search strategies for the branch-and-bound algorithm with storage restrictions by Karp, Saks and Widgerson (FOCS '86), who gave deterministic and randomized n • exp(O( √ log n)) time algorithms using O(log(n) 2.5 ) and O( √ log n) space respectively. We present a new randomized algorithm with running time O(n log(n) 3 ) using O(log n) space, substantially improving the previous best randomized running time at the expense of slightly increased space usage. We also show an Ω(log(n)n/ log(log(n))) for any algorithm that solves the problem in the same amount of space, indicating that our algorithm is nearly optimal. * This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement QIP-805241)1 [KSW86] did not give the problem a name, so we have attempted to give a descriptive one here.