Greedy Algorithms for Steiner Forest

In the Steiner Forest problem, we are given terminal pairs $\{s_i, t_i\}$, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson gave primal-dual constant-factor approximation algorithms for this problem; until now, the only constant-factor approximations we know are via linear programming relaxations. We consider the following greedy algorithm: Given terminal pairs in a metric space, call a terminal "active" if its distance to its partner is non-zero. Pick the two closest active terminals (say $s_i, t_j$), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. Our main result is that this algorithm is a constant-factor approximation. We also use this algorithm to give new, simpler constructions of cost-sharing schemes for Steiner forest. In particular, the first "group-strict" cost-shares for this problem implies a very simple combinatorial sampling-based algorithm for stochastic Steiner forest

Pathologies of the Brauer-Manin obstruction

We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the \'etale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.Comment: 22 pages, to appear in Mathematische Zeitschrif

The generalized identification of truly interfacial molecules (ITIM) algorithm for nonplanar interfaces

We present a generalized version of the ITIM algorithm for the identification of interfacial molecules, which is able to treat arbitrarily shaped interfaces. The algorithm exploits the similarities between the concept of probe sphere used in ITIM and the circumsphere criterion used in the α-shapes approach, and can be regarded either as a reference-frame independent version of the former, or as an extended version of the latter that includes the atomic excluded volume. The new algorithm is applied to compute the intrinsic orientational order parameters of water around a dodecylphosphocholine and a cholic acid micelle in aqueous environment, and to the identification of solvent-reachable sites in four model structures for soot. The additional algorithm introduced for the calculation of intrinsic density profiles in arbitrary geometries proved to be extremely useful also for planar interfaces, as it allows to solve the paradox of smeared intrinsic profiles far from the interface. © 2013 American Institute of Physics

Transport on randomly evolving trees

The time process of transport on randomly evolving trees is investigated. By introducing the notions of living and dead nodes a model of random tree evolution is constructed which describes the spreading in time of objects corresponding to nodes. By using the method of the age-dependent branching processes we derive the joint distribution function of the number of living and dead nodes, and determine the correlation between these node numbers as a function of time. Also analyzed are the stochastic properties of the end-nodes; and the correlation between the numbers of living and dead end-nodes is shown to change its character suddenly at the very beginning of the evolution process. The survival probability of random trees is investigated and expressions are derived for this probability.Comment: 16 pages, 8 figures, published in Phys. Rev. E 72, 051101 (2005

Partitioning Hypergraphs is Hard: Models, Inapproximability, and Applications

We study the balanced $k$-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on $n$ nodes, our goal is to partition the node set into $k$ parts of size at most $(1+\epsilon)\cdot \frac{n}{k}$ each, while minimizing the cost of the partitioning, defined as the number of cut hyperedges, possibly also weighted by the number of partitions they intersect. We show that this problem cannot be approximated to within a $n^{1/\text{poly} \log\log n}$ factor of the optimal solution in polynomial time if the Exponential Time Hypothesis holds, even for hypergraphs of maximal degree 2. We also study the hardness of the partitioning problem from a parameterized complexity perspective, and in the more general case when we have multiple balance constraints. Furthermore, we consider two extensions of the partitioning problem that are motivated from practical considerations. Firstly, we introduce the concept of hyperDAGs to model precedence-constrained computations as hypergraphs, and we analyze the adaptation of the balanced partitioning problem to this case. Secondly, we study the hierarchical partitioning problem to model hierarchical NUMA (non-uniform memory access) effects in modern computer architectures, and we show that ignoring this hierarchical aspect of the communication cost can yield significantly weaker solutions.Comment: Published in the 35th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2023