We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^\gamma$ where $\gamma\leq 1$ and $U$ is the uniform $[0,1]$ random variable. There is also a constraint that the spanning tree $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,\gamma$, the asymptotic value of the optimum weight via the consideration of a dual problem.
more »
« less
A Randomly Weighted Minimum Arborescence with a Random Cost Constraint
We study the minimum spanning arborescence problem on the complete digraph [Formula: see text], where an edge e has a weight W e and a cost C e , each of which is an independent uniform random variable U s , where [Formula: see text] and U is uniform [Formula: see text]. There is also a constraint that the spanning arborescence T must satisfy [Formula: see text]. We establish, for a range of values for [Formula: see text], the asymptotic value of the optimum weight via the consideration of a dual problem.
more »
« less
 Award ID(s):
 1955175
 NSFPAR ID:
 10414683
 Date Published:
 Journal Name:
 Mathematics of Operations Research
 Volume:
 47
 Issue:
 2
 ISSN:
 0364765X
 Page Range / eLocation ID:
 1664 to 1680
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


null (Ed.)Given an element set E of order n, a collection of subsets [Formula: see text], a cost c S on each set [Formula: see text], a covering requirement r e for each element [Formula: see text], and an integer k, the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection [Formula: see text] to fully cover at least k elements such that the cost of [Formula: see text] is as small as possible and element e is fully covered by [Formula: see text] if it belongs to at least r e sets of [Formula: see text]. This problem generalizes the minimum kunion problem (MinkU) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a [Formula: see text]approximation algorithm for MinPSMC, in which [Formula: see text] is the maximum size of a set in S. And when [Formula: see text], we present a bicriteria algorithm fully covering at least [Formula: see text] elements with approximation ratio [Formula: see text], where [Formula: see text] is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself.more » « less

We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an rrooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasipolynomial time [Formula: see text]approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]approximation algorithm for directed (polymatroid) Steiner tree in quasipolynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]approximation algorithms for the singlesource buyatbulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and singlesource buyatbulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors).more » « less

Let [Formula: see text] be a directed graph associated with a weight [Formula: see text]. For an edgecut [Formula: see text] of [Formula: see text], the average weight of [Formula: see text] is denoted and defined as [Formula: see text]. An optimal edgecut with average weight is an edgecut [Formula: see text] such that [Formula: see text] is maximum among all edgecuts (or minimum, symmetrically). In this paper, a polynomial algorithm for this problem is proposed for finding an optimal edgecut in a rooted tree separating the root and the set of all leafs. This algorithm enables us to develop an automatic clustering method with more accurate detection of community output.more » « less

The Golden–Thompson trace inequality, which states that Tr e H+ K ≤ Tr e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr X s Y t X 1− s Y 1− t ≤ Tr XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem.more » « less