Optimal Resource Allocation for Batch Testing

Here we introduce the Optimal Tree Completion Problem, a general optimization problem that involves completing an unrooted binary tree (i.e., adding missing leaves) so as to minimize its distance from a reference tree on a superset of the leaves. More formally, given a pair of unrooted binary trees (T,t) where T has leaf set S and t has leaf set R, a subset of S, we wish to add all the leaves from S R to t so as to produce a new tree t' on leaf set S that has the minimum distance to T. We show that when the distance is defined by the Robinson-Foulds (RF) distance, an optimal solution can be found in polynomial time. We also present OCTAL, an algorithm that solves this RF Optimal Tree Completion Problem exactly in quadratic time. We report on a simulation study where we complete estimated gene trees using a reference tree that is based on a species tree estimated from a multi-locus dataset. OCTAL produces completed gene trees that are closer to the true gene trees than an existing heuristic approach, but the accuracy of the completed gene trees computed by OCTAL depends on how topologically similar the estimated species tree is to the true gene tree. Hence, under conditions with relatively low gene tree heterogeneity, OCTAL can be used to provide highly accurate completions of estimated gene trees. We close with a discussion of future research.

Mobile edge computing (MEC) brings a breakthrough for Internet of Things (IoT) for its ability of offloading tasks from user equipments (UEs) to nearby servers which have rich computation resource. 5G network brings a huge breakthrough on transmission rate. Together with MEC and 5G, both execution delay of tasks and time delay from downloading would be shorter and the quality of experience (QoE) of UEs can be improved. Considering practical conditions, the computation resource of an MEC server is finite to some extent. Therefore, how to prevent the abuse of MEC resource and further allocate the resource reasonably becomes a key point for an MEC system. In this paper, an MEC system with multi-user is considered where a base station (BS) with an MEC server, which can not only provide computation offloading service but also data cache service. Especially, we take the charge for both data transmission and task computation as one part of total cost of UEs, and then explore a joint optimization for downlink resource allocation, offloading decision and computation resource allocation to minimize the total cost in terms of the time delay and the charge to UEs. The proposed problem is formulated as a mixed integer programming (MIP) one which is NP-hard. Therefore, we decouple the original problem into two subproblems which are downlink resource allocation problem and joint offloading decision and computation resource allocation problem. Then we address these two subproblems by using convex and nonconvex optimization techniques, respectively. An iterative algorithm is proposed to obtain a suboptimal solution in polynomial time. Simulation results show that our proposed algorithm performs better than benchmark algorithms.

Recently, an unconventional clock distribution scheme, called Branch-and-Combine (BaC) was proposed. The scheme is the first to guarantee constant skew upper bound irrespective of the clocked network's size. In BaC clocking, a set of interconnected nodes perform simple processing on clock signals such that the path from the source to any node is automatically and adaptively selected such that it is the shortest delay path. The graph underlying a BaC network is constrained by the requirement that each pair of adjacent nodes is in a cycle of length /spl les/k, where k is the feature cycle length. The graph representing such a network is called a BaC(k) graph. The feature cycle length (k) is an important parameter upon which skew bound and node function depend. We study the complexity of the general problem of designing a minimum cost BaC network for clocking a data processing network of arbitrary topology so that a certain feature cycle length is satisfied. We define two versions of the problem, differing in the way we are allowed to place edges in the graph representing the BaC network. We show that, in both cases, the general optimization problem is NP hard. We also provide efficient heuristic algorithms for both versions of the optimization problem. When k=2, the two versions of the optimization problem become the same and can be solved in polynomial time. For k=3, the complexity is still unknown.

Lattice-based cryptography has recently emerged as a prime candidate for efficient and secure post-quantum cryptography. The two main hard problems underlying its security are the shortest vector problem (SVP) and the closest vector problem (CVP). Various algorithms have been studied for solving these problems, and for SVP, lattice sieving currently dominates in terms of the asymptotic time complexity: one can heuristically solve SVP in time $2^{0.292d}$ in high dimensions $d$ [BDGL'16]. Although several SVP algorithms can also be used to solve CVP, it is not clear whether this also holds for heuristic lattice sieving methods. The best time complexity for CVP is currently $2^{0.377d}$ [BGJ'14]. In this paper we revisit sieving algorithms for solving SVP, and study how these algorithms can be modified to solve CVP and its variants as well. Our first method is aimed at solving one problem instance and minimizes the overall time complexity for a single CVP instance with a time complexity of $2^{0.292d}$. Our second method minimizes the amortized time complexity for several instances on the same lattice, at the cost of a larger preprocessing cost. We can solve the closest vector problem with preprocessing (CVPP) with $2^{0.636d}$ space and preprocessing, in $2^{0.136d}$ time, while the query complexity can even be reduced to $2^{\epsilon d}$ at the cost of preprocessing time and memory complexities of $(1/\epsilon)^{O(d)}$. For easier variants of CVP, such as approximate CVP and bounded distance decoding (BDD), we further show how the preprocessing method achieves even better complexities. For instance, we can solve approximate CVPP with large approximation factors $k$ with polynomial-sized advice in polynomial time if $k = \Omega(\sqrt{d/\log d})$, heuristically closing the gap between the decision-CVPP result of [AR'04] and the search-CVPP result of [DRS'14].

5G new radio (NR) introduced flexible numerology to provide the necessary flexibility for multiplexing the communication of heterogeneous services on a shared channel. One of the fundamental challenges of 5G NR is to develop resource allocation schemes to efficiently exploit such flexibility to optimize resource allocation of ultra-reliable low-latency communications (URLLC) in coexistence with enhanced mobile broadband (eMBB) while ensuring their colliding performance requirements. This problem of 5G NR resource allocation to URLLC services coexisted with eMMB services is an NP -hard problem. We present a new formulation of this problem by considering the interplay of eMBB and URLLC services’ constraints and targets. The objective of the formulated problem is to probabilistically meet quality of service (QoS) requirements of both eMBB and URLLC services when the optimal global solution is infeasible. To this end, we express the problem as an integer linear programming problem and consider two formulations with hard and soft URLLC throughput constraints. The overall problem is then treated as a specific instance of bin packing optimization, whose objective is to minimize the placements of URLLC services. We apply a greedy resource allocation algorithm to provide a near-optimal solution in polynomial-time to this problem. Finally, we numerically evaluate the optimal solutions of the two different formulations. Performance results demonstrate and confirm that the proposed 5G NR resource allocation solutions can provide efficient resource utilization while satisfying the performance requirements of the eMBB and URLLC services.

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of omega-tree languages derived from weak regular omega-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived omega-tree languages to learning the underlying class of omega-word languages, for any class of omega-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [Maler and Pneuli, Inform. Comput., 1995] for the class of weak regular omega-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi omega-word acceptors. A previous claim of an algorithm to learn regular omega-trees due to Jayasrirani, Begam and Thomas [Jayasrirani et al., ICGI, 2008] is unfortunately incorrect, as shown in [Angluin, YALEU/DCS/TR-1528, 2016].

Co-reservations are an efficient means to support guarantees on the allocation of resources in order to execute complex distributed application scenarios. Scheduling multiple moldable co-reservations involves several steps leading to guarantees of resource allocation. Previous work has provided either mechanisms for individual steps only or for selecting co-reservations, but with more limited application scenarios. In this work, we extend previous work by studying the problem as generic optimization problem, by developing a mixed integer linear programming model which allows more freedom in the specification of the goals of requests, resources and the broker, and by performing extensive experiments to analyze important parameters on the time to solve various problem instances.

Notes on Computational Complexity of GE Inequalities

The Dial-a-Ride problem may contain various constraints for pickup-delivery requests, such as time windows and ride time constraints. For a tour, given as a sequence of pickup and delivery stops, there exist polynomial time algorithms to find a schedule respecting these constraints, provided that there exists one. However, if no feasible schedule exists, the natural question is to find a schedule minimising constraint violations. We model a generic fixed-sequence scheduling problem, allowing lateness and ride time violations with linear penalty functions and prove its APX-hardness. We also present an approach leading to a polynomial time algorithm if only the time window constraints can be violated (by late visits). Then, we show that the problem can be solved in polynomial time if all the ride time constraints are bounded by a constant. Lastly, we give a polynomial time algorithm for the instances where all the pickups precede all the deliveries in the sequence of stops.

Mobile Edge Computing (MEC) is a promising computing paradigm, which can provide mobile devices with powerful computing capability and low delay by offloading resource-intensive tasks to the MEC Server. However, due to the varying wireless channel conditions and different task delay requirements, the resource allocation decision made by MEC Server cannot always achieve the lowest system energy consumption. In this paper, we consider resource allocation problem in a multi-user MEC system. The objective is to achieve minimum system energy consumption under delay constraint by suitably allocating communication and computing resources. We solve the communication resources (subcarrier) allocation problem through regression algorithm, make use of three regression algorithms to predict subcarrier allocation. And then calculate the computing resources (CPU frequency) allocation and system energy consumption. Simulation results show that machine learning can achieve near-optimal system energy consumption, where the system energy consumption of the ANN(Artificial Neural Network), Xgboost (Extreme Gradient Boosting), and RF(Random Forest) algorithms is 1.30%, 7.49%, and 23.29% higher than the optimal system energy consumption respectively. In the view of prediction loss, ANN and Xgboost perform better than RF.

We consider a finite buffer shared by multiple packet queues. Throughput can be considerably improved by partitioning the buffer space among the queues judiciously, especially under a high load regime. We formulate optimal buffer partitioning as a resource allocation problem, the solution of which is found through a greedy incremental algorithm in polynomial time. The rest of the work is devoted to applying the optimal buffer allocation strategy in different scenarios modeling a wireless downlink. First, the strategy is applied in a general parallel M/M/1/m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> system and a numerical study verifies that the strategy may boost the throughput considerably. Then, a multichannel extension of this system is considered when the users have different arrival rates and channels have different outage probabilities. Jointly optimal buffer space allocation and channel assignment problems in this scenario are shown to be separable. Lastly, buffer allocation is considered in a system where users need to be multiplexed and scheduled based on channel state. It is shown that this system can be modeled as a set of parallel M/G/1/m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> queues to which the optimum buffer allocation strategy is again applicable. The improvement brought by optimal buffer allocation to scheduling based solely on channel-state is explored. It is observed that buffer optimization can result in remarkable throughput increase on top of channel-based user selection.

The set ℓ Pol of complex numbers ζ , which can be approximated in a polynomial time cζ ·n k depending from the precision 2−n, is a field. The operations of the field can be approximated in a time<c · n 2 and better relative to the time one needs to approximate the operands ζ 1 ,ζ 2 with precision 2−n. In the case of addition and multiplication it can be done for |ζ 1 |, |ζ 2 |<a with a c depending from a and the constants in the running time of the polynomial time approximation of ζ 1,ζ 2, but in the case of division the constant depends additionally from |ζ | . The same holds for the ring of the polynomials ℓ Pol [x ] and the field of the rational functions f ∈ ℓ Pol (x ). The constant factor in the polynomial time bound of the approximation of f (ζ ) depend additionally from the distance of ζ from the singularities of f . The field ℓ Pol is algebraically closed. In the case that the polynomials have constant coefficients there exist algorithms to approximate the roots of the polynomials in a linear time with the Newton method if the roots are simple and an isolation of the roots is precomputed [Sch82]. But this precomputation needs much time [ESY06]. The best known algorithms to solve this problem need a time O (m 5(τ +logm )2), m the degree of the polynomial and τ the length of the binary representation of the coefficients. The problem with not constant coefficients has been attacked on base of bitstreams in [Eig08], [ESY06]. In [Eig08] on can find an excellent overview of the state of art. In the case the coefficients are not fixed constants but given by approximations the situation is much more complicated because the precomputation depends from the precision of the approximation. It is clear that in general the approximation of a root of a polynomial with coefficients, which cannot be approximated in a time O (n k ), cannot be done in this time. Ker-I Ko discusses the approximation problems on base of the unary representation of the numbers [Ko91]. Polynomial time algorithms relative to dyadic and unary representation are obviously not the same. The motivation of this paper is the question how far the mentioned problems can be solved by global polynomial time algorithms. This are algorithms with a running time <c·n k with the constant c only depending from the constants of the polynomial time approximations of the arguments and the domain of the function to be computed. We prove that there exist analytic functions, which are computable on all finite domains in this sense and that analytic functions for which this is true can be represented by a power series in z.

Given a graph G=(V,E) with node weight w:V→R+ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(nO(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+e)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+e)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.

Considering the proliferation in the number of cloud users on an everyday basis, the task of resource provisioning in order to support all these users becomes a challenging problem. When resource allocation is non-optimal, users may face high costs or performance issues. So, in order to maximize profit and resource utilization while satisfying all client requests, it is essential for Cloud Service Providers to come up with ways to allocate resources adaptively for diverse conditions. This is a constrained optimization problem. Each client that submits a request to the cloud has its own best interests in mind. But each of these clients competes with other clients in the quest to obtain required quantum of resources. Hence, every client is a participant in this competition. So, a preliminary analysis of the problem reveals that it can be modelled as a game between clients. A game theoretic modelling of this problem provides us an ability to find an optimal resource allocation by employing game theoretic concepts. Resource allocation problems are NP-Hard, involving VM allocation and migration within and possibly, among data centres. Owing to the dynamic nature and number of requests, static methods fail to surmount race conditions. Using a Min-Max Game approach, we propose an algorithm that can overcome the problems mentioned. We propose to employ a utility maximization approach to solve the resource provisioning and allocation problem. We implement a new factor into the game called the utility factor which considers the time and budget constraints of every user. Resources are provisioned for tasks having the highest utility for the corresponding resource.

A tournament T = (V , A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ϵ > 0, the kernelized instance has at most (2 + ϵ)k vertices. Our result improves the previous known bound of O(k2 ) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST.