Polynomial-time Approximation Algorithm Research Articles

A Flow-Network-Based Polynomial-Time Approximation Algorithm for the Minimum Constrained Input Structural Controllability Problem

This paper deals with minimum cost constrained input selection (minCCIS) for state-space structured systems. Our goal is to optimally select an input set from the given inputs such that the system is structurally controllable when the set of states that each input can influence is prespecified and each input has a cost associated with it. This problem is known to be NP-hard. First, we give a flow-network-based novel necessary and sufficient graph theoretic condition for checking structural controllability. Using this condition, we propose a polynomial time reduction of the problem to a known NP-hard problem: the minimum-cost fixed-flow problem (MCFF). Subsequently, we show that approximation schemes of MCFF directly apply to the minCCIS problem. Using the special structure of the flow-network constructed for the structured system, we formulate a polynomial time approximation algorithm based on minimum weight bipartite matching problem and a greedy scheme for solving the MCFF problem on the system flow-network. The proposed algorithm gives a so-called $\Delta$ -approximate solution to MCFF, where $\Delta$ denotes the maximum in-degree of input vertices in the flow-network of the structured system.

An Improved Bound for Minimizing the Total Weighted Completion Time of Coflows in Datacenters

In data-parallel computing frameworks, intermediate parallel data is often produced at various stages which needs to be transferred among servers in the datacenter network (e.g., the shuffle phase in MapReduce). A stage often cannot start or be completed unless all the required data pieces from the preceding stage are received. Coflow is a recently proposed networking abstraction to capture such communication patterns. We consider the problem of efficiently scheduling coflows with release dates in a shared datacenter network so as to minimize the total weighted completion time of coflows. Several heuristics have been proposed recently to address this problem, as well as a few polynomial-time approximation algorithms with provable performance guarantees. Our main result in this paper is a polynomial-time deterministic algorithm that improves the prior known results. Specifically, we propose a deterministic algorithm with approximation ratio of 5, which improves the prior best known ratio of 12. For the special case when all coflows are released at time zero, our deterministic algorithm obtains approximation ratio of 4 which improves the prior best known ratio of 8. The key ingredient of our approach is an improved linear program formulation for sorting the coflows followed by a simple list scheduling policy. Extensive simulation results, using both synthetic and real traffic traces, are presented that verify the performance of our algorithm and show improvement over the prior approaches.

Approximation algorithms for tours of height-varying view cones

We introduce a novel coverage problem that arises in aerial surveying applications. The goal is to compute a shortest path that visits a given set of cones. The apex of each cone is restricted to lie on the ground plane. The common angle of the cones represent the field of view of the onboard camera. The cone heights, which can be varying, correspond with the desired observation quality (e.g. resolution). This problem is a novel variant of the traveling salesman problem with neighborhoods (TSPN). We name it Cone-TSPN. Our main contribution is a polynomial time approximation algorithm for Cone-TPSN. We analyze its theoretical performance and show that it returns a solution whose length is at most times the length of the optimal solution where and are the heights of the tallest and shortest input cones, respectively.We demonstrate the use of our algorithm in a representative precision agriculture application. Wefurther study its performance in simulation using randomly generated cone sets. Our results indicate that the performance of our algorithm is superior to standard solutions.

A Polynomial-Time Approximation Algorithm for One Problem Simulating the Search in a Time Series for the Largest Subsequence of Similar Elements

We analyze the mathematical aspects of the data analysis problem consisting in the search (selection) for a subset of similar elements in a group of objects. The problem arises, in particular, in connection with the analysis of data in the form of time series (discrete signals). One of the problems in modeling this challenge is considered, namely, the problem of searching in a finite sequence of points from the Euclidean space for the subsequence with the greatest number of terms such that the quadratic spread of the elements of this subsequence with respect to its unknown centroid does not exceed a given percentage of the quadratic spread of elements of the input sequence with respect to its centroid. It is shown that the problem is strongly NP-hard. A polynomial-time approximation algorithm is proposed. This algorithm either establishes that the problem has no solution or finds a 1/2-approximate solution if the length M* of the optimal subsequence is even, or it yields a \(\frac{1}{2}\left(\begin{array}{c}1-\frac{1}{M^*}\\ \end{array}\right)\)-approximate solution if M* is odd. The time complexity of the algorithm is O(N3(N2+q)), where N is the number of points in the input sequence and q is the space dimension. The results of numerical simulation that demonstrate the effectiveness of the algorithm are presented.

Growing Linear Dynamical Networks Endowed by Spectral Systemic Performance Measures

We propose an axiomatic approach for design and performance analysis of noisy linear consensus networks by introducing a notion of systemic performance measure. This class of measures are spectral functions of Laplacian eigenvalues of the network that are monotone, convex, and orthogonally invariant with respect to the Laplacian matrix of the network. It is shown that several existing gold-standard and widely used performance measures in the literature belong to this new class of measures. We build upon this new notion and investigate a general form of the combinatorial problem of growing a linear consensus network via minimizing a given systemic performance measure. Two efficient polynomial-time approximation algorithms are devised to tackle this network synthesis problem: a linearization-based method and a simple greedy algorithm based on rank-one updates. Several theoretical fundamental limits on the best achievable performance for the combinatorial problem are derived that assist us to evaluate optimality gaps of our proposed algorithms. A detailed complexity analysis confirms the effectiveness and viability of our algorithms to handle large-scale consensus networks.

Algorithm and complexity of the two disjoint connected dominating sets problem on trees

In this paper, we consider a variation of the classic dominating set problem - The Two Disjoint Connected Dominating Sets (DCDS) problem, which finds applications in many real domains including wireless sensor networks. In the DCDS problem, we are given a graph G=(V,E) and required to find a new edge set E′ with minimum cardinality such that the resulting new graph after the adding of E′ has a pair of disjoint connected dominating sets. This problem is very hard in general graphs, and we show that it is NP-hard even restricted to trees. We also present a polynomial time approximation algorithm for the DCDS problem for arbitrary trees with performance ratio 32 asymptotically.

Scheduling a Single Machine with Primary and Secondary Objectives

We study a scheduling problem in which jobs with release times and due dates are to be processed on a single machine. With the primary objective to minimize the maximum job lateness, the problem is strongly NP-hard. We describe a general algorithmic scheme to minimize the maximum job lateness, with the secondary objective to minimize the maximum job completion time. The problem of finding the Pareto-optimal set of feasible solutions with these two objective criteria is strongly NP-hard. We give the dominance properties and conditions when the Pareto-optimal set can be formed in polynomial time. These properties, together with our general framework, provide the theoretical background, so that the basic framework can be expanded to (exponential-time) implicit enumeration algorithms and polynomial-time approximation algorithms (generating the Pareto sub-optimal frontier with a fair balance between the two objectives). Some available in the literature experimental results confirm the practical efficiency of the proposed framework.

Jointly Optimal Routing and Caching for Arbitrary Network Topologies

We study a problem of minimizing routing costs by jointly optimizing caching and routing decisions over an arbitrary network topology. We cast this as an equivalent caching gain maximization problem, and consider both source routing and hop-by-hop routing settings. The respective offline problems are NP-hard. Nevertheless, we show that there exist polynomial time approximation algorithms producing solutions within a constant approximation from the optimal. We also produce distributed, adaptive algorithms with the same approximation guarantees. We simulate our adaptive algorithms over a broad array of different topologies. Our algorithms reduce routing costs by several orders of magnitude compared with prior art, including algorithms optimizing caching under fixed routing.

A PTAS for Scheduling Unrelated Machines of Few Different Types

Scheduling on Unrelated Machines is a classical optimization problem where [Formula: see text] jobs have to be distributed to [Formula: see text] machines. Each of the jobs [Formula: see text] has on machine [Formula: see text] a processing time [Formula: see text]. The goal is to minimize the makespan, i.e., the maximum completion time of the longest-running machine. Unless [Formula: see text], this problem does not allow for a polynomial-time approximation algorithm with a ratio better than [Formula: see text]. A natural scenario is however that many machines are of the same type, like a CPU and GPU cluster: for each of the [Formula: see text] machine types, the machines [Formula: see text] of the same type [Formula: see text] satisfy [Formula: see text] for all jobs [Formula: see text]. For the case where the number [Formula: see text] of machine types is constant, this paper presents an approximation scheme, i.e., an algorithm of approximation ratio [Formula: see text] for [Formula: see text], with an improved running time only single exponential in [Formula: see text].

Mixed-radix Naccache–Stern encryption

In this work, we explore a combinatorial optimization problem stemming from the Naccache–Stern cryptosystem. We show that solving this problem results in bandwidth improvements, and suggest a polynomial-time approximation algorithm to find an optimal solution. Our work suggests that using optimal radix encoding results in an asymptotic 50% increase in bandwidth.

Factorization Norms and Hereditary Discrepancy

The $\gamma_2$ norm of a real $m\times n$ matrix $A$ is the minimum number $t$ such that the column vectors of $A$ are contained in a $0$-centered ellipsoid $E\subseteq\mathbb{R}^m$ which in turn is contained in the hypercube $[-t, t]^m$. We prove that this classical quantity approximates the \emph{hereditary discrepancy} $\mathrm{herdisc} A$ as follows: $\gamma_2(A) = {O(\log m)}\cdot \mathrm{herdisc} A$ and $\mathrm{herdisc} A = O(\sqrt{\log m}\,)\cdot\gamma_2(A) $. Since $\gamma_2$ is polynomial-time computable, this gives a polynomial-time approximation algorithm for hereditary discrepancy. Both inequalities are shown to be asymptotically tight. We then demonstrate on several examples the power of the $\gamma_2$ norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of $\Omega(\log^{d-1} n)$ for the \emph{$d$-dimensional Tusnady problem}, asking for the combinatorial discrepancy of an $n$-point set in $\mathbb{R}^d$ with respect to axis-parallel boxes. For $d>2$, this improves the previous best lower bound, which was of order approximately $\log^{(d-1)/2}n$, and it comes close to the best known upper bound of $O(\log^{d+1/2}n)$, for which we also obtain a new, very simple proof.

Computing maxmin edge length triangulations

In 1991, Edelsbrunner and Tan gave an $O(n^2)$ algorithm for finding the MinMax Length triangulation of a set of points in the plane, but stated the complexity of finding a MaxMin Edge Length Triangulation (MELT) as a natural open problem. We resolve this long-standing problem by showing that computing a MELT is NP-complete. Moreover, we prove that (unless P=NP), there is no polynomial-time approximation algorithm that can approximate MELT within any polynomial factor. While this may be taken as conclusive evidence from a theoretical point of view that the problem is hopelessly intractable, it still makes sense to consider powerful optimization methods, such as integer programming (IP), in order to obtain provably optimal solutions for intances of non-trivial size. A straightforward IP based on pairwise disjointness of the $\Theta(n^2)$ segments between the n points has $\Theta(n^4)$ constraints, making this IP hopelessly intractable from a practical point of view, even for relatively small $n$. The main algorithm engineering twist of this paper is to demonstrate how the combination of geometric insights with refined methods of combinatorial optimization can still help to put together an exact method capable of computing optimal MELT solutions for planar point sets up to $n = 200$. Our key idea is to exploit specific geometric properties in combination with more compact IP formulations, such that we are able to drastically reduce the IPs. On the practical side, we combine two of the most powerful software packages for the individual components: CGAL for carrying out the geometric computations, and CPLEX for solving the IPs. In addition, we discuss specific analytic aspects of the speedup for random point sets.

Colocating tasks in data centers using a side-effects performance model

In data centers, many tasks (services, virtual machines or computational jobs) share a single physical machine. We explore a new resource management model for such colocation. Our model uses two parameters of a task—its size and its type—to characterize how a task influences the performance of the other tasks allocated on the same machine. As typically a data center hosts many similar, recurring tasks (e.g. a webserver, a database, a CPU-intensive computation), the resource manager should be able to construct these types and their performance interactions. In particular, we minimize the total cost in a model in which each task’s cost is a function of the total sizes of tasks allocated on the same machine (each type is counted separately). We show that for a linear cost function the problem is strongly NP-hard, but polynomially-solvable in some particular cases. We propose an algorithm polynomial in the number of tasks (but exponential in the number of types and machines) and another algorithm polynomial in the number of tasks and machines (but exponential in the number of types and admissible sizes of tasks). We also propose a polynomial time approximation algorithm, and, in the case of a single type, a polynomial time exact algorithm. For convex costs, we prove that, even for a single type, the problem becomes NP-hard, and we propose an approximation algorithm. We experimentally verify our algorithms on instances derived from a real-world data center trace. While the exact algorithms are infeasible for large instances, the approximations and heuristics deliver reasonable performance.

An approximation algorithm for maximum internal spanning tree

Given a graph G, the maximum internal spanning tree problem (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of $$\frac{3}{4}$$ . In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, $$\frac{13}{17}$$ ) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.

Energy Optimal Task Scheduling with Normally-off Local Memory and Sleep-aware Shared Memory with Access Conflict

The rapid development of the Real-Time and Embedded System (RTES) has increased the requirement on the processing capabilities of sensors, mobiles and smart devices, etc. Meanwhile, energy efficiency techniques are in desperate need as most devices in RTES are battery powered. Following the above trends, this work explores the memory system energy efficiency for a general multi-core architecture. This architecture integrates a local memory in each processing core, with a large off-chip memory shared among multiple cores. Decisions need to be made on whether tasks will be executed with the shared memory or the local memory to minimize the total energy consumption within real-time constraints. This paper proposes optimal schemes as well as a polynomial-time approximation algorithm with constant ratio. The problem complexity analysis for different task and system models is also presented. Experimental results show that the proposed approximation scheme performs close to the optimal solution in average.

Optimal Pricing in Markets with Non-Convex Costs

We consider a market run by an operator, who seeks to satisfy a given consumer demand for a commodity by purchasing the needed amount from a group of competing suppliers with non-convex cost functions. The operator knows the suppliers' cost functions and announces a price/payment function for each supplier, which determines the payment to that supplier for producing different quantities. Each supplier then makes an individual decision about how much to produce, in order to maximize its own profit. The key question is how to design the price functions. To that end, we propose a new pricing scheme, which is applicable to general non-convex costs, and allows using general parametric pricing functions. Optimizing for the quantities and the price parameters simultaneously, and the ability to use general parametric pricing functions allows our scheme to find prices that are typically economically more efficient and less discriminatory than those of the existing schemes. In addition, we supplement the proposed method with a polynomial-time approximation algorithm, which can be used to approximate the optimal quantities and prices. Our framework extends to the case of networked markets, which, to the best of our knowledge, has not been considered in previous work.

Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center

A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed.

Multi-Product Newsvendor Problem with Customer-Driven Demand Substitution: A Stochastic Integer Program Perspective

This paper studies a multi-product newsvendor problem with customer-driven demand substitution, where each product, once run out of stock, can be proportionally substituted by the others. This problem has been widely studied in the literature, however, due to nonconvexity and intractability, only limited analytical properties have been reported and no efficient approaches have been proposed. This paper first completely characterizes the optimal order policy when the demand is known and reformulates this nonconvex problem as a binary quadratic program. When the demand is random, we formulate the problem as a two-stage stochastic integer program, derive several necessary optimality conditions, prove the submodularity of the profit function, and also develop polynomial-time approximation algorithms and show their performance guarantees. We further propose a tight upper bound via nonanticipativity dual, which is proven to be very close to the optimal value and can yield a good-quality feasible solution under a mild condition. Our numerical investigation demonstrates effectiveness of the proposed algorithms. Moreover, several useful findings and managerial insights are revealed from a series of sensitivity analyses.

Randomization-Embedded Greed: Pursuing More Platform Profits in Mobile Crowd Sensing

The recent proliferation of sensor-equipped mobile phones, together with the intrinsic mobility of their users, has enabled mobile crowdsensing (MCS) to spring up. In a typical MCS system, the MCS platform recruits mobile phone users (workers) to perform the sensing tasks published by requesters. The requesters are interested in some urban events and are willing to pay for the sensing data returned by the workers. A lot of incentive mechanisms have been presented in the literature, in order to stimulate workers to participate into MCS campaigns or to maximize the social welfare. However, the community has not yet paid much attention to optimizing the platform profit, which is highly valued by the profit-making MCS organizers. In this paper, we consider a realistic MCS scenario that depends on probabilistic collaboration among workers and has constrained capacity of platform; and we focus on the platform profit maximization (PPM) in this MCS scenario. The PPM problem is NP-hard, and the key challenge stems mainly from the non-monotonicity caused by the probabilistic collaboration and the quality-based payment of the requester. First, we propose two polynomial-time approximation algorithms for the PPM problem: MaxG and RandG . Our main effort is dedicated to design RandG , which involves a randomization policy of selecting workers in its greedy framework, aimed at pursuing chances of skipping over incompetent local optima. We also prove that RandG can achieve a constant approximation ratio in expectation. Second, we present algorithm RandCom for general PPM problems, which combines MaxG and RandG to struggle to make as high platform profit as possible. Finally, we conduct extensive simulation to evaluate our designs in terms of platform profit.

Chamberlin--Courant Rule with Approval Ballots: Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

We consider the problem of winner determination under Chamberlin--Courant's multiwinner voting rule with approval utilities. This problem is equivalent to the well-known NP-complete MaxCover problem and, so, the best polynomial-time approximation algorithm for it has approximation ratio 1 - 1/e. We show exponential-time/FPT approximation algorithms that, on one hand, achieve arbitrarily good approximation ratios and, on the other hand, have running times much better than known exact algorithms. We focus on the cases where the voters have to approve of at most/at least a given number of candidates.