Polynomial-time Approximation Algorithm Research Articles

Toward Efficient City-Scale Patrol Planning Using Decomposition and Grafting

Motivated by the increasing need of the real-world patrolling, this paper studies a practical city-scale patrolling (CSP) variant. In CSP, the police are scheduled to patrol city regions, and the objective is not only to protect public security but also to respond to incidents timely. We use an integer program (IP) to formulate the CSP problem, with the objective of maximizing the police visibility rate (PVR) to improve public safety and the additional constraint of response time guarantee to handle incidents timely. For such an NP-hard problem, existing studies either cannot scale-up or do not provide a bound from optimum. To fill the research gap, we propose a decomposition and grafting approach. We first decompose the original CSP into two weakly-coupled subproblems, minimizing police problem (MinP) and maximizing PVR (MaxP) problem. By exploiting the subproblem structures, a polynomial time approximation algorithm is proposed for MinP, and a polynomial time optimal algorithm is proposed for MaxP. We prove that such a decomposition can provide the 1 - α approximation ratio, where α is the percentage of the police used in MinP. To further improve patrolling efficiency, a grafting mechanism is proposed to integrate the two subproblems' solutions. Finally, we conduct extensive experiments on the real dataset of Foshan, a modern Chinese city. The results demonstrate that compared with benchmarks, our approach scales well to city-scale problem instances with fine-grained periods, hundreds of regions, and hundreds of police officers.

Cost-Fair Task Allocation in Mobile Crowd Sensing With Probabilistic Users

Mobile crowd sensing (MCS) is a new paradigm for urban-scale monitoring. This article concentrates on the Cost-Fair Task Allocation (CFA) problem for the MCS scenario where the collaboration of multiple probabilistic mobilephone users is needed to yield more reliable observation. CFA aims to allocate sensing tasks to users so that the sensing costs undertaken by all users are as balancing as possible, while the requirement of the requester for data reliability can be satisfied. CFA is greatly important to MCS campaigns in terms of reliability and sustainability. We design two algorithms to solve the CFA problem in the offline and online cases, respectively. Specifically, we propose a novel penalty-based model to reformulate the offline CFA problem, and based on this model, we design an offline algorithm, which can yield a computation-efficient $\varepsilon$ ɛ -solution with any small $\varepsilon >0$ ɛ > 0 . For the online case, we design a polynomial-time approximation algorithm, which struggles to allocate each of the sequentially arriving tasks to users as fairly as possible, and can achieve an upper-bounded competitiveness relative to the optimal CFA solution. Finally, we conduct extensive numeric analyses to validate the performance of our algorithms under diverse experimental setups.

Efficient algorithms for approximating quantum partition functions

We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.

Minimum constellation covers: hardness, approximability and polynomial cases

This paper considers two graph covering problems, the Minimum Constellation Cover (CC) and the Minimum k-Split Constellation Cover (k-SCC). The input of these problems consists on a graph $$G=\left( V,E\right) $$ and a set $${\mathcal {C}}$$ of stars of G, and the output is a minimum cardinality set of stars C, such that any two different stars of C are edge-disjoint and the union of the stars of C covers all edges of G. For CC, the set C must be compound by edges of G or stars of $${\mathcal {C}}$$ while, for k-SCC, an integer k is given and the elements of C must be k-stars obtained by splitting stars of $${\mathcal {C}}$$ . This work proves that unless $$P=NP$$ , CC does not admit polynomial time $$\left| {\mathcal {C}}\right| ^{{\mathcal {O}}\left( 1\right) }$$ -approximation algorithms and k-SCC cannot be $$\left( \left( 1-\epsilon \right) \ln \left| E\right| \right) $$ -approximated in polynomial time, for any $$\epsilon >0$$ . Additionally, approximation ratios are given for the worst feasible solutions of the problems and, for k-SCC, polynomial time approximation algorithms are proposed, achieving a $$\left( \ln \left| E\right| -\ln \ln \left| E\right| +\varTheta \left( 1\right) \right) $$ approximation ratio. Also, polynomial time algorithms are presented for special classes of graphs that include bounded degree trees and cacti.

(1 + ϵ)2-and Polynomial-Time Approximation Algorithms for Network Lifetime Maximization With Relay Hop Bounded Connected Target Coverage in WSNs

Scheduling sensor activity to prolong the network lifetime while guaranteeing coverage and connectivity is a fundamental and critical issue in handling wireless sensor networks. Although many efforts have been made in this area, none of the prior works considers the relay hop bound constraint. As a result, the existing scheduling algorithms can provide only connectivity with uncontrollable latency to the sinks and can't be applied to delay-sensitive applications. In this paper, we are the first one coping with the scheduling problem for network lifetime maximization under the requirements of full target coverage and connectivity with bounded relay hop. We first propose an exact LP (i.e., Linear Programming) formulation to determine the optimal solution. Then, to reduce the time complexity, we develop a (1 + E)2-approximation algorithm based on the partitioning and shifting technique. Furthermore, we propose an approximate LP formulation whose variable size is polynomial to the number of targets and sensors and whose performance ratio is M, where M is the maximum number of targets covered by a sensor. The experiment results show that when the number of sensors is sufficiently large, our algorithms extend the network lifetime up to 3.68 times compared to the existing approaches. Moreover, the proposed algorithms shorten time complexity significantly. Specifically, our algorithms' time complexity is always less than 20% that of the benchmarks.

Distance measures for embedded graphs

We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fréchet distance if the graphs are planar embedded and if the embedding meets a certain geometric restriction. Moreover, we prove that deciding the distances based on the Fréchet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.

A Differentially Private Incentive Design for Traffic Offload to Public Transportation

Increasingly large trip demands have strained urban transportation capacity, which consequently leads to traffic congestion and rapid growth of greenhouse gas emissions. In this work, we focus on achieving sustainable transportation by incentivizing passengers to switch from private cars to public transport. We address the following challenges. First, the passengers incur inconvenience costs when changing their transit behaviors due to delay and discomfort, and thus need to be reimbursed. Second, the inconvenience cost, however, is unknown to the government when choosing the incentives. Furthermore, changing transit behaviors raises privacy concerns from passengers. An adversary could infer personal information (e.g., daily routine, region of interest, and wealth) by observing the decisions made by the government, which are known to the public. We adopt the concept of differential privacy and propose privacy-preserving incentive designs under two settings, denoted as two-way communication and one-way communication. Under two-way communication, passengers submit bids and then the government determines the incentives, whereas in one-way communication, the government simply sets a price without acquiring information from the passengers. We formulate the problem under two-way communication as a mixed integer linear program and propose a polynomial-time approximation algorithm. We show the proposed approach achieves truthfulness, individual rationality, social optimality, and differential privacy. Under one-way communication, we focus on how the government should design the incentives without revealing passengers’ inconvenience costs while still preserving differential privacy. We formulate the problem as a convex program and propose a differentially private and near-optimal solution algorithm. A numerical case study using the Caltrans Performance Measurement System (PeMS) data source is presented as evaluation. The results show that the proposed approaches achieve a win-win situation in which both the government and passengers obtain non-negative utilities.

STAR

Timed association rules (TARs) generalize classical association rules (ARs) so that we can express temporal dependencies of the form “If X is true at time t , then Y will likely be true at time ( t +τ).” As with ARs, solving the TAR mining problem can generate huge numbers of rules. We show that methods to summarize ARs cannot work directly with TARs, and we develop two notions— strong and weak summaries—to summarize a set of TARs. We show that the problems of finding strong/weak summaries are NP-hard, and we provide polynomial-time approximation algorithms. We show experimentally that the coverage provided by our summarization methods is very high. Both technical measures based on coverage and human experiments on six World Bank datasets using 100 subjects from Mechanical Turk and a separate experiment with terrorism experts on a terrorism dataset show that while both summarization methods perform well, weak summaries are preferred, despite their taking more time to compute than strong summaries.

Hybrid Flow Table Installation: Optimizing Remote Placements of Flow Tables on Servers to Enhance PDP Switches for In-Network Computing

Recently, the programmable data plane (PDP) switches have been considered as the key enablers for in-network computing. However, the limited memory resources in them for flow tables might restrict their performance. This work addresses this challenge by studying how to optimize the placements of flow tables in the external memory on multiple servers, and to access them with remote direct memory access (RDMA) for ensuring low latency. Specifically, we consider a data-center network (DCN) that uses PDP switches as top-of-rack (ToR) switches, and propose and optimize the hybrid flow table installation (hFT-INST) on each ToR switch. With hFT-INST, the switch can either store flow tables in its local memory or use RDMA to install and access them remotely in its rack servers. We first design the protocol and operation procedure of hFT-INST. Then, regarding the key problem of hFT-INST, i.e., how to place the flow tables on the external memory on different servers, we take a few practical parameters into account, and formulate a mixed integer linear programming (MILP) model to tackle it. Next, the optimization in the MILP is transformed into a capacitated facility location problem (CFLP) with additional constraints. We further transform it into a ${k}$ -median problem through pre-processing, and design a polynomial-time approximation algorithm to solve the problem. Extensive simulations confirm the performance of our proposed algorithm. We also prototype our design of the hFT-INST, and conduct experiments to demonstrate its feasibility.

On the Cross-Layer Network Planning for Flexible Ethernet Over Elastic Optical Networks

This article studies the cross-layer network planning that tries to combine flexible Ethernet (FlexE) and elastic optical networks (EONs), for FlexE-over-EONs. We focus our investigation on the most challenging setting, i.e., the FlexE-over-EONs based on the FlexE-aware architecture, and consider both single-hop and multi-hop scenarios for the cross-layer planning. For the single-hop scenario, we assume that all the client flows are routed over end-to-end lightpaths in the EON. We formulate a mixed integer linear programming (MILP) model for this problem, transform it into the class constrained bin packing problem (CCBP), and leverage the primal-dual interior-point (PDIP) method to propose a polynomial-time approximation algorithm for it. Then, for the multi-hop scenario, we use a more realistic assumption that each client flow can be routed over multiple lightpaths in the EON. We show that after solving the virtual topology design, the cross-layer planning in this scenario can be transformed into that in the single-hop scenario. Therefore, an integer linear programming (ILP) model is formulated to tackle the virtual topology design, and we design a polynomial-time approximation algorithm for it by modifying the well-known branch-and-bond method. To evaluate the performance of our two-step method for the multi-hop scenario, we also propose a heuristic algorithm. Extensive simulations verify that regarding large-scale cross-layer planning for FlexE-over-EONs, our approximation algorithms are significantly more time-efficient than the ILP/MILP models, and their solutions have bounded gaps to the optimal ones and are much better than those of the heuristic.

On the Bilevel Optimization to Design Control Plane for SDONs in Consideration of Planned Physical-Layer Attacks

In the network planning of software-defined optical networks (SDONs), the control plane design is of great importance because it directly affects the performance and reliability of network control and management (NC&M). In this article, we consider the planned physical-layer attacks from a rational attacker, which can analyze the control plane of an SDON and target its attacks to the most vulnerable part. To address such attacks, we model the control plane design as a bilevel optimization, where the upper-level optimization is for the network planner to design the control plane whose vulnerability to planned attacks is minimized, while the lower-level optimization is for the attacker to plan its attacks such that the control plane can be disturbed as severely as possible. We first develop two approaches to solve the bilevel model exactly. Specifically, we first leverage the cutting plane method to solve it directly, and then transform it into a single-level mixed integer linear programming (MILP) model with the Bellman method for problem solving. To improve the time efficiency for large-scale problems, we also propose a polynomial-time approximation algorithm based on linear programming (LP) relaxation and randomized rounding. Extensive simulations with various physical topologies verify the effectiveness of our proposals.

Application-Driven Provisioning of Service Function Chains Over Heterogeneous NFV Platforms

Although network function virtualization (NFV) has been proven to be beneficial in terms of equipment cost, service delivery flexibility, and time-to-market, most of the studies in this area only addressed homogeneous NFV platforms (e.g., with virtual machines (VMs) only). In this work, we argue that by leveraging heterogeneous NFV platforms such as VMs, docker containers, and programmable hardware accelerators (e.g., SmartNICs), one could achieve better flexibility and cost-effectiveness to support virtual network function service chains (vNF-SCs) with various quality-of-service (QoS) requirements. Therefore, we study application-driven provisioning of vNF-SCs over heterogeneous NFV platforms, and design a polynomial-time approximation algorithm to tackle the problem for near-optimal solutions. We first introduce a layered auxiliary graph (LAG) based approach to model the problem of vNF-SC provisioning, and then formulate a novel integer linear programming (ILP) model based on it. Specifically, the ILP model minimizes the total cost of vNF-SC deployment while ensuring that the QoS requirements of all the vNF-SCs are satisfied. To solve the ILP time-efficiently, we propose an approximation algorithm based on linear programming (LP) relaxation and randomized rounding. Extensive simulations confirm that with significantly improved time-efficiency, our proposed algorithm can provide near-optimal solutions whose gaps to the exact ones are bounded.

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

This paper studies a multiproduct 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, because of 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.

Network Utility Maximization Under Maximum Delay Constraints and Throughput Requirements

We consider a multi-path routing problem of maximizing the aggregate user utility over a multi-hop network, subject to link capacity constraints, maximum end-to-end delay constraints, and user throughput requirements. A user's utility is a concave function of the achieved throughput or the experienced maximum delay. The problem is important for supporting real-time multimedia traffic and is uniquely challenging due to the need of simultaneously considering maximum delay constraints and throughput requirements. In this paper, we first show that it is NP-complete either (i) to construct a feasible solution strictly meeting all constraints, or (ii) to obtain an optimal solution after relaxing either the maximum delay constraints or the throughput requirements. We then develop a polynomial-time approximation algorithm named PASS. The design of PASS leverages a novel understanding between non-convex maximum-delay-aware problems and their convex average-delay-aware counterparts, which can be of independent interest and suggests a new avenue for solving maximum-delay-aware network optimization problems. We prove that PASS always obtains approximate solutions (i.e., with theoretical performance guarantees), at the cost of violating both the maximum delay constraints and the throughput requirements by up to constant ratios. We also develop two variants of PASS, named PASS-M and PASS-T, to generate approximate solutions at the cost of violating either the maximum delay constraints or the throughput requirements by up to problem-dependent ratios. We evaluate our solutions using extensive simulations on Amazon EC2 datacenters supporting video-conferencing traffic. Compared to the existing algorithms and a conceivable baseline, our solutions obtain up to 100% improvement of utilities, by meeting the throughput requirements but relaxing the maximum delay constraints to the extent acceptable for practical video conferencing applications.

You Calculate and I Provision: A DRL-Assisted Service Framework to Realize Distributed and Tenant-Driven Virtual Network Slicing

This article studies the problem of virtual network (VNT) slicing in datacenter interconnections (DCIs), and proposes a novel service framework to better balance the tradeoff between cost-effectiveness, and time-efficiency. Our idea is to partition a DCI into non-overlapped subgraphs, divide the VNT slicing in each subgraph into four collaborative steps, and get tenants involved in the calculation of virtual network embedding (VNE) schemes. With our proposal, an agent of infrastructure provider (InP) leverages deep reinforcement learning (DRL) to price, and advertise the substrate resources in one subgraph, motivates tenants to request resources in a load-balanced manner, and accepts VNE schemes from the tenants to avoid resource conflicts. Meanwhile, the tenants' task is to compute their own VNE schemes independently, and distributedly according to the resource information (i.e., the available resources, and their prices) advertised by the agent. We first design the DRL model based on the deep deterministic policy gradient (DDPG), and develop a VNT compression method based on auto-encoder (AE) to generalize the DRL's operation. Then, we study how to resolve resource conflicts among the distributedly-calculated VNE schemes, build a conflict graph (CG) to transform the VNE selection into finding the maximum weighted independent set (MWIS) in the CG, and design a polynomial-time approximation algorithm to solve the problem. Extensive simulations confirm that compared with the centralized service framework relying solely on the InP for VNE calculation, our proposed DRL-assisted distributed framework provisions VNT requests with significantly shorter computation time, and comparable blocking performance.

Approximation of set multi-cover via hypergraph matching

For a given b∈Nn and a hypergraph H=(V,E) with maximum degree Δ, the b-set multicover problem which we are concerned with in this paper may be stated as follows: find a minimum cardinality subset C⊆E such that no vertex vi∈V is contained in less than bi hyperedges of C. Peleg, Schechtman, and Wool (1997) conjectured that for any fixed Δ and b=mini⁡bi, the problem cannot be approximated with a ratio strictly smaller than δ:=Δ−b+1, unless P=NP. In this paper, we show that the conjecture of Peleg et al. is not true on ℓ-uniform hypergraphs by presenting a polynomial-time approximation algorithm based on a matching/covering duality for hypergraphs due to Ray-Chaudhuri (1960), which we convert into an approximative form. The given algorithm yields a ratio smaller than (1−b(ℓ+1)Δ)Δb. Moreover, we prove that the lower bound conjectured by Peleg et al. holds for regular hypergraphs under the unique games conjecture.

Structural parameters for scheduling with assignment restrictions

We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than 1.5 unless P=NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than 1.5 exist. We introduce a graph framework based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of the resulting graphs, in particular, their tree- and cliquewidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms generalizing and extending previous results in this area.

On Virtual Network Reconfiguration in Hybrid Optical/Electrical Datacenter Networks

Hybrid optical/electrical datacenter networks (HOE-DCNs) build inter-rack networks with both electrical Ethernet switches and optical cross-connects (OXCs), and have been considered as a promising DCN architecture. However, to adapt to the dynamic network environment, the reconfiguration of virtual networks (VNTs) in an HOE-DCN still faces the unique difficulty that the HOE-DCN's topology can change because of the one-to-one connectivity of OXCs. To the best of our knowledge, this problem still has not been fully explored. In this article, we address this problem, and consider how to achieve effective VNT reconfiguration in an HOE-DCN such that the IT resource usages in racks can be re-balanced with the migration of virtual machines (VMs). We first formulate a mixed integer linear programming (MILP) to describe the VNT reconfiguration. Then, we solve the problem with two steps, 1) obtaining the VM migration schemes to balance the loads on racks and 2) determining the reconfiguration schemes of related virtual links (VLs) and the OXC. For the first step, we propose a polynomial-time approximation algorithm by leveraging linear relaxation. Then, we tackle the optimization of the second step by developing an algorithm that involves a linear-time dynamic programming and an integer linear programming (ILP). To solve the ILP time-efficiently, we propose another polynomial-time approximation algorithm based on Lagrangian relaxation. Our simulations confirm the effectiveness of the proposed approximation algorithms and verify that the overall procedure including them outperforms the existing approach.

Polynomial-Time Algorithms for Multiple-Arm Identification with Full-Bandit Feedback

We study the problem of stochastic multiple-arm identification, where an agent sequentially explores a size- subset of arms (also known as a super arm) from given arms and tries to identify the best super arm. Most work so far has considered the semi-bandit setting, where the agent can observe the reward of each pulled arm or assumed each arm can be queried at each round. However, in real-world applications, it is costly or sometimes impossible to observe a reward of individual arms. In this study, we tackle the full-bandit setting, where only a noisy observation of the total sum of a super arm is given at each pull. Although our problem can be regarded as an instance of the best arm identification in linear bandits, a naive approach based on linear bandits is computationally infeasible since the number of super arms is exponential. To cope with this problem, we first design a polynomial-time approximation algorithm for a 0-1 quadratic programming problem arising in confidence ellipsoid maximization. Based on our approximation algorithm, we propose a bandit algorithm whose computation time is (log ), thereby achieving an exponential speedup over linear bandit algorithms. We provide a sample complexity upper bound that is still worst-case optimal. Finally, we conduct experiments on large-scale data sets with more than 10 super arms, demonstrating the superiority of our algorithms in terms of both the computation time and the sample complexity.

Graph Matching Via the Lens of Supermodularity

Graph matching, the problem of aligning a pair of graphs so as to minimize their edge disagreements, has received widespread attention owing to its broad spectrum of applications in data science. As the problem is NP–hard in the worst-case, a variety of approximation algorithms have been proposed for obtaining high quality, suboptimal solutions. In this article, we approach the task of designing an efficient polynomial-time approximation algorithm for graph matching from a previously unconsidered perspective. Our key result is that graph matching can be formulated as maximizing a monotone, supermodular set function subject to matroid intersection constraints. We leverage this fact to apply a discrete optimization variant of the minorization-maximization algorithm which exploits supermodularity of the objective function to iteratively construct and maximize a sequence of global lower bounds on the objective. At each step, we solve a maximum weight matching problem in a bipartite graph. Differing from prior approaches, the algorithm exploits the combinatorial structure inherent in the problem to generate a sequence of iterates featuring monotonically non-decreasing objective value while always adhering to the combinatorial matching constraints. Experiments on real-world data demonstrate the empirical effectiveness of the algorithm relative to the prevailing state-of-the-art.