Primal-dual Approximation Algorithm Research Articles

A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties

A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties

A primal–dual approximation algorithm for Minsat

We characterize the optimal solution to the linear programming relaxation of the standard formulation for the minimum satisfiability problem. We give a O ( n m 2 ) combinatorial algorithm to solve the fractional version of the minimum satisfiability problem optimally where n ( m ) is the number of variables (clauses). As a by-product, we obtain a 2 ( 1 − 1 / 2 k ) approximation algorithm for the minimum satisfiability problem where k is the maximum number of literals in any clause.

Truthful auction mechanisms for VNF chain provisioning and allocation across geo-distributed datacenters

With the fast development of network function virtualization (NFV), NFV markets are emerging that allows network service providers (NSPs) to trade different virtual network function (VNF) chains among users. Therefore, efficient mechanisms for VNF chain provisioning and allocation are key to guarantee efficient operations of the NFV markets. One fundamental issue is how to maximize the social welfare of the market such that selfish users have incentives to acquire VNF chains from the markets. To overcome this issue, in this paper, we propose a truthful randomized combinatorial auction mechanism (TRCAM), which integrates the primal–dual approximation algorithm design technique and efficient pricing strategy. This mechanism considers dynamic VNF chain provisioning of the NSP and flexible VNF chain deployment of users. To be specific, we first formulate the social welfare maximization problem as an integer linear program, which is proven to be NP-hard. Then, we propose a primal–dual approximate algorithm which is near-optimal with polynomial-time complexity and employ it as a building block to design the TRCAM. Furthermore, we develop a novel binary-search-based algorithm to boost the average-case performance of the TRCAM. Both strict theoretical analysis and extensive experimental studies demonstrate the effectiveness of our proposed TRCAM, which achieves approximate truthfulness, near-optimal social welfare, individual rationality, and computational efficiency.

A Primal-Dual Approximation Algorithm for the Minimum Soft Capacitated Power Cover Problem

A Primal-Dual Approximation Algorithm for the Minimum Soft Capacitated Power Cover Problem

A primal-dual approximation algorithm for the k-prize-collecting minimum power cover problem

A primal-dual approximation algorithm for the k-prize-collecting minimum power cover problem

An improved primal-dual approximation algorithm for the k-means problem with penalties

AbstractIn the k-means problem with penalties, we are given a data set $${\cal D} \subseteq \mathbb{R}^\ell $$ of n points where each point $$j \in {\cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set $${\rm{C}}S \subseteq {{\cal R}^\ell }$$ with |CS| ≤ k and a penalized subset $${{\cal D}_p} \subseteq {\cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely $$\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).

Approximation algorithms for the dynamic k-level facility location problems

In this paper, we first consider a dynamic k-level facility location problem, which is a generalization of the k-level facility location problem when considering time factor. We present a combinatorial primal-dual approximation algorithm for this problem which finds a constant factor approximate solution. Then, we investigative the dynamic k-level facility location problem with submodular penalties and outliers, which extend the existing problem on two fronts, namely from static to dynamic and from without penalties (outliers) to penalties (outliers) allowed. Based on primal-dual technique and the triangle inequality property, we also give two constant factor approximation algorithms for the dynamic problem with submodular penalties and outliers, respectively.

Online Auction-Based Resource Allocation for Service-Oriented Network Slicing

Future mobile communication networks (5G) are envisioned to support a wide range of use cases with diverse service requirements. Network slicing as a key architectural enabling technology for 5G can provide users with customized services and meet the requirement of economic efficiency, where resource allocation is of vital importance. In this paper, we propose an online auction-based resource allocation scheme for network slicing (ORANS) to meet the diverse service requirements and improve network economic efficiency. We first propose a bidding information construction method to provide customized bidding information for users based on service types. Next, we model the slicing resource allocation problem as an online winner determination problem, with aim to maximize the social welfare of auction participants. We then implement a resource allocation strategy and payment rule to accomplish the service-oriented optimal resource allocation. Extensive simulation experiments are conducted to evaluate the performance of the proposed ORANS and the numerical results show that ORANS significantly outperforms the greedy algorithm and the primal-dual approximation algorithm in network economic efficiency while satisfying the diverse service requirements of users.

An Approximation Algorithm for Threshold Voltage Optimization

We present a primal-dual approximation algorithm for minimizing the leakage power of an integrated circuit by assigning gate threshold voltages. While most existing techniques do not provide a performance guarantee, we prove an upper bound on the power consumption. The algorithm is practical and works with an industrial sign-off timer. It can be used for post-routing power reduction or for optimizing leakage power throughout the design flow. We demonstrate the practical performance on recent microprocessor units. Our implementation obtains significant leakage power reductions of up to 8% on top of one of the most successful algorithms for gate sizing and threshold voltage optimization. After timing-aware global routing, we achieve leakage power reductions of up to 34%.

A Note on Submodular Function Minimization with Covering Type Linear Constraints

In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by $$\varDelta _i$$Δi the number of non-zero coefficients in the ith constraints. Furthermore, we assume that $$\varDelta _1 \ge \varDelta _2 \ge \cdots \ge \varDelta _m$$Δ1źΔ2źźźΔm. For this problem, Koufogiannakis and Young proposed a polynomial-time $$\varDelta _1$$Δ1-approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with $$\{0,1\}$${0,1}-variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is $$\max \{\varDelta _2, \min \{\varDelta _1, 1 + \varPi \}\}$$max{Δ2,min{Δ1,1+ź}}, where $$\varPi $$ź is the maximum size of a connected component of the input submodular function.

Multilayer matching of metric structures using hierarchically well-separated trees

We consider the matching problem between two metric distributions where establishing a one-to-one matching of features may not always be possible. Although many-to-many graph matching techniques achieve the desired multi map between features, they ignore the spatial structure of the nodes. We propose a novel technique, multilayer matching, for solving the matching problem which utilizes both the individual node features and the clustering information of nodes. Our method uses the hierarchically well-separated trees (HSTs) to represent the metric distribution such that non-leaf nodes in the tree representation corresponds to a constellation of features in the original structure. By using HSTs in a linear programming setup, we obtain a matching between features through finding a mapping between non-leaf nodes among the two HSTs. We further provide a primal-dual approximation algorithm for the multilayer matching which runs several order of magnitudes faster while achieving comparable success rates. Application of the method to the image matching problem is also presented in the paper. Empirical evaluation of the method and its primal-dual extension on a set of recognition tests show the robustness and efficiency of the overall approach.

A primal-dual approximation algorithm for a two depot heterogeneous traveling salesman problem

Surveillance applications require a collection of heterogeneous vehicles to visit a set of targets. We consider a fundamental routing problem that arises in these applications involving two vehicles. Specifically, we consider a routing problem where there are two heterogeneous vehicles that start from distinct initial locations and a set of targets. The objective is to find a tour for each vehicle such that each of the targets is visited at least once by a vehicle and the sum of the distances traveled by the vehicles is minimal. We consider an important special case of this routing problem where the travel costs satisfy the triangle inequality and the following monotonicity property: the first vehicle’s cost of traveling between any two targets is at most equal to the second vehicle’s cost of traveling between the same targets. We present a primal-dual algorithm for this case that provides an approximation ratio of 2.

Joint Time-Domain Resource Partitioning, Rate Allocation, and Video Quality Adaptation in Heterogeneous Cellular Networks

Heterogenous cellular networks (HCN) introduce small cells within the transmission range of a macrocell. For the efficient operation of HCNs it is essential that the high-power macrocell shuts off its transmissions for an appropriate amount of time in order for the low-power small cells to transmit. This is a mechanism that allows time-domain resource partitioning (TDRP) and is critical to be optimized for maximizing the throughput of the complete HCN. In this paper, we investigate video communication in HCNs when TDRP is employed. After defining a detailed system model for video streaming in such an HCN, we consider the problem of maximizing the experienced video quality at all the users, by jointly optimizing the TDRP for the HCN, the rate allocated to each specific user, and the selected video quality transmitted to a user. The NP-hard problem is solved with a primal-dual approximation algorithm that decomposes the problem into simpler subproblems, making them amenable to fast well-known solution algorithms. Consequently , the calculated solution can be enforced in the time scale of real-life video streaming sessions. This last observation motivates the enhancement of the proposed framework to support video delivery with dynamic adaptive streaming over HTTP (DASH). Our extensive simulation results demonstrate clearly the need for our holistic approach for improving the video quality and playback performance of the video streaming users in HCNs.

Primal–dual approximation algorithm for the two-level facility location problem via a dual quasi-greedy approach

The main contribution of this work is to propose a primal–dual combinatorial 3(1+ε)-approximation algorithm for the two-level facility location problem (2-LFLP) by exploring the approximation oracle concept. This result improves the previous primal–dual 6-approximation algorithm for the multilevel facility location problem, and also matches the previous primal–dual approximation ratio for the single-level facility location problem. One of the major merits of primal–dual type algorithms is their easy adaption to other variants of the facility location problems. As a demonstration, our primal–dual approximation algorithm can be easily adapted to several variants of the 2-LFLP, including models with stochastic scenario, dynamically arrived demands, and linear facility cost.

Approximation Algorithm for Facility Location Problems

Significant research effort has been devoted in the study of approximation algorithms for NP-hard problems. In this work we modify a known primal-dual approximation algorithm for facility location problem. Although we fail to give a performance guarantee for the new approach but we show that our method performs better in a tight case.

A primal-dual approximation algorithm for the Asymmetric Prize-Collecting TSP

We present a primal-dual Â?log(n)Â?-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294---301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.'s heuristic (Frieze et al. in Networks 12:23---39, 1982) or the recent Asadpour et al.'s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1+Â?log(n)Â? and $3+ 8\frac{\log(n)}{\log(\log(n))}$ as an approximation ratio. Our algorithm achieves an approximation ratio of Â?log(n)Â? which is weaker than $3+ 8\frac{\log(n)}{\log(\log(n))}$ but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.

A primal–dual approximation algorithm for the vertex cover [formula omitted] problem

We introduce the vertex cover P n ( V C P n ) problem, that is, the problem of finding a minimum weight set F ⊂ V such that the graph G [ V − F ] has no P n , where P n is a path with n vertices. The problem also has its application background. In this paper, we first show that the V C P n problem is NP-hard for any integer n ≥ 2 . Then we restrict our attention to the V C P 3 problem and give a 2-approximation algorithm using the primal–dual method.

A primal-dual approximation algorithm for stochastic facility location problem with service installation costs

We consider the stochastic version of the facility location problem with service installation costs. Using the primal-dual technique, we obtain a 7-approximation algorithm.

A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties

We consider the facility location problem with submodular penalties (FLPSP), introduced by Hayrapetyan et al. (Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 933–942, 2005), who presented a 2.50-approximation algorithm that is non-combinatorial because this algorithm has to solve the LP-relaxation of an integer program with exponential number of variables. The only known polynomial algorithm for this exponential LP is via the ellipsoid algorithm as the corresponding separation problem for its dual program can be solved in polynomial time. By exploring the properties of the submodular function, we offer a primal-dual 3-approximation combinatorial algorithm for this problem.

Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm.