Improved Approximation Ratio Research Articles

An Improved Approximation Algorithm for the Capacitated Correlation Clustering Problem

Correlation clustering problem (CorCP) is a classical clustering problem, which clusters data based on the similarity of data set, and has many applications in interaction networks, cross-lingual link detection, and communication networks, etc. In this paper, we study a practical generalization of the CorCP, called the capacitated correlation clustering problem (the capacitated CorCP), by constructing a labeled complete graph. On this labeled complete graph, each vertex represents a piece of data. If two pieces of data are similar, then the edge between the corresponding vertices is marked by a positive label [Formula: see text]. Otherwise, this edge is marked by a negative label −. The objective of the capacitated CorCP is to group some similar data sets into one cluster as far as possible, while satisfying the cluster capacity constraint. To achieve this objective, we shall partition the vertex set of the labeled complete graph into several clusters, each cluster’s size subjecting to an upper bound, so as to minimize the number of disagreements. Here the number of disagreements is defined as the total number of the edges with positive labels between clusters and the edges with negative labels within clusters. Different with the previous algorithm in [18], which subjects to the constraint on the cluster size by a penalty measure, we design an algorithm for the capacitated CorCP to directly output a feasible solution by iteratively constructing clusters based on a preset threshold. Through carefully setting the threshold and sophisticatedly analyzing, our algorithm is proved to have an improved approximation ratio of 5.37. In addition, we also conduct a series of numerical experiments to demonstrate the effectiveness of our algorithm.

Scheduling multiple two-stage flowshops with a deadline

Recently, motivated by applications in cloud computing, scheduling multiple two-stage flowshops with the objective of minimizing the makespan has drawn increasing attention. Motivated by the same applications, the current paper studies the problem of scheduling multiple two-stage flowshops with a deadline, with the objective of maximizing the profit of the jobs that can be completed by the deadline. Algorithms and complexity of the problem are investigated. For the case where the number of two-stage flowshops is part of the input, we present an efficient approximation algorithm with a constant ratio. The approximation ratio is improved via a study of the relationship between the problem and the multiple knapsack problem, combined with the recent progresses in the research in approximation algorithms for the multiple knapsack problem. By integrating techniques in the study of the classical Knapsack problem and the classical Makespan problem on multiple processors, plus additional new techniques, a polynomial-time approximation algorithm with a further improved approximation ratio is developed for the case where the number of flowshops is a fixed constant.

Benchmarking Quantum Coprocessors in an Application-Centric, Hardware-Agnostic, and Scalable Way

Existing protocols for benchmarking current quantum co-processors fail to meet the usual standards for assessing the performance of High-Performance-Computing platforms. After a synthetic review of these protocols -- whether at the gate, circuit or application level -- we introduce a new benchmark, dubbed Atos Q-score (TM), that is application-centric, hardware-agnostic and scalable to quantum advantage processor sizes and beyond. The Q-score measures the maximum number of qubits that can be used effectively to solve the MaxCut combinatorial optimization problem with the Quantum Approximate Optimization Algorithm. We give a robust definition of the notion of effective performance by introducing an improved approximation ratio based on the scaling of random and optimal algorithms. We illustrate the behavior of Q-score using perfect and noisy simulations of quantum processors. Finally, we provide an open-source implementation of Q-score that makes it easy to compute the Q-score of any quantum hardware.

Computing coverage kernels under restricted settings

Given a set B of d-dimensional boxes (i.e., axis-aligned hyperrectangles), a minimum coverage kernel is a subset of B of minimum size covering the same region as B. Computing it is NP-hard, but as for many similar NP-hard problems (e.g., Box Cover, and Orthogonal Polygon Covering), the problem becomes solvable in polynomial time under restrictions on B. We show that computing minimum coverage kernels remains NP-hard even when restricting the graph induced by the input to a highly constrained class of graphs. Alternatively, we present two polynomial-time approximation algorithms for this problem: one deterministic with an approximation ratio within O(log⁡n), and one randomized with an improved approximation ratio within O(lg⁡OPT) (with high probability).

Improved approximation for Fractionally Subadditive Network Design

Recently, Guruganesh et al. [1] proposed the Fractionally Subadditive Network Design (f-SAND) problem. In this problem, we are given a weighted graph G with a special root vertex r together with k subsets of its vertices called scenarios. We need to assign capacities to the edges of G so that for every scenario, the assigned capacities support a unit flow from the root to all vertices in the scenario, simultaneously. The goal is to minimize the total cost of the solution.f-SAND is a fairly natural problem and it generalizes several well-studied problems, such as the Steiner Tree problem. The added interest to this problem stems from the fact that standard tools used for approximating network design problems do not seem to apply to f-SAND. Despite this, Guruganesh et al. conjecture the existence of an algorithm with an approximation ratio bounded by a constant, i.e. independent of k. They make a step towards this goal by giving a 32-approximation algorithm for k=2.In this paper, we present an algorithm similar to that of [1], together with an analysis that not only yields an improved approximation ratio of 43, but is also significantly simpler. We also generalize it to arbitrary k, giving a 23k-approximation. On the negative side, we argue that without additional insights, the techniques used are not likely to produce a constant factor approximation algorithm.

Improved approximation for two dimensional Strip Packing with polynomial bounded width

We study the well-known two-dimensional Strip Packing problem. Given a set of rectangular axis-parallel items and a strip of width W with infinite height, the objective is to find a packing of all items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of 3/2 of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type (nW)f(1/ε). If W is polynomially bounded by the number of items, this is a polynomial running-time. The currently best pseudo-polynomial approximation algorithm by Nadiradze and Wiese achieves an approximation ratio of 1.4+ε. We present a pseudo-polynomial algorithm with improved approximation ratio 4/3+ε. Furthermore, the presented algorithm has a significantly smaller running-time as the 1.4+ε approximation algorithm.

Maximizing a Monotone Submodular Function with a Bounded Curvature under a Knapsack Constraint

We consider the problem of maximizing a monotone submodular function under a knapsack constraint. We show that, for any fixed $\epsilon > 0$, there exists a polynomial-time algorithm with an approximation ratio $1-c/e-\epsilon$, where $c \in [0,1]$ is the (total) curvature of the input function. This approximation ratio is tight up to $\epsilon$ for any $c \in [0,1]$. To the best of our knowledge, this is the first result for a knapsack constraint that incorporates the curvature to obtain an approximation ratio better than $1-1/e$, which is tight for general submodular functions. As an application of our result, we present a polynomial-time algorithm for the budget allocation problem with an improved approximation ratio.

Space-Constrained Interval Selection

We study streaming algorithms for the interval selection problem: finding a maximum cardinality subset of disjoint intervals on the line. A deterministic 2-approximation streaming algorithm for this problem is developed, together with an algorithm for the special case of proper intervals, achieving improved approximation ratio of 3/2. We complement these upper bounds by proving that they are essentially the best possible in the streaming setting: It is shown that an approximation ratio of 2 − ϵ (or 3/2 − ϵ for proper intervals) cannot be achieved unless the space is linear in the input size. In passing, we also answer an open question of Adler and Azar (J. Scheduling 2003) regarding the space complexity of constant-competitive randomized preemptive online algorithms for the same problem.

Approximating $k$-Median via Pseudo-Approximation

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our improved approximation ratio is achieved by exploiting the power of pseudo-approximation. More specifically, our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an $\alpha$-approximation algorithm for $k$-median, it is sufficient to give a pseudo-approximation algorithm that finds an $\alpha$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significantly smaller cost than that of opening only $k$ facilities. Second, we give such a pseudo-approximation algorithm with $\alpha=1+\sqrt{3}+\epsilon$. Prior to our work, it was not even known whether opening $k+o(k)$ facilities would help improve the approximation ratio.

Improving Christofides' Algorithm for the s-t Path TSP

We present a deterministic (1+√5/2)-approximation algorithm for the s - t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old ratio set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this article can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s - t path problem and the unit-weight graphical metric s - t path TSP.

Optimal Association in Wireless Mesh Networks

The wireless multihop backhaul is a unique feature of wireless mesh networks (WMNs), which necessitates redesign of association control algorithms. In this paper, we formulate and propose approximation algorithms for the problem of optimal joint association and bandwidth allocation in WMNs, considering max–min fairness (MM) and proportional fairness (PF) objectives. We first relax the integral association constraint and get an optimal fractional association solution. Then, we propose two rounding algorithms, namely, largest fraction rounding and bipartite graph rounding, to get an integral solution and analyze their theoretical approximation ratios. Finally, we propose two approximation ratio improvement algorithms so that the improved approximation ratio can more accurately reflect the true performance gap between the produced solution and the optimal one. Our simulation results show that the proposed algorithms achieve performance that is close to the optimal and outperform popular heuristic algorithms. We also compare the performance of PF and MM in WMNs in terms of network throughput and fairness in user bandwidth. Finally, we compare the performances of the proposed rounding algorithms and show that the approximation ratio can be reduced to 1–2 by the proposed ratio improvement algorithms. Therefore, our proposed algorithm is able to achieve nearly optimal association control, as well as bandwidth allocation, considering MM or PF, with small approximation ratios, in WMNs.

An Improved Approximation Ratio to the Partial-Terminal Steiner Tree Problem

We consider a generalization of both the classic Steiner tree problem and the terminal Steiner tree problem. Given a complete graph <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${ G = (V,E)}$</tex> </formula> with a metric cost function <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${ c:E \rightarrow {\BBQ_ \geq }}$</tex> </formula> and two proper subsets <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ R \subset V$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ R^\prime \subseteq R$</tex> </formula> , a partial-terminal Steiner tree is a Steiner tree which contains all vertices in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\it R$</tex> </formula> such that all vertices in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$R^\prime$</tex> </formula> must be leaves. The partial-terminal Steiner tree problem is to find a partial-terminal Steiner tree of the minimum cost in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$G$</tex> </formula> . The previously best-known approximation ratio of the problem is <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ 2\rho$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\bf \rho$</tex> </formula> is the approximation ratio of the Steiner tree problem. In this paper, we improve the ratio from <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ 2\rho$</tex> </formula> to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ 2\rho - {\rho \over {3\rho - 2}} - f$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$f$</tex> </formula> is a non-negative function whose value is between 0 and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$ \rho - {\rho \over {3\rho - 2}}$</tex> </formula> .

An improved approximation ratio for the jump number problem on interval orders

The jump number problem is to find a linear extension of a given poset that minimizes the number of incomparable adjacent elements. The problem is NP-hard even in the class of interval orders and so three different 3/2-approximation algorithms for this case have been proposed respectively by Felsner, Sysło, and Mitas. We build upon the approach of Mitas, where the problem is reduced to packing vertex-disjoint edges and odd cycles in a certain graph. We prove that the requirement of independence between edges and cycles may be abandoned. In effect, the jump number problem is reduced to the set cover problem, and the 4/3-approximation algorithm of Duh and Fürer for the 3-set cover problem is used to show that the approximation ratio of the jump number problem on interval orders is tighter than 89/60.

Corrigendum

In Gandhi et al. [2006], we gave an algorithm for the data migration and non-deterministic open shop scheduling problems in the minimum sum version, that was claimed to achieve a 5.06-approximation. Unfortunately, it was pointed to us by Maxim Sviridenko that the argument contained an unfounded assumption that has eluded all of its readers until now. We detail in this document how this error can be amended. A side effect is an improved approximation ratio of 4.96.

Improved approximation algorithms for the robust fault-tolerant facility location problem

We consider the robust α fault-tolerant facility location problem (α-RFLP), recently introduced by Chechik and Peleg (2010) [6]. We present an improved approximation algorithm with ratio 5.236 for the 1-RFLP comparing to 6.5 by Chechik and Pelegʼs. For the general α-RFLP (fixed α⩾2), the same algorithm with a different subroutine tailored for α⩾2 provides an improved approximation ratio 1.005+6.015α comparing to 1.5+7.5α by Chechik and Pelegʼs. The key component of our algorithm is the resolution of an auxiliary facility location problem (FLP) by a variant of the LP-rounding technique of Byrka and Aardal (2010) [2] to estimate the total weighted facility open cost and shipping cost.

An improved approximation ratio for the minimum linear arrangement problem

We observe that combining the techniques of Arora, Rao, and Vazirani, with the rounding algorithm of Rao and Richa yields an O ( log n log log n ) -approximation for the minimum-linear arrangement problem. This improves over the O ( log n ) -approximation of Rao and Richa.

Job Shop Scheduling with Unit Processing Times

We consider randomized algorithms for the preemptive job shop problem, or equivalently, the case in which all operations have unit length. We give an α-approximation for the case of two machines where α &lt; 1.45, an improved approximation ratio of O(log m/log log m) for an arbitrary number m of machines, and the first (2 + ε)-approximation for a constant number of machines. The first result is via an approximation algorithm for a string matching problem that is of independent interest.

A Polylogarithmic Approximation of the Minimum Bisection

A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size $n/2$. The bisection cost is the number of edges connecting the two sets. The problem of finding a bisection of minimum cost is prototypical to graph partitioning problems, which arise in numerous contexts. This problem is NP-hard. We present an algorithm that finds a bisection whose cost is within a factor of $O(\log^{1.5} n)$ from the minimum. For graphs excluding any fixed graph as a minor (e.g., planar graphs) we obtain an improved approximation ratio of $O(\log n)$. The previously known approximation ratio for bisection was roughly $\sqrt{n}$.

An Improved Approximation Ratio for the Covering Steiner Problem

In the Covering Steiner problem, we are given an undirected graph with edge- costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-MST and the Group Steiner problems; indeed, when all the vertices of the graph lie in one group with a requirement of k, we get the k-MST problem, and when there are multiple groups with unit requirements, we obtain the Group Steiner problem. While many covering problems (e.g., the covering integer programs such as set cover) become easier to approximate as the requirements increase, the Covering Steiner problem

Selecting Forwarding Neighbors in Wireless Ad Hoc Networks

Broadcasting is a fundamental operation which is frequent in wireless ad hoc networks. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding is very inefficient and can result in high redundancy, contention, and collision. One approach to reducing the redundancy is to let each node forward the message only to a small subset of 1-hop neighbors that cover all of the node's 2-hop neighbors. In this paper we propose two practical heuristics for selecting the minimum number of forwarding neighbors: an O(nlog n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an O(nlog 2 n) time algorithm with an improved approximation ratio of 3, where n is the number of 1- and 2-hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n 3 log n) time.