Worst-case Approximation Ratio Research Articles

Average-Case Analysis of Greedy Matching for Large-Scale D2D Resource Sharing

Given the proximity of many wireless users and their diversity in consuming local resources (e.g., data-plans, computation and energy resources), device-to-device (D2D) resource sharing is a promising approach towards realizing a sharing economy. This paper adopts an easy-to-implement greedy matching algorithm with distributed fashion and only sub-linear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(\log n)$</tex-math></inline-formula> parallel complexity (in user number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> ) for large-scale D2D sharing. Practical cases indicate that the greedy matching's average performance is far better than the worst-case approximation ratio 50% as compared to the optimum. However, there is no rigorous average-case analysis in the literature to back up such encouraging findings and this paper is the first to present such analysis for multiple representative classes of graphs. For 1D linear networks, we prove that our greedy algorithm performs better than 86.5% of the optimum. For 2D grids, though dynamic programming cannot be directly applied, we still prove this average performance ratio to be above 76%. For the more challenging Erdos-Rényi random graphs, we equivalently reduce to the asymptotic analysis of random trees and successfully prove a ratio up to 79%. Finally, we conduct experiments using real data to simulate realistic D2D networks, and show that our analytical performance measure approximates well practical cases.

Joint Dataset Reconstruction and Power Control for Distributed Training in D2D Edge Network

The intrinsic nature of non-independent and identically distributed datasets on heterogeneous devices slows down the distributed model training process and reduces the training accuracy. To settle this problem, we propose a dataset reconstruction scheme to transform the data distribution of training device’s dataset into independent and identically distributed dataset via data exchange among trusted devices. For energy efficiency, we further consider power control for the devices. We then formulate an optimization problem, which is a mixed integer non-linear programming problem, to minimize the total energy consumption for each round of distributed training. Due to the NP-hardness and coupling property of the optimization problem, we decompose it into two subproblems for dataset reconstruction and power control, respectively. An approximation algorithm is designed to obtain a near-optimal auxiliary devices set for dataset reconstruction with minimum energy consumption, while meeting the variance constraint of the optimization problem. We prove that approximation algorithm has a worst-case approximation ratio of 1+ln|Ωi(t)|, where |Ωi(t)| is the required data samples for dataset reconstruction of each training device. For power control, we design a dynamic programming algorithm to further reduce the energy consumption. For comparison, we propose three benchmark schemes that adopt either one of the algorithms or neither. We also customize three baseline algorithms based on the state-of-the-arts to compare with our proposed algorithm. Numerical results show that, our proposed algorithm outperforms three benchmarks on the average energy consumption for one round for different cases. When varying the labels that each device owns, our proposed algorithm outperforms the other three baseline algorithms on training accuracy. Besides, when setting a target accuracy, our proposed algorithm always has the lowest energy consumption.

Width Helps and Hinders Splitting Flows

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation X on a directed graph G into weighted source-to-sink paths whose weighted sum equals X . We show that, for acyclic graphs, considering the width of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of width-stable graphs, for which a popular heuristic is a O (log Val(X))-approximation (Val(X) being the total flow of X ), and strengthen its worst-case approximation ratio from \(\Omega (\sqrt {m}) \) to Ω ( m /log m ) for sparse graphs, where m is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a (⌈log ‖ X ‖⌉ + 1)-approximation (‖ X ‖ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations (‖ X ‖ ≤ 1), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by [17], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

Optimality of the coordinate-wise median mechanism for strategyproof facility location in two dimensions

We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in \(\mathbb {R}^2\). We show that for the p-norm (\(p \ge 1\)) objective, the coordinate-wise median mechanism (CM) has the lowest worst-case approximation ratio in the class of deterministic, anonymous, and strategyproof mechanisms. For the minisum objective and an odd number of agents n, we show that CM has a worst-case approximation ratio (AR) of \(\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}\). For the p-norm social cost objective (\(p\ge 2\)), we find that the AR for CM is bounded above by \(2^{\frac{3}{2}-\frac{2}{p}}\). We conjecture that the AR of CM actually equals the lower bound \(2^{1-\frac{1}{p}}\) (as is the case for \(p=2\) and \(p=\infty\)) for any \(p\ge 2\).

The metric distortion of multiwinner voting

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when q⩽k/3, asymptotically linear in the number of agents when k/3<q⩽k/2, and constant when q>k/2.

The Metric Distortion of Multiwinner Voting

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when q

Scheduling with complete multipartite incompatibility graph on parallel machines: Complexity and algorithms

In this paper, the problem of scheduling on parallel machines with a presence of incompatibilities between jobs is considered. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine.The paper provides several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). The problems are considered for a variety of machine types: identical, uniform and unrelated. The results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. Also, polynomial-time exact algorithms are provided for easier problems and algorithms with a guaranteed constant worst-case approximation ratio for harder ones.In particular, there is provided a polynomial-time approximation scheme (PTAS) for scheduling with respect to ∑Cj on uniform machines, when the number of parts (i.e. sets of vertices constituting the mentioned graph) is bounded. The problem is addressed by developing a linear programming relaxation technique with appropriate rounding. This technique together with an exhaustive search (albeit in a manner controlled by the precision parameter) allows to provide the desired algorithm. For Cmax a PTAS is provided for the case of unit time jobs, but when the number of parts is part of the input. Interestingly, the latter result shows a connection between the considered problem of scheduling and covering problems.

Greedy Matching in Bipartite Random Graphs

This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].

The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε &gt; 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε &gt; 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

In this paper we consider a problem of job scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). We consider a variety of machine types in our paper: identical, uniform, and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for the easier problems, or algorithms with a guaranteed constant worst-case approximation ratio. In particular, we fill the gap on research for the problem of finding a schedule with the smallest ∑Cj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting.

MaxCut quantum approximate optimization algorithm performance guarantees for p&gt;1

We obtain worst case performance guarantees for $p=2$ and $3$ QAOA for MAXCUT on uniform 3-regular graphs. Previous work by Farhi et al obtained a lower bound on the approximation ratio of $0.692$ for $p=1$. We find a lower bound of $0.7559$ for $p=2$, where worst case graphs are those with no cycles $\leq 5$. This bound holds for any 3 regular graph evaluated at particular fixed parameters. We conjecture a hierarchy for all $p$, where worst case graphs have with no cycles $\leq 2p+1$. Under this conjecture, the approximation ratio is at least $0.7924$ for all 3 regular graphs and $p=3$. In addition, using a simple indistinguishability argument we find an upper bound on the worst case approximation ratio for all $p$, which indicates classes of graphs for which there can be no quantum advantage for at least $p<6$.

On the effectiveness of the genetic paradigm for polygonization

A polygon is simple if it is a closed chain of straight line segments that do not self-intersect. Given a finite set P of input points in the Euclidean plane, the search for a simple polygon with vertex set P is a very well-known and studied computational problem, referred to as the simple polygonization. Its optimization version, that requires to find a simple polygon having either minimum or maximum area, is known to be NP-hard. Moreover, no bounded approximation algorithm is known for the minimization flavor, while the maximization one admits an algorithm guaranteeing 12 worst-case approximation ratio. In this work, we design a new algorithm to practically attack both the optimization problems, based on the genetic paradigm. We demonstrate its effectiveness through an extensive experimental evaluation that employs a reference test-bed set of input instances.

Approximation Ratios of RePair, LongestMatch and Greedy on Unary Strings

A grammar-based compressor is an algorithm that receives a word and outputs a context-free grammar that only produces this word. The approximation ratio for a single input word is the size of the grammar produced for this word divided by the size of a smallest grammar for this word. The worst-case approximation ratio of a grammar-based compressor for a given word length is the largest approximation ratio over all input words of that length. In this work, we study the worst-case approximation ratio of the algorithms Greedy, RePair and LongestMatch on unary strings, i.e., strings that only make use of a single symbol. Our main contribution is to show the improved upper bound of O((logn)8·(loglogn)3) for the worst-case approximation ratio of Greedy. In addition, we also show the lower bound of 1.34847194⋯ for the worst-case approximation ratio of Greedy, and that RePair and LongestMatch have a worst-case approximation ratio of log2(3).

Influence Maximization with Priority in Online Social Networks

The Influence Maximization (IM) problem, which finds a set of k nodes (called seedset) in a social network to initiate the influence spread so that the number of influenced nodes after propagation process is maximized, is an important problem in information propagation and social network analysis. However, previous studies ignored the constraint of priority that led to inefficient seed collections. In some real situations, companies or organizations often prioritize influencing potential users during their influence diffusion campaigns. With a new approach to these existing works, we propose a new problem called Influence Maximization with Priority (IMP) which finds out a set seed of k nodes in a social network to be able to influence the largest number of nodes subject to the influence spread to a specific set of nodes U (called priority set) at least a given threshold T in this paper. We show that the problem is NP-hard under well-known IC model. To find the solution, we propose two efficient algorithms, called Integrated Greedy (IG) and Integrated Greedy Sampling (IGS) with provable theoretical guarantees. IG provides a 1−(1−1k)t-approximation solution with t is an outcome of algorithm and t≥1. The worst-case approximation ratio is obtained when t=1 and it is equal to 1/k. In addition, IGS is an efficient randomized approximation algorithm based on sampling method that provides a 1−(1−1k)t−ϵ-approximation solution with probability at least 1−δ with ϵ&gt;0,δ∈(0,1) as input parameters of the problem. We conduct extensive experiments on various real networks to compare our IGS algorithm to the state-of-the-art algorithms in IM problem. The results indicate that our algorithm provides better solutions interns of influence on the priority sets when approximately give twice to ten times higher than threshold T while running time, memory usage and the influence spread also give considerable results compared to the others.

Strategyproof auction mechanisms for network procurement

Deferred-acceptance auctions can be seen as heuristic algorithms to solve {{mathcal {N}}}{{mathcal {P}}}-hard allocation problems. Such auctions have been used in the context of the Incentive Auction by the US Federal Communications Commission in 2017, and they have remarkable incentive properties. Besides being strategyproof, they also prevent collusion among participants. Unfortunately, the worst-case approximation ratio of these algorithms is very low in general, but it was observed that they lead to near-optimal solutions in experiments on the specific allocation problem of the Incentive Auction. In this work, which is inspired by the telecommunications industry, we focus on a strategic version of the minimum Steiner tree problem, where the edges are owned by bidders with private costs. We design several deferred-acceptance auctions (DAAs) and compare their performance to the Vickrey–Clarke–Groves (VCG) mechanism as well as several other approximation mechanisms. We observe that, even for medium-sized inputs, the VCG mechanisms experiences impractical runtimes and that the DAAs match the approximation ratios of even the best strategy-proof mechanisms in the average case. We thus provide another example of an important practical mechanism design problem, where empirics suggest that carefully designed deferred-acceptance auctions with their superior incentive properties need not come at a cost in terms of allocative efficiency. Our experiments provide insights into the trade-off between solution quality and runtime and into the additional premium to be paid in DAAs to gain weak group-strategyproofness rather than just strategyproofness.

Strategic facility location problems with linear single-dipped and single-peaked preferences

We consider the design of mechanisms for locating facilities on an interval. There are multiple agents on the interval, each receiving a utility determined by their distances to the facilities. The objectives considered are maximization of social welfare (sum of utilities) and egalitarian welfare (minimum utility). Agents can misreport their locations, and so we require the mechanisms to be strategyproof—no agent should be able to benefit from misreporting; subject to strategyproofness, we attempt to design mechanisms that are approximately optimal (have small worst-case approximation ratios). The novelty of our work is the consideration of models in which single-dipped and single-peaked preferences exist simultaneously. We consider two models. In the first model, there is a single facility, and agents may disagree about its nature: some agents prefer to be near the facility, while others prefer to be far from it. In the second model, there are two facilities: a desirable facility that all agents want near, and an undesirable facility that all agents want far. We design a variety of approximately optimal strategyproof mechanisms for both models, and prove several lower bounds as well. For the social welfare objective, we provide best-possible deterministic strategyproof mechanisms in the first model and the second model. We then provide improved randomized strategyproof mechanisms for each model, as well as a non-tight lower bound on the worst-case approximation ratio attainable by such mechanisms for the first model. For the egalitarian welfare objective, we provide a lower bound on randomized strategyproof mechanisms for the first model, as well as an optimal (non-approximate) strategyproof mechanism for the second model. All of our mechanisms are also group strategyproof: no coalition of agents can unanimously benefit from misreporting.

Approximating combinatorial optimization problems with the ordered weighted averaging criterion

This paper deals with combinatorial optimization problems with K cost scenarios, inducing K linear objectives. The popular Ordered Weighted Averaging (OWA) criterion is used to aggregate the objectives and compute a solution. It is well-known that minimizing OWA for most basic combinatorial problems is weakly NP-hard even if the number of scenarios K equals two, and strongly NP-hard when K is a part of the input. In this paper, the problem with nonincreasing weights in the OWA criterion and a large K is first considered. A method of reducing the number of scenarios, by appropriately aggregating the costs before solving the problem, is proposed. It is shown that an optimal solution to the reduced problem has a guaranteed worst-case approximation ratio. Some new approximation results for the Hurwicz criterion, which is a special case of OWA, are also presented.

An Experimental Study of Algorithms for Online Bipartite Matching

We perform an experimental study of algorithms for online bipartite matching under the known i.i.d input model with integral types. In the last decade, there has been substantial effort in designing complex algorithms to improve worst-case approximation ratios. Our goal is to determine how these algorithms perform on more practical instances rather than worst-case instances. In particular, we are interested in whether the ranking of the algorithms by their worst-case performance is consistent with the ranking of the algorithms by their average-case/practical performance. We are also interested in whether preprocessing times and implementation difficulties that are introduced by these algorithms are justified in practice. To that end, we evaluate these algorithms on different random inputs as well as real-life instances obtained from publicly available repositories. We compare these algorithms against several simple greedy-style algorithms. Most of the complex algorithms in the literature are presented as being non-greedy (i.e., an algorithm can intentionally skip matching a node that has available neighbors) to simplify the analysis. Every such algorithm can be turned into a greedy one without hurting its worst-case performance. On our benchmarks, non-greedy versions of these algorithms perform much worse than their greedy versions. Greedy versions perform about as well as the simplest greedy algorithm by itself. This, together with our other findings, suggests that simplest greedy algorithms are competitive with the state-of-the-art worst-case algorithms for online bipartite matching on many average-case and practical input families. Greediness is by far the most important property of online algorithms for bipartite matching.

Identical parallel machine scheduling with assurance of maximum waiting time for an emergency job

Currently, customer satisfaction is playing an increasingly vital role in both manufacturing and service industries. Assuring an acceptable waiting time to customers is considered as an effective approach to improve customer satisfaction. In this study, an identical parallel machine scheduling problem assuring the maximum waiting time for an emergency job which arrives at any time is investigated. A mixed integer programming model is formulated, based on which a variant formulation is generated. The formulations are further enhanced by various techniques, which forms two formulation-based methods. Two objectives, makespan and total completion time, are considered separately. Regarding the makespan, the worst-case approximation ratios of the classical heuristic rules are deduced. For the total completion time, efficient bounds are provided and the NP-hardness of the problem is proved. Heuristic methods based on the classical dispatch rules are developed, for both the cases. Extensive computational experiments are conducted, based on which the performances of the formulation-based methods and heuristics are compared, the relationship between the objective values and the assured maximum waiting time for an emergency job is explored, and a few observations and managerial insights are obtained.

Approximately Optimal Mechanisms for Strategyproof Facility Location: Minimizing Lp Norm of Costs

This paper is concerned with the problem of locating a facility on a line in the presence of strategic agents, also located on that line. Each agent incurs a cost equal to her distance to the facility whereas the planner wishes to minimize the Lp norm of the vector of agent costs. The location of each agent is only privately known, and the goal is to design a strategyproof mechanism that approximates the optimal cost well. It is shown that the median mechanism provides a 21−1/p approximation ratio, and that this is the optimal approximation ratio among all deterministic strategyproof mechanisms. For randomized mechanisms, two results are shown: First, for any integer p larger than 2, no mechanism—from a rather large class of randomized mechanisms—has an approximation ratio better than that of the median mechanism. This is in contrast to the case of p = 2 and p = ∞ where a randomized mechanism provably helps improve the worst case approximation ratio. Second, for the case of 2 agents, the Left-Right-Middle (LRM) mechanism, first designed by Procaccia and Tennenholtz for the special case of infinity norm, provides the optimal approximation ratio among all randomized mechanisms.