Approximation Algorithms for Computing Maximin Share Allocations

The problem of fair division of indivisible goods is a fundamental problem of resource allocation in multi-agent systems, also studied extensively in social choice. Recently, the problem was generalized to the case when goods form a graph and the goal is to allocate goods to agents so that each agent’s bundle forms a connected subgraph. For the maximin share fairness criterion, researchers proved that if goods form a tree, an allocation offering each agent a bundle of at least her maximin share value always exists. Moreover, it can be found in polynomial time. In this paper we consider the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out to be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide in this case results on allocations guaranteeing each agent a certain fraction of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting, the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding 1/2-approximate MMS allocations---an improvement from the previously best available guarantee of 11/30.

We study the problem of fairly dividing indivisible goods among a set of agents under the fairness notion of Any Price Share (APS). APS is known to dominate the widely studied Maximin share (MMS). Since an exact APS allocation may not exist, the focus has traditionally been on the computation of approximate APS allocations. [4] studied the problem under additive valuations, and asked (i) how large can the APS value be compared to the MMS value? and (ii) what guarantees can one achieve beyond additive functions. We partly answer these questions by considering valuations beyond additive, namely submodular and XOS functions, with binary marginals. For the submodular functions with binary marginals, also known as matroid rank functions (MRFs), we show that APS is exactly equal to MMS. Consequently, following [5] we show that an exact APS allocation exists and can be computed efficiently while maximizing the social welfare. Complementing this result, we show that it is NP-hard to compute the APS value within a factor of 5/6 for submodular valuations with three distinct marginals of {0, 1/2, 1.} We then consider binary XOS functions, which are immediate generalizations of binary submodular functions in the complement free hierarchy. In contrast to the MRFs setting, MMS and APS values are not equal under this case. Nevertheless, we can show that they are only a constant factor apart. In particular, we show that under binary XOS valuations, MMS ≤= APS ≤= 2 x MMS + 1. Further, we show that this is almost the tightest bound we can get using MMS, by giving an instance where APS ≥= 2 x MMS. The upper bound on APS, combined with [17], implies a 0.1222-approximation for APS under binary XOS valuations. And the lower bound implies the non-existence of better than 0.5-APS even when agents have identical valuations, which is in sharp contrast to the guaranteed existence of exact MMS allocation when agent valuations are identical.

We consider the problem of fair allocation of indivisible items among $n$ agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting the proportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the to one paradigm, such as envy-free up to one item (EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constant fraction of his maximin share (MMS). The up to one item paradigm fails to give such a guarantee, and it is not difficult to present examples in which previous BoBW mechanisms give agents only a $\frac{1}{n}$ fraction of their MMS. Our main result is a deterministic polynomial time algorithm that computes a distribution over allocations that is ex-ante proportional, and ex-post, every allocation gives every agent at least his proportional share up to one item, and more importantly, at least half of his MMS. Moreover, this last ex-post guarantee holds even with respect to a more demanding notion of a share, introduced in this paper, that we refer to as the truncated proportional share (TPS). Our guarantees are nearly best possible, in the sense that one cannot guarantee agents more than their proportional share ex-ante, and one cannot guarantee agents more than a $\frac{n}{2n-1}$ fraction of their TPS ex-post.

We consider the problem of fairly allocating a set of indivisible items under the criteria of the maximin share guarantee. Specifically, we study approximation of maximin share allocations under hereditary set system valuations, in which each valuation function is based on the independent sets of an underlying hereditary set system. Using a lone divider approach, we show the existence of 1/2-approximate MMS allocations, improving on the 11/30 guarantee of Li and Vetta [ 37 ]. Moreover, we prove that (2/3 + ε)-approximate MMS allocations do not always exist in this model for every ε > 0, an improvement from the recent 3/4 + ε result of Li and Deng [ 36 ]. Our existence proof is constructive but does not directly yield a polynomial-time approximation algorithm. However, we show that a 2/5-approximate MMS allocation can be found in polynomial time, given valuation oracles. Finally, we show that our existence and approximation results transfer to a variety of problems within constrained fair allocation, improving on existing results in some of these settings.

We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, which is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations, a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the article is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.

The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a “fair” manner, where every agent has a valuation for each subset of goods. We assume monotone valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is “envy-freeness up to any good,” EFX, where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$, where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have (1) envy-freeness up to any good, (2) no agent values $P$ higher than her own bundle, and (3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive and leads to a pseudopolynomial time algorithm to find such an allocation. When agents have additive valuations and $|{P}|$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation that is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. (Very recently and independently, Amanatidis, Ntokos, and Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], also showed the existence of a 4/7-GMMS allocation.)

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form a hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3666 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.

We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of (3/4 + 1/12n). Most of these results are based on complicated analyses, especially those providing better than 2/3 factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of (3/4 + min(1/36, 3/(16n-4))). For small n, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.

In combinatorial optimization, a popular approach toNP-hard problems is the design of approximation algorithms. These algorithms typically run in polynomial time and are guaranteed to produce a solution which is within a known multiplicative factor of optimal. Unfortunately, the known factor is often known to be large in pathological instances. Conventional wisdom holds that, in practice, approximation algorithms will produce solutions closer to optimal than their proven guarantees. In this paper, we use the rigorous-analysis-of-heuristics framework to investigate this conventional wisdom.We analyze the performance of 3 related approximation algorithms for the uncapacitated facility location problem (from [Jain, Mahdian, Markakis, Saberi, Vazirani, 2003] and [Mahdian, Ye, Zhang, 2002]) when each is applied to an instances created by placing n points uniformly at random in the unit square. We find that, with high probability, these 3 algorithms do not find asymptotically optimal solutions, and, also with high probability, a simple plane partitioning heuristic does find an asymptotically optimal solution.

The Hamming center problem for a set S of k binary strings, each of length n, is to find a binary string fl of length n that minimizes the maximum Hamming distance between p and any string in S. Its decision version is known to be NP-complete [2]. We provide several approximation algorithms for the Hamming center problem. Our main result is a randomized ($ + E)approximation algorithm running in polynomial time if the Hamming radius of S is at least superlogarithmic in k. Furthermore, we show how to find in polynomial time a set B of O(log k) strings of length n such that for each string in S there is at least one string in B within Hamming distance not exceeding the radius of S.

We study the following all-or-nothing multicommodity flow problem in planar graphs. Input: A graph G with n vertices and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Find: A largest subset W of {1, ...., k such that for every i in W, we can send one unit of flow between si and ti. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri et al. (STOC'04) give a poly-logarithmic factor approximation algorithm and show that a natural LP-relaxation has a poly-logarithmic integrality gap. This result is in contrast with the integrality gap Ω(√n) for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an O(1)-approximation algorithm and show that the integrality gap is O(1). In particular, in polynomial time, we can find an index set W with |W| = Ω(OPT) and eight si-ti paths for each i in W such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the recent result by S'eguin-Charbonneau and Shepherd (FOCS'11) who give an O(1)-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

• NP-Completeness and APX-Hardness • We prove that the problems MinFixMask OPT and MinFixMask BOUND are NP-complete. • We establish that FixMasks + is NP-complete and APX-hard. • We show NP-completeness of DFixMasks + . • Exact, FPT, and Approximation Algorithms for DFixMasks + • We design a fixed-parameter tractable (FPT) algorithm running in O *(2 M ) time, parameterized by the number of shield configurations M . • We present a polynomial-time 1 M -approximation algorithm. • We propose an exponential k M -approximation method (for a treatment plan length k ). • We provide a tight ( 1 − 1 e ) -approximation algorithm by exploiting monotonicity and submodularity of the objective function. • Quasi-Polynomial Algorithms and Logarithmic Approximations • We develop quasi-polynomial-time algorithms for MinFixMask OPT , MinFixMask OPT + , MinFixMask BOUND , and MinFixMask BOUND + , assuming that the maximum prescribed dose d ^ max is polynomially bounded. • Under the same assumption, we achieve polynomial-time approximation algorithms with a logarithmic factor log ( d ^ max ) for MinFixMask OPT and MinFixMask OPT + . • Polynomial-Time Solvable Special Cases • We identify polynomial-time solvable cases: FixMask and FixMasks + when M = 1 . • We show that the problems MinFixMask BOUND and MinFixMask BOUND + can be solved in polynomial time when T max 1, N ;. In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2009 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)An almost O(log k)-approximation for k-connected subgraphsZeev NutovZeev Nutovpp.912 - 921Chapter DOI:https://doi.org/10.1137/1.9781611973068.99PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider two cases of the Survivable Network Design (SND) problem: given a complete graph Gn = (V, En) with costs on the edges and connectivity requirements {r(u, v):u,v ∊ V}, find a minimum cost subgraph G of Gn that contains r(u, v) internally disjoint uv-paths for all u, v ∊ V. Our main result is an -approximation algorithm for the k-Connected Subgraph problem (the case r(u, v) = k for all u, v ∊ V), for both directed and undirected graphs, where n = |V|. Our ratio is O(log k), unless k = n – o(n). Previously, the best known approximation guarantees for this problem were O(log2 k) for directed/undirected graphs [Kortsarz and Nutov STOC 2004, Fakcharoenphol and Laekhanukit STOC 2008], and O(log k) for undirected graphs with [Cheriyan, Vempala, and Vetta STOC 2002]. As in previous work, we consider the k-Connectivity Augmentation problem of increasing at minimum cost the connectivity of a given graph J from k − 1 to k; a ρ-approximation for it is used to derive an O(ρ · log k)-approximation for k-Connected Subgraph. Fakcharoenphol and Laekhanukit showed that k-Connectivity Augmentation admits an O(log v)-approximation algorithm, where v is the number of minimal “violated” sets in J. However, we may have v = Θ(n), so this gives only an O(log n)-approximation. We design a novel primal-dual algorithm that adds an edge set of cost ≤ opt to get . Combined with the algorithm of Fakcharoenphol and Laekhanukit, this gives the ratio for k-Connectivity Augmentation, which is O(1), unless k = n – o(n). Our additional result is for the (undirected) Rooted SND, where for a “root” s ∊ V, the connectivity requirements are {r(s, t) = r(t): t ∊ T ⊆ V}, and the solution graph should contain r(t) internally disjoint st-paths for all t ∊ T. For large values of k = maxt ∊ T r(t) Rooted SND is at least as hard to approximate as Directed Steiner Tree [Lando and Nutov APPROX 2008]. For Rooted SND [Chakraborty, Chuzhoy, Khanna STOC 08] gave recently a kO(k2) log4 n-approximation algorithm. Slightly later [Chuzhoy and Khanna FOCS 08] improved the ratio to O(k2 log n), and also gave an O(k8 log2 n)-approximation algorithm for the case of node-costs. Independently, we obtained a simple approximation algorithm with ratios O(k2 log n) for edge-costs, and O(k4 log2 n) for node-costs. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-680-1eISBN:978-1-61197-306-8 https://doi.org/10.1137/1.9781611973068Book Series Name:ProceedingsBook Code:PR132Book Pages:xviii + 1288