Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents’ attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), the well-known NP-hard matching problem Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belongs to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. This tractability result can be extended to the setting in which each agent promotes at most one “exceptional” candidate to the top of his her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists.
Settings in which agents are partitioned into different types, or derive their preferences based on a set of attributes assigned to each candidate, have been considered for the problems of sampling and counting stable matchings in instances of SM or SR (see, e.g., @cite_40 @cite_11 @cite_16 ). In @cite_3 , the authors study the problem of characterising matchings that are rationalisable as stable matchings when agents' preferences are unobserved. They focus on a restricted setting that translates into assigning each agent a type based on several attributes, and assuming that agents of the same type are identical and have identical preferences. They remark that empirical studies on marriage typically make such an assumption @cite_25 . Bounded agent types have been considered in @cite_30 @cite_14 to derive polynomial-time results for the coalition structure generation problem, an important issue in cooperative games when the goal is to partition the participants into exhaustive and disjoint coalitions in order to maximise the social welfare.
@cite_30 @cite_14 @cite_3 @cite_40 @cite_16 @cite_25 @cite_11
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In the stable roommates (SR) problem we have n agents, where each agent ranks all other agents in strict order of preference. The problem is then to match agents into pairs such that no two agents prefer each other to their matched partners, and this is a stable matching. The stable marriage (SM) problem is a special case of SR, where we have two equal sized sets of agents, men and women, where men rank only women and women rank only men. Every instance of SM admits at least one stable matching, whereas for SR as the number of agents increases the number of instances with stable matchings decreases. So, what will happen if in SM we allow men to rank men and women to rank women, i.e. we relax gender separation? Will stability abruptly disappear? And what happens in a stable roommates scenario if agents do not rank all other agents? Again, is stability uncommon? And finally, what happens if there are an odd number of agents? We present empirical evidence to answer these questions. An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within @math , for any e>0, unless P=NP. If the preference lists contain ties, we improve this result to n1−e. Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an sr instance I. We study the problem of computing maximin share allocations, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of an agent is the best she can guarantee to herself, if she is allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then is to find a partition, where each agent is guaranteed her maximin share. Such allocations do not always exist, hence we resort to approximation algorithms. Our main result is a 2 3-approximation that runs in polynomial time for any number of agents and goods. This improves upon the algorithm of Procaccia and Wang (2014), which is also a 2 3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in Procaccia and Wang (2014), exploiting the construction of carefully selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty in establishing lower bounds, we undertake a probabilistic analysis. We prove that in randomly generated instances, maximin share allocations exist with high probability. This can be seen as a justification of previously reported experimental evidence. Finally, we provide further positive results for two special cases arising from previous works. The first is the intriguing case of three agents, where we provide an improved 7 8-approximation. The second case is when all item values belong to 0, 1, 2 , where we obtain an exact algorithm. We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants. Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is ''as stable as possible''). We show that this problem is NP-hard and not approximable within n^1^-^@e, for any @e>0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within @d, for some @d>1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimize the number of agents involved in a blocking pair. Forming effective coalitions is a major research challenge in the field of multi-agent systems. Central to this endeavour is the problem of partitioning the set of agents into exhaustive and disjoint coalitions such that the social welfare is maximized. This coalition structure generation problem is extremely challenging due to the exponential number of partitions that need to be examined. Specifically, given n agents, there are O(nn) possible partitions. To date, the only algorithm that can find an optimal solution in O(3n) is the Dynamic Programming (DP) algorithm, due to However, one of the main limitations of DP is that it requires a significant amount of memory. In this paper, we devise an Improved Dynamic Programming algorithm (IDP) that is proved to perform fewer operations than DP (e.g. 38.7 of the operations given 25 agents), and is shown to use only 33.3 of the memory in the best case, and 66.6 in the worst. We consider how to combine the preferences of multiple agents despite the presence of incompleteness and incomparability in their preference orderings. An agent's preference ordering may be incomplete because, for example, there is an ongoing preference elicitation process. It may also contain incomparability as this is useful, for example, in multi-criteria scenarios. We focus on the problem of computing the possible and necessary winners, that is, those outcomes which can be or always are the most preferred for the agents. Possible and necessary winners are useful in many scenarios including preference elicitation. First we show that computing the sets of possible and necessary winners is in general a difficult problem as is providing a good approximation of such sets. Then we identify general properties of the preference aggregation function which are sufficient for such sets to be computed in polynomial time. Finally, we show how possible and necessary winners can be used to focus preference elicitation. As an important coordination and cooperation mechanism in multi-agent systems, coalition of agents exhibits some excellent characteristics and draws researchers' attention increasingly. Cooperation formation has been a very active area of research in multi-agent systems. An efficient algorithm is needed for this topic since the numbers of the possible coalitions are exponential in the number of agents. Genetic algorithm (GA) has been widely reckoned as a useful tool for obtaining high quality and optimal solutions for a broad range of combinatorial optimization problems due to its intelligent advantages of self-organization, self-adaptation and inherent parallelism. This paper proposes a GA-based algorithm for coalition structure formation which aims at achieving goals of high performance, scalability, and fast convergence rate simultaneously. A novel 2D binary chromosome encoding approach and corresponding crossover and mutation operators are presented in this paper. Two valid parental chromosomes are certain to produce a valid offspring under the operation of the crossover operator. This improves the efficiency and shortens the running time greatly. The proposed algorithm is evaluated through a robust comparison with heuristic search algorithms. We have confirmed that our new algorithm is robust, self-adaptive and very efficient by experiments. The results of the proposed algorithm are found to be satisfactory. Procuring multiple agents with different ability levels to independently solve the same task is common in labor markets, crowdsourcing environments and research and development projects due to two reasons: some agents may fail to provide a satisfactory solution, and the redundancy increases the quality of the best solution found. However, incentivizing large number of agents to compete for one task is difficult; agents need fair ex-ante guaranteed payoffs that consider their ability levels and failure rates to exert efforts. We model such domains as a cooperative game called the Max-Game, where each agent has a weight representing its ability level, and the value of an agent coalition is the maximal weight of the agents in the coalition. When agents may fail, we redefine the value of a coalition as the expected maximal weight of its surviving members. We analyze the core, the Shapley value, and the Banzhaf index as methods of payoff division. Surprisingly, the latter two, which are usually computationally hard, can be computed in polynomial time. Finally, we initiate the study of a new form of sabotage where agents may be incentivized to influence the failure probabilities of their peers, and show that no such incentive is present in a restricted case of Max-Games. We consider strategic games that are inspired by Schelling's model of residential segregation. In our model, the agents are partitioned into k types and need to select locations on an undirected graph. Agents can be either stubborn, in which case they will always choose their preferred location, or strategic, in which case they aim to maximize the fraction of agents of their own type in their neighborhood. We investigate the existence of equilibria in these games, study the complexity of finding an equilibrium outcome or an outcome with high social welfare, and also provide upper and lower bounds on the price of anarchy and stability. Some of our results extend to the setting where the preferences of the agents over their neighbors are defined by a social network rather than a partition into types. Consider a matching problem on a graph where disjoint sets of vertices are privately owned by self-interested agents. An edge between a pair of vertices indicates compatibility and allows the vertices to match. We seek a mechanism to maximize the number of matches despite self-interest, with agents that each want to maximize the number of their own vertices that match. Each agent can choose to hide some of its vertices, and then privately match the hidden vertices with any of its own vertices that go unmatched by the mechanism. A prominent application of this model is to kidney exchange, where agents correspond to hospitals and vertices to donor-patient pairs. Here hospitals may game an exchange by holding back pairs and harm social welfare. In this paper we seek to design mechanisms that are strategyproof, in the sense that agents cannot benefit from hiding vertices, and approximately maximize efficiency, i.e., produce a matching that is close in cardinality to the maximum cardinality matching. Our main result is the design and analysis of the eponymous Mix-and-Match mechanism; we show that this randomized mechanism is strategyproof and provides a 2-approximation. Lower bounds establish that the mechanism is near optimal. We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procacciafirst (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1- @math allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a @math - @math allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a @math - @math allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of ! to a @math factor in the additive setting. The main idea for our @math - @math allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time. We consider matching markets where a centralized authority must find a matching between the agents on one side of the market, and the items on the other side. Such settings occur, for example, in mail-based DVD rental services such as NetFlix or in some job markets. The objective is to find a popular matching, or a matching that is preferred by a majority of the agents to any other matching. This concept was first defined and studied by The main drawback of this concept is that popular matchings sometimes do not exist. We partially address this issue in this paper, by proving that in a probabilistic setting where preference lists are drawn at random and the number of items is more than the number of agents by a small multiplicative factor, popular matchings almost surely exist. More precisely, we prove that there is a threshold α ≈ 1.42 such that if the number of items divided by the number of agents exceeds this threshold, then a solution almost always exists. Our proof uses a characterization result by , and a number of tools from the theory of random graphs and phase transitions. We study truthful mechanisms for matching and related problems in a partial information setting, where the agents' true utilities are hidden, and the algorithm only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph of agent utilities, but the algorithm can only elicit the agents' private information in the form of a preference ordering for each agent induced by the underlying weights. Against this backdrop, we design truthful algorithms to approximate the true optimum solution with respect to the hidden weights. Our techniques yield universally truthful algorithms for a number of graph problems: a 1.76-approximation algorithm for Max-Weight Matching, 2-approximation algorithm for Max k-matching, a 6-approximation algorithm for Densest k-subgraph, and a 2-approximation algorithm for Max Traveling Salesman as long as the hidden weights constitute a metric. We also provide improved approximation algorithms for such problems when the agents are not able to lie about their preferences. Our results are the first non-trivial truthful approximation algorithms for these problems, and indicate that in many situations, we can design robust algorithms even when the agents may lie and only provide ordinal information instead of precise utilities. This paper considers some game-theoretic aspects of matching problems and procedures, of the sort which involve matching the members of one group of agents with one or more members of a second, disjoint group of agents, ail of whom have preferences over the possible resulting matches. The main focus of this paper is on determining the extent to which matching procedures can be designed which give agents the incentive to honestly reveal their preferences, and which produce stable matches.Two principal results are demonstrated. The first is that no matching procedure exists which always yields a stable outcome and gives players the incentive to reveal their true preferences, even though procedures exist which accomplish either of these goals separately. The second result is that matching procedures do exist, however, which always yield a stable outcome and which always give all the agents in one of the two disjoint sets of agents the incentive to reveal their true preferences. Coalitions and cooperation are key topics in multi-agent systems (mas). They enable agents to achieve goals that they may not have been able to achieve independently. A range of previous studies have found that many problems in coalitional games tend to be computationally intractable - that is, the computational complexity grows rapidly as a function of the number of participating agents. However, these hardness results generally require that each agent is of a different type. Here, we observe that in many mas settings, while the number of agents may grow, the number of different types of agents remains small. We formally define the notion of agent types in cooperative games. We then re-examine the computational complexity of the different coalition formation problems when assuming that the number of agent types is fixed. We show that most of the previously hard problems become polynomial when the number of agent types is fixed. We consider multiple different game formulations and representations (characteristic function with subadditive utilities, crg, and graphical representations) and several different computational problems (including stability, core-emptiness, and Shapley value). We present the first complete algorithm for the SMTI problem, the stable marriage problem with ties and incomplete lists. We do this in the form of a constraint programming encoding of the problem. With this we are able to carry out the first empirical study of the complete solution of SMTI instances. In the stable marriage problem (SM) [5] we have n men and n women. Each man ranks the women, giving himself a preference list. Similarly each woman ranks the men, giving herself a preference list. The problem is then to marry men and women such that they are stable i.e. such that there is no incentive for individuals to divorce and elope. This problem is polynomial time solvable. However, when preference lists contain ties and are incomplete (SMTI) the problem of determining if there is a stable matching of size n is then NP-complete, as is the optimisation problem of finding the largest or smallest stable matching [6, 10]. In this paper we present constraint programming solutions for the SMTI decision and optimisation problems, a problem generator for random instances of SMTI, and an empirical study of this problem. Our objective is to investigate the implications of the “replacement principle” for the fair allocation of an infinitely divisible commodity among agents with single-peaked preferences. The principle says that when one of the components of the data entering the description of the problem to be solved changes, all of the relevant agents should be affected in the same direction: they all gain or they all lose. We apply it to situations in which the preferences of one of the agents may change, under the name ofwelfare-domination under preference-replacement. We show that there is no selection from the no-envy and Pareto solution satisfying it. Then, we weaken it by limiting its application to situations in which the change is not so disruptive that it turns the economy from one in which there is “too little” of the commodity to one in which there is “too much,” or conversely. We show that there is only one selection from the no-envy and Pareto solution satisfying this property andreplication-invariance. It is the “uniform rule,” a solution that has played a central role in previous analyses of the problem.Journal of Economic LiteratureClassification Numbers: D51, D63, D70. We study two-sided matching markets with couples and show that for a natural preference domain for couples, the domain of weakly responsive preferences, stable outcomes can always be reached by means of decentralized decision making. Starting from an arbitrary matching, we construct a path of matchings obtained from satisfying' blocking coalitions that yields a stable matching. Hence, we establish a generalization of Roth and Vande Vate's (1990) result on path convergence to stability for decentralized singles markets. Furthermore, we show that when stable matchings exist, but preferences are not weakly responsive, for some initial matchings there may not exist any path obtained from satisfying' blocking coalitions that yields a stable matching. In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using protocols where determining a beneficial manipulation is hard. Especially among computational agents, it is reasonable to measure this hardness by computational complexity. Some earlier work has been done in this area, but it was assumed that the number of voters and candidates is unbounded. Such hardness results lose relevance when the number of candidates is small, because manipulation algorithms that are exponential only in the number of candidates (and only slightly so) might be available. We give such an algorithm for an individual agent to manipulate the Single Transferable Vote (STV) protocol, which has been shown hard to manipulate in the above sense. This motivates the core of this article, which derives hardness results for realistic elections where the number of candidates is a small constant (but the number of voters can be large). The main manipulation question we study is that of coalitional manipulation by weighted voters. (We show that for simpler manipulation problems, manipulation cannot be hard with few candidates.) We study both constructive manipulation (making a given candidate win) and destructive manipulation (making a given candidate not win). We characterize the exact number of candidates for which manipulation becomes hard for the plurality, Borda, STV, Copeland, maximin, veto, plurality with runoff, regular cup, and randomized cup protocols. We also show that hardness of manipulation in this setting implies hardness of manipulation by an individual in unweighted settings when there is uncertainty about the others' votes (but not vice-versa). To our knowledge, these are the first results on the hardness of manipulation when there is uncertainty about the others' votes.