No abstract

Motivated by certain applications from physics, biochemistry, economics, and computer science in which the objects under investigation are unknown or not directly accessible because of various limitations, we propose a trial-and-error model to examine search problems in which inputs are unknown. More specifically, we consider constraint satisfaction problems i Ci, where the constraints Ci are hidden, and the goal is to find a solution satisfying all constraints. We can adaptively propose a candidate solution (i.e., trial), and there is a verification oracle that either confirms that it is a valid solution, or returns the index i of a violated constraint (i.e., error), with the exact content of Ci still hidden.We studied the time and trial complexities of a number of natural CSPs, summarized as follows. On one hand, despite the seemingly very little information provided by the oracle, efficient algorithms do exist for Nash, Core, Stable Matching, and SAT problems, whose unknown-input versions are shown to be as hard as the corresponding known-input versions up to a factor of polynomial. The techniques employed vary considerably, including, e.g., order theory and the ellipsoid method with a strong separation oracle.On the other hand, there are problems whose complexities are substantially increased in the unknown-input model. In particular, no time-efficient algorithms exist for Graph Isomorphism and Group Isomorphism (unless PH collapses or P = NP). The proofs use quite nonstandard reductions, in which an efficient simulator is carefully designed to simulate a desirable but computationally unaffordable oracle.Our model investigates the value of input information, and our results demonstrate that the lack of input information can introduce various levels of extra difficulty. The model accommodates a wide range of combinatorial and algebraic structures, and exhibits intimate connections with (and hopefully can also serve as a useful supplement to) certain existing learning and complexity theories.

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envyfreeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions.

We study envy-free cake cutting with strategic agents, where each agent may manipulate his private information in order to receive a better allocation. We focus on piecewise constant utility functions and consider two scenarios: the general setting without any restriction on the allocations and the restricted setting where each agent has to receive a connected piece. We show that no deterministic truthful envy-free mechanism exists in the connected piece scenario, and the same impossibility result for the general setting with some additional mild assumptions on the allocations. Finally, we study a large market model where the economy is replicated and demonstrate that truth-telling converges to a Nash equilibrium.

We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality to this graphical setting. Given a simple undirected graph G, an allocation is envy-free on G if no agent envies any of her neighbor's share, and is proportional on G if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations open new research directions in developing simple and efficient algorithms that can produce fair allocations under specific graph structures. On the algorithmic frontier, we first propose a moving-knife algorithm that outputs an envy-free allocation on trees. The algorithm is significantly simpler than the discrete and bounded envy-free algorithm recently designed in [Aziz and Mackenzie, 2016a] for complete graphs. Next, we give a discrete and bounded algorithm for computing a proportional allocation on descendant graphs, a class of graphs by taking a rooted tree and connecting all its ancestor-descendant pairs.

We study the problem of fairly dividing a heterogeneous resource, commonly known as cake cutting and chore division, in the presence of strategic agents. While a number of results in this setting have been established in previous works, they rely crucially on the free disposal assumption, meaning that the mechanism is allowed to throw away part of the resource at no cost. In the present work, we remove this assumption and focus on mechanisms that always allocate the entire resource. We exhibit a truthful and envy-free mechanism for cake cutting and chore division for two agents with piecewise uniform valuations, and we complement our result by showing that such a mechanism does not exist when certain additional constraints are imposed on the mechanisms. Moreover, we provide bounds on the efficiency of mechanisms satisfying various properties, and give truthful mechanisms for multiple agents with restricted classes of valuations.

Budget feasible mechanism design is the study of procurement combinatorial auctions in which the sellers have private costs to produce items, and the buyer (auctioneer) aims to maximize her valuation function on a subset of purchased items under the budget constraint on the total payment. One of the most important questions in the field is "which valuation domains admit truthful budget feasible mechanisms with 'small' approximations to the social optimum?" Singer [Proceedings of the 51st FOCS, IEEE Press, Piscataway, NJ, 2010, pp. 765-774] showed that submodular functions have a constant approximation mechanism. Dobzinski, Papadimitriou, and Singer [Proceedings of the 12th ACM Conference on Electronic Commerce, ACM, New York, 2011, pp. 273-282] gave an O(log 2 n) approximation mechanism for subadditive functions and remarked that "A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive functions." In this paper, we give an affirmative answer to this question. To this end we relax the prior-free mechanism design framework to the Bayesian mechanism design framework (these are two standard approaches from computer science and economics, respectively). Then we convert our results in the Bayesian setting back to the prior-free framework by employing Yao's minimax principle. Along the way, we obtain the following results: (i) a polynomial time constant approximation for XOS valuations (a.k.a. fractionally subadditive valuations, a superset of submodular functions), (ii) a polynomial time O(log n/ log log n)-approximation for general subadditive valuations, (iii) a constant approximation for general subadditive functions in the Bayesian framework-we allow correlation in the distribution of sellers' costs and provide a universally truthful mechanism, (iv) the existence of a prior-free constant approximation mechanism via Yao's minimax principle. . Results (i)-(iii) were presented at STOC 12 [8]; result (iv) is new.

We study the online allocation problem under a roommate market model introduced in [Chan et al., 2016]. Consider a fixed supply of n rooms and a list of 2n applicants arriving online in random order. The problem is to assign a room to each person upon her arrival, such that after the algorithm terminates, each room is shared by exactly two people. We focus on two objectives: (1) maximizing the social welfare, which is defined as the sum of valuations that applicants have for their rooms, plus the happiness value between each pair of roommates; (2) satisfying the stability property that no small group of people would be willing to switch roommates or rooms. We first show a polynomial-time online algorithm that achieves a constant competitive ratio for social welfare maximization. We then extend it to the case where each room is assigned to c > 2 people, and achieve a competitive ratio of Ω(1/c 2 ). Finally, we show both positive and negative results in satisfying various stability conditions in this online setting.