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].

Multi-Type Resource Allocation with Partial Preferences

We propose multi-type probabilistic serial (MPS) and multi-type random priority (MRP) as extensions of the well-known PS and RP mechanisms to the multi-type resource allocation problems (MTRAs) with partial preferences. In our setting, there are multiple types of divisible items, and a group of agents who have partial order preferences over bundles consisting of one item of each type. We show that for the unrestricted domain of partial order preferences, no mechanism satisfies both sd-efficiency and sd-envy-freeness. Notwithstanding this impossibility result, our main message is positive: When agents' preferences are represented by acyclic CP-nets, MPS satisfies sd-efficiency, sd-envy-freeness, ordinal fairness, and upper invariance, while MRP satisfies ex-post-efficiency, sd-strategyproofness, and upper invariance, recovering the properties of PS and RP. Besides, we propose a hybrid mechanism, multi-type general dictatorship (MGD), combining the ideas of MPS and MRP, which satisfies sd-efficiency, equal treatment of equals and decomposability under the unrestricted domain of partial order preferences.

Average-case Analysis of the Assignment Problem with Independent Preferences

The fundamental assignment problem is in search of welfare maximization mechanisms to allocate items to agents when the private preferences over indivisible items are provided by self-interested agents. The mainstream mechanism \textit{Random Priority} is asymptotically the best mechanism for this purpose, when comparing its welfare to the optimal social welfare using the canonical \textit{worst-case approximation ratio}. Surprisingly, the efficiency loss indicated by the worst-case ratio does not have a constant bound \cite{FFZ:14}.Recently, \cite{DBLP:conf/mfcs/DengG017} shows that when the agents' preferences are drawn from a uniform distribution, its \textit{average-case approximation ratio} is upper bounded by 3.718. They left it as an open question of whether a constant ratio holds for general scenarios. In this paper, we offer an affirmative answer to this question by showing that the ratio is bounded by $1/\mu$ when the preference values are independent and identically distributed random variables, where $\mu$ is the expectation of the value distribution. This upper bound improves the results in \cite{DBLP:conf/mfcs/DengG017} for the Uniform distribution as well. Moreover, under mild conditions, the ratio has a \textit{constant} bound for any independent random values. En route to these results, we develop powerful tools to show the insights that for most valuation inputs, the efficiency loss is small.