We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Economics Of Discrimination

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Two-person Cooperative Games

We study competition in matching markets with random heterogeneous preferences and an unequal number of agents on either side. First, we show that even the slightest imbalance yields an essentially unique stable matching. Second, we give a tight description of stable outcomes, showing that matching markets are extremely competitive. Each agent on the short side of the market is matched with one of his top choices, and each agent on the long side either is unmatched or does almost no better than being matched with a random partner. Our results suggest that any matching market is likely to have a small core, explaining why small cores are empirically ubiquitous.

/pdf/unbalanced-random-matching-markets-the-stark-effect-of-4lif3dp78c.pdf

Unbalanced Random Matching Markets: The Stark Effect of Competition

http://web.stanford.edu/~iashlagi/papers/NeadChains1.pdf

Nonsimultaneous Chains and Dominos in Kidney-Paired Donation-Revisited

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.

Mix and match

As of May 2014 there were more than 100,000 patients on the waiting list for a kidney transplant from a deceased donor. Although the preferred treatment is a kidney transplant, every year there are fewer donors than new patients, so the wait for a transplant continues to grow. To address this shortage, kidney paired donation (KPD) programs allow patients with living but biologically incompatible donors to exchange donors through cycles or chains initiated by altruistic (nondirected) donors, thereby increasing the supply of kidneys in the system. In many KPD programs a centralized algorithm determines which exchanges will take place to maximize the total number of transplants performed. This optimization problem has proven challenging both in theory, because it is NP-hard, and in practice, because the algorithms previously used were unable to optimally search over all long chains. We give two new algorithms that use integer programming to optimally solve this problem, one of which is inspired by the techniques used to solve the traveling salesman problem. These algorithms provide the tools needed to find optimal solutions in practice.

/pdf/finding-long-chains-in-kidney-exchange-using-the-traveling-2gshjx0b4f.pdf

Finding long chains in kidney exchange using the traveling salesman problem

As multi-hospital kidney exchange has grown, the set of players has grown from patients and surgeons to include hospitals. Hospitals can enroll only their hard-tomatch patient-donor pairs, while conducting easily-arranged exchanges internally. This behavior has already been observed. We show that the cost of making it individually rational for hospitals to participate fully is low in almost every large exchange pool (although the worst-case cost is very high), while the cost of failing to guarantee individual rationality could be high, in lost transplants. We identify a mechanism that achieves high eciency while giving hospitals incentives to reveal all patient-donor pairs.

/pdf/free-riding-and-participation-in-large-scale-multi-hospital-1oeui5b4nz.pdf

Itai Ashlagi

Papers

Unbalanced Random Matching Markets: The Stark Effect of Competition

Nonsimultaneous Chains and Dominos in Kidney-Paired Donation-Revisited

Mix and match

Finding long chains in kidney exchange using the traveling salesman problem

Free riding and participation in large scale, multi‐hospital kidney exchange