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Journal on Data Semantics VI
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In this paper we study the problem of online market clearing where there is one commodity in the market, being bought and sold by multiple buyers and sellers who submit buy and sell bids that arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing surplus, we present a (randomized) online algorithm with competitive ratio ln(pmax - pmin) + 1, when bids are in a range [pmin,pmax], which we show is the best possible. A simpler algorithm has ratio twice this, and call can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no money-losing trades are allowed. Interestingly, we show that if the online algorithm is allowed to subsidize matches --- match money-losing pairs if it has already collected enough money from previous pairs to pay for them --- then it can be 1-competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, the ability to subsidize is at least as valuable as knowing the future. For the problems of maximizing buy or sell volume, we present algorithms that achieve a competitive ratio of 2(ln(pmax/pmin) + 1) without subsidization. We also present algorithms that achieve a competitive ratio of ln(pmax/pmin) + 1 with subsidization with respect to the optimal offline algorithm that cannot use subsidies. This is the best possible competitive ratio for the setting. We present all of our results as corollaries of theorems on online matching in an incomplete interval graph.