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In this article we consider combinatorial markets with valuations only forsingletons and pairs of buy/sell-orders for swapping two items in equalquantity. We provide an algorithm that permits polynomial time market-clearingand -pricing. The results are presented in the context of our main application:the futures opening auction problem.  Futures contracts are an important tool to mitigate market risk andcounterparty credit risk. In futures markets these contracts can be traded withvarying expiration dates and underlyings. A common hedging strategy is to rollpositions forward into the next expiration date, however this strategy comeswith significant operational risk. To address this risk, exchanges started tooffer so-called futures contract combinations, which allow the traders forswapping two futures contracts with different expiration dates or for swappingtwo futures contracts with different underlyings. In theory, the price is inboth cases the difference of the two involved futures contracts. However, inparticular in the opening auctions price inefficiencies often occur due tosuboptimal clearing, leading to potential arbitrage opportunities.  We present a minimum cost flow formulation of the futures opening auctionproblem that guarantees consistent prices. The core ideas are to model ordersas arcs in a network, to enforce the equilibrium conditions with the help oftwo hierarchical objectives, and to combine these objectives into a singleweighted objective while preserving the price information of dual optimalsolutions. The resulting optimization problem can be solved in polynomial timeand computational tests establish an empirical performance suitable forproduction environments.

Pricing and clearing combinatorial markets with singleton and swap orders: Efficient algorithms for the futures opening auction problem