Binary Steiner Trees: Structural Results and an Exact Solution Approach - Liers, Frauke; Martin, Alexander; Pape, Susanne (Discrete Optimization, 2016)
In this paper we study the Steiner tree problem with degree constraints. Motivated by an application in computational biology we focus on binary Steiner trees in which all node degrees are required to be at most three. It is shown that finding a binary Steiner tree is <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S157252861630038X&_mathId=si109.gif&_user=111111111&_pii=S157252861630038X&_rdoc=1&_issn=15725286&md5=85a1ec838ea644fb1a19f4cb3b694ac4" title="Click to view the MathML source">NP-complete for arbitrary graphs. We relate the problem to Steiner trees without degree constraints as well as degree-constrained spanning trees by proving approximation ratios. Further, we give integer programming formulations for this problem on undirected and directed graphs and study the associated polytopes for both cases. Some classes of facets are introduced. Based on this study a branch-&-cut approach is developed and evaluated on biological instances coming from the reconstruction of phylogenetic trees. We are able to solve nearly all instances up to 200 nodes to optimality within a limited amount of time. This shows the effectiveness of our approach.
Keywords: Steiner tree; Degree constraint; Steiner ratio; Integer programming; Polytope; Branch-&-cut

Pricing and clearing combinatorial markets with singleton and swap orders: Efficient algorithms for the futures opening auction problem - Müller, Johannes; Pokutta, Sebastian; Martin, Alexander; Pape, Susanne; Peter, Andrea; Winter, Thomas (Mathematical Methods of Operations Research, 2017)
In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and -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 and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the price information of dual optimal solutions. The resulting optimization problem can be solved in polynomial time and computational tests establish an empirical performance suitable for production environments.
Keywords: Equilibrium problems; Hierarchical objectives; Linear programming; Network flows; Combinatorial auctions; Futures exchanges