{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:39Z","timestamp":1753893819445,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Various algebraic and geometric problems reduce to the sink-finding problem in unique sink orientations (USOs), which are orientations of the hypercube graph that have a unique sink in every subcube. A USO is called realizable if it can arise from applying one of these reductions. We study how realizability influences the query complexity of the sink-finding problem. To this end, we consider a specific subclass of USOs, the so-called Matou\u0161ek USOs. The Matou\u0161ek USOs are a family of USOs which are a translation of the LP-type problems used by Matou\u0161ek to show that the Sharir-Welzl algorithm for LP-type problems may require at least subexponential time [Matou\u0161ek, 1994]. G\u00e4rtner showed that the Random Facet algorithm for USO sink-finding requires at least subexponentially many vertex evaluation queries on Matou\u0161ek USOs, but at most quadratically many queries on realizable Matou\u0161ek USOs [G\u00e4rtner, 2002]. However, G\u00e4rtner did not fully characterize this realizable subset. In this paper, we fully characterize the realizable subset of the Matou\u0161ek-type USOs (the USOs isomorphic to a Matou\u0161ek USO) and also provide concrete realizations using instances of the P-Matrix Linear Complementarity Problem, basing our work on the so-called cyclic-P-matroids studied by Fukuda, Klaus, and Miyata. We further extend the results of Matou\u0161ek and G\u00e4rtner for the Random Facet algorithm to the query complexity of the sink-finding problem itself: we show that sink-finding is strictly easier on realizable Matou\u0161ek-type USOs than on all Matou\u0161ek-type USOs. We show that in the realizable case $O(\\log^2 n)$ queries suffice, while in general exactly n queries are needed.<\/jats:p>","DOI":"10.37236\/12773","type":"journal-article","created":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T14:03:03Z","timestamp":1744380183000},"source":"Crossref","is-referenced-by-count":0,"title":["Realizability in Matou\u0161ek Unique Sink Orientations: Characterization and Complexity Gap"],"prefix":"10.37236","volume":"32","author":[{"given":"Bernd","family":"G\u00e4rtner","sequence":"first","affiliation":[]},{"given":"Simon","family":"Weber","sequence":"additional","affiliation":[]},{"given":"Joel","family":"Widmer","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2025,4,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T14:03:03Z","timestamp":1744380183000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,11]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/12773","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,4,11]]},"article-number":"P2.7"}}