{"id":1189051,"date":"2026-10-07T13:55:08","date_gmt":"2026-10-07T20:55:08","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/hardness-of-online-directed-steiner-network\/"},"modified":"2026-10-15T12:58:49","modified_gmt":"2026-10-15T19:58:49","slug":"hardness-of-online-directed-steiner-network","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/hardness-of-online-directed-steiner-network\/","title":{"rendered":"Hardness of Online Directed Steiner Network"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">In the Directed Steiner Network (DSN) problem we are given a directed graph and a set of demands <math><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>s<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><msub><mi>t<\/mi><mi>i<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>, and asked to find a cheap subgraph connecting each terminal pair. In its online version, the demands arrive online and must be served by buying edges irrevocably. DSN is a fundamental hard problem in network design, heavily studied in both the offline and the online setting. Offline, it has a superpolylogarithmic hardness of approximation. However, offline hardness says nothing about online algorithms, which are computationally unrestricted. It has been an open question whether uncertainty itself (needing to commit to a solution without knowing future demands) rules out polylogarithmic-competitive online algorithms. In this work, we show the first such unconditional, information-theoretic hardness. Namely, we give an <math><semantics><mrow><mtext>exp!bigl(Omega(sqrt{log n})bigr)<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">exp!bigl(Omega(sqrt{log n})bigr)<\/annotation><\/semantics><\/math> bound on the competitive ratio, which holds even for randomized algorithms against an oblivious adversary, and on unit-cost DAGs. Our proof uses a novel connection between online network design and algebraic coding theory. We encode requests using a hidden low-degree polynomial, whose past evaluations reveal nothing about future ones. We then use list-recovery bounds to show that an algorithm cannot make cheaply reusable decisions without knowing those future evaluations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the Directed Steiner Network (DSN) problem we are given a directed graph and a set of demands (si,ti), and asked to find a cheap subgraph connecting each terminal pair. In its online version, the demands arrive online and must be served by buying edges irrevocably. DSN is a fundamental hard problem in network design, [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"text","value":"Gary Hoppenworth","user_id":0},{"type":"text","value":"Yao-Wei Long","user_id":0},{"type":"user_nicename","value":"Sepideh Mahabadi","user_id":"40780"},{"type":"user_nicename","value":"Jakub 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