{"id":1188549,"date":"2025-11-23T00:00:00","date_gmt":"2025-11-23T08:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=1188549"},"modified":"2026-10-02T14:58:26","modified_gmt":"2026-10-02T21:58:26","slug":"weighted-chairman-assignment-and-flow-time-scheduling","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/weighted-chairman-assignment-and-flow-time-scheduling\/","title":{"rendered":"Weighted Chairman Assignment and Flow-Time Scheduling"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">Given positive integers <math><mrow><mi>m<\/mi><mo>,<\/mo><mi>n<\/mi><\/mrow><\/math>, a fractional assignment <math><mrow><mi>x<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mrow><mrow><mi>m<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><\/mrow><\/mrow><\/math> and weights <math><mrow><mi>d<\/mi><mo>\u2208<\/mo><msubsup><mi>\u211d<\/mi><mrow><mrow><mo>&<\/mo><mi>g<\/mi><mi>t<\/mi><mo>;<\/mo><mn>0<\/mn><\/mrow><\/mrow><mi>n<\/mi><\/msubsup><\/mrow><\/math>, we show that there exists an assignment <math><semantics><mrow><mtext>y in \\{0,1\\}^{m times n}<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">y in \\{0,1\\}^{m times n}<\/annotation><\/semantics><\/math> so that for every <math><mrow><mi>i<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mi>m<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/math> and <math><mrow><mi>t<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mi>n<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><\/math>, <math display=\"block\"><semantics><mrow><mtext>Big|sum_{j in [t]} d_j (x_{ij} &#8211; y_{ij}) Big|<max_{j in [n]} d_j.<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">Big|sum_{j in [t]} d_j (x_{ij} &#8211; y_{ij}) Big|<max_{j in [n]} d_j.<\/annotation><\/semantics><\/math> This generalizes a result of Tijdeman (1973) on the unweighted version, known as the chairman assignment problem. This also confirms a special case of the single-source unsplittable flow conjecture with arc-wise lower and upper bounds due to Morell and Skutella (IPCO 2020). As an application, we consider a scheduling problem where jobs have release times and machines have closing times, and a job can only be scheduled on a machine if it is released before the machine closes. We give a <math><mn>3<\/mn><\/math>-approximation algorithm for maximum flow-time minimization.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given positive integers m,n, a fractional assignment x\u2208[0,1]m\u00d7n and weights d\u2208\u211d>0n, we show that there exists an assignment y in \\{0,1\\}^{m times n}y in \\{0,1\\}^{m times n} so that for every i\u2208[m] and t\u2208[n], Big|sum_{j in [t]} d_j (x_{ij} &#8211; y_{ij}) Big|<\/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":"Siyue Liu","user_id":0},{"type":"user_nicename","value":"Victor Reis","user_id":"43536"}],"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Information Technology Convergence and 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