▸case-01 I have a workforce scheduling scenario where 15 technicians need to be assigned to 15 maintenance shifts, but each technician is qualified only for specific shifts and each assignment incurs a different travel cost. Can you model this as a flow graph optimization problem? Please provide the resulting network setup including the defined nodes, directed edges, identified source and sink, the reduction method used, and the chosen solving algorithm. | fail→fail | 13,091 | 11,503 | -12% | 1 | 1 | 0% | 2,554 | 2,863 | +12% | 0 | 0 | — |
▸case-02 We are trying to find the minimal capacity bottleneck to separate a network of communication nodes into two disconnected subnetworks while minimizing total severed bandwidth. Please transform this scenario into a graph optimization formulation and output the network layout detailing the graph nodes, weighted edges, terminal source/sink pair, reduction strategy, and suggested flow algorithm. | pass→pass | 16,520 | 10,770 | -35% | 1 | 1 | 0% | 2,741 | 2,545 | -7% | 0 | 0 | — |
▸case-03 A university needs to pair 20 mentors with 20 mentees based on subject compatibility without any weights or costs. Formulate this as a network flow problem and output a structured JSON response with nodes, edges, source, sink, reduction, and algorithm. | fail→fail | 9,176 | 8,482 | -8% | 1 | 1 | 0% | 1,981 | 1,920 | -3% | 0 | 0 | — |
▸case-04 We are performing binary image segmentation where pixels must be assigned to foreground or background based on pixel intensity preferences and neighborhood penalty costs. Formulate this as a network flow graph model and output a JSON response containing network nodes, edges, source, sink, algorithm, and reduction. | fail→fail | 11,201 | 8,365 | -25% | 1 | 1 | 0% | 2,475 | 2,093 | -15% | 0 | 0 | — |
▸case-05 A water distribution system has pipe capacities along with minimum flow requirements (lower bounds) for each pipe to prevent freezing. Formulate this network flow problem and output a JSON response with the graph structure, reduction, and algorithm. | fail→fail | 15,069 | 9,202 | -39% | 1 | 1 | 0% | 3,248 | 2,298 | -29% | 0 | 0 | — |
▸case-06 A logistics company needs to ship goods from 3 factories with specific production limits to 5 warehouses with specific demands over a road network where each road segment has a maximum capacity and a per-unit transit cost. Formulate this optimization as a flow network and return a structured JSON response. | fail→pass | 11,100 | 6,992 | -37% | 1 | 1 | 0% | 2,450 | 2,060 | -16% | 0 | 0 | — |
▸case-07 We need to find the maximum number of completely edge-disjoint paths between a server and a client router in a data network. Formulate this using network flow theory and output a JSON response with the nodes, edges, source, sink, algorithm, and reduction. | fail→fail | 11,543 | 7,486 | -35% | 1 | 1 | 0% | 2,369 | 1,621 | -32% | 0 | 0 | — |
▸case-08 We need to find the maximum number of path-disjoint routes between node A and node B in a telecom network such that no intermediate node is shared across paths. Model this network flow optimization problem and return the JSON network description. | fail→pass | 13,010 | 5,621 | -57% | 1 | 1 | 0% | 2,849 | 1,603 | -44% | 0 | 0 | — |
▸case-09 A sports analyst wants to determine if a team is mathematically eliminated from winning the division based on remaining games between all teams. Formulate this as a network flow problem and return a JSON structure detailing source, sink, nodes, edges, reduction, and algorithm. | fail→fail | 13,159 | 6,334 | -52% | 1 | 1 | 0% | 2,317 | 1,815 | -22% | 0 | 0 | — |
▸case-10 An airline must cover a set of scheduled flight legs using the minimum number of flight crews, where each crew can perform a sequence of compatible flights. Formulate this as a network flow reduction and output a JSON object with the network layout. | fail→fail | 14,458 | 6,591 | -54% | 1 | 1 | 0% | 3,581 | 1,765 | -51% | 0 | 0 | — |
▸case-11 A mining company must choose a set of open-pit mining blocks to extract. Certain blocks require removing overlying blocks first (dependencies), and each block has either positive revenue or negative extraction cost. Formulate this optimal closure problem as a flow graph in JSON format. | fail→fail | 11,761 | 9,063 | -23% | 1 | 1 | 0% | 2,644 | 2,004 | -24% | 0 | 0 | — |
▸case-12 A disaster relief agency has 4 distribution centers supplying emergency kits and 6 regional shelters receiving them through a road network with bottleneck capacities. Formulate this as a standard single-source single-sink flow network in JSON format. | fail→fail | 13,361 | 6,812 | -49% | 1 | 1 | 0% | 3,305 | 1,927 | -42% | 0 | 0 | — |
▸case-13 A census bureau needs to round fractional matrix entries to integers such that row and column sums are preserved within upper and lower bounds. Formulate this as a network flow problem and return a JSON network description. | fail→fail | 13,579 | 7,511 | -45% | 1 | 1 | 0% | 3,005 | 2,004 | -33% | 0 | 0 | — |
▸case-14 An airline wants to assign specific aircraft types to flight segments across weekly schedules while respecting aircraft capacity constraints and minimizing total operational cost. Formulate this flow problem and return the JSON graph description. | fail→pass | 16,143 | 11,470 | -29% | 1 | 1 | 0% | 3,507 | 2,729 | -22% | 0 | 0 | — |
▸case-15 A software firm has 50 tasks and 50 developers. Each developer can do a subset of tasks, and each pairing produces a distinct utility score. We want to maximize total utility. Formulate this as a network flow problem and return the JSON payload. | fail→fail | 13,623 | 5,942 | -56% | 1 | 1 | 0% | 3,039 | 1,688 | -44% | 0 | 0 | — |
▸case-16 An event host needs to seat N delegation groups across M dining tables such that no two members of the same delegation sit at the same table, subject to table seating capacities. Model this as a flow problem and output the JSON configuration. | fail→fail | 12,352 | 5,848 | -53% | 1 | 1 | 0% | 2,704 | 1,689 | -38% | 0 | 0 | — |
▸case-17 An evacuation planning system needs to route people out of a building through corridors where each corridor has a traversal time in addition to capacity. Formulate this dynamic flow over time problem into a network representation in JSON. | fail→pass | 13,198 | 8,824 | -33% | 1 | 1 | 0% | 2,743 | 2,189 | -20% | 0 | 0 | — |
▸case-18 For an open-pit mine block selection problem modeled as a minimum cut on a network with 100,000 nodes and 500,000 edges, choose an efficient maximum flow algorithm and return the network setup in JSON format. | fail→fail | 11,846 | 7,884 | -33% | 1 | 1 | 0% | 2,441 | 1,976 | -19% | 0 | 0 | — |
▸case-19 We need to simultaneously segment foreground objects across two related images by enforcing consistency penalties between corresponding pixels across both images. Formulate this as a flow network in JSON. | fail→fail | 15,311 | 9,119 | -40% | 1 | 1 | 0% | 3,152 | 2,374 | -25% | 0 | 0 | — |
▸case-20 Given a weighted directed graph representing city intersections with non-negative road travel times, find the single-source shortest path from Intersection A to all other intersections. Return the step-by-step distance table. | fail→fail | 21,777 | 12,986 | -40% | 1 | 1 | 0% | 2,552 | 3,288 | +29% | 0 | 0 | — |
▸case-21 We need to connect 10 electrical sub-stations with power lines such that all stations are connected in a single network with minimum total wire length, with no capacity constraints or directed flows. Provide the network design. | fail→fail | 8,083 | 12,435 | +54% | 1 | 1 | 0% | 1,530 | 2,736 | +79% | 0 | 0 | — |
▸case-22 A delivery driver must visit 12 specific customer locations exactly once and return to the depot, minimizing total travel distance. Recommend the exact or heuristic optimization algorithm for this route. | fail→fail | 12,711 | 9,310 | -27% | 1 | 1 | 0% | 1,873 | 2,187 | +17% | 0 | 0 | — |
▸case-23 Given a geographic map of 15 adjacent regions, assign color indices to each region such that no two adjacent regions share the same color using the minimum number of colors. | fail→fail | 12,222 | 15,296 | +25% | 1 | 1 | 0% | 2,402 | 3,077 | +28% | 0 | 0 | — |