logistics coordinators

Abstract: We study randomized mechanisms for scheduling indivisible jobs on $n$ unrelated machines. Each machine is controlled by a selfish agent who privately knows its processing time for each job. A mechanism assigns every job to a machine and makes payments. We require truthfulness in expectation: no agent can increase its expected payment minus its expected processing cost by misreporting, regardless of the other agents' reports. The makespan is the maximum total processing time assigned to a machine; the objective is to minimize its expectation. We prove that the optimal approximation ratio is $(n+1)/2$ for every $n\ge7$, relative to the minimum makespan of an integral assignment. The lower bound holds for every $n$ and allows unrestricted computation and arbitrary correlations among assignments, assuming finite expected payments and a guarantee uniform over all finite job counts. The matching mechanism uses correlated rounding and runs in expected polynomial bit time on rational inputs. These guarantees hold for all finite nonnegative processing times. For two machines, we characterize the optimal approximation ratio exactly as an analytic constant, approximately $1.505995705434$, and give a mechanism attaining it. For three machines the optimal ratio is $2$. For four, five, and six machines we obtain upper bounds of $3$, $105/32$, and $29/8$, respectively, compared with the general lower bound $(n+1)/2$. The proofs for $n\ge7$ do not rely on computer-assisted certificates; the two- and three-machine results use exact rational computer-assisted certificates.

Abstract: Large language models (LLMs) increasingly help users arrange appointments, plan journeys, and allocate limited resources. These tasks require models to satisfy constraints on time, cost, capacity, and order, or determine that no feasible solution exists. Bangla is the world's seventh most widely spoken language, and existing benchmarks cover many language and reasoning tasks. However, benchmarks for evaluating planning and decision-making in Bangla under explicit constraints remain lacking. We introduce BANGLAPLAN, a benchmark of 1,000 problems across ten task families grounded in everyday life and institutions in Bangladesh. It includes feasible and infeasible problems in four language registers: everyday, formal, colloquial, and Bangla--English mixed. Answers cover eight structured types, including schedules, routes, allocations, and ordered steps, in a common JSON format. We construct the benchmark through language-model-assisted drafting and independent human validation. We evaluate 11 open-weight and proprietary model configurations using exact match, separately reporting feasible-case accuracy, infeasibility detection, and format errors. We find a large performance gap: GPT-6.1 Sol achieves 98.7% accuracy and Qwen3.5-27B with thinking reaches 80.3%, while direct-answer configurations score at most 30.9%. Seven of eleven configurations perform below the always-``infeasible'' baseline. Our findings also suggest that harder problems primarily reduce plan-construction accuracy, while infeasibility detection remains more robust. BANGLAPLAN provides a reproducible basis for evaluating planning and decision-making in Bangla. BANGLAPLAN is available at: https://github.com/mrayhanulmasud/banglaplan.

Abstract: We study the problem of scheduling jobs on a serial-batch machine with the aim of minimising the total weighted late work. In a serial-batch setting, jobs within a batch are processed sequentially, and none are removed from the machine until the last job in the batch completes its processing. The processing time of a batch is the sum of the processing times of the jobs within it, and the completion time for each job in the batch is equal to the makespan of the jobs in the batch. When a new batch begins, a constant setup time is required for the machine. We show that minimising the total weighted late work in this environment is $NP$-hard even if all jobs have a common due date and unit weight. For the general problem, we present a pseudo-polynomial time dynamic programming algorithm. Additionally, we explore two special cases, i.e., one with a common due date and another with an agreeable condition among due dates, processing times and weights. For both special cases, we develop specialised pseudo-polynomial time dynamic programming algorithms. The proposed approaches are equipped with specialised acceleration techniques to enhance their computational performance. The extended experiments demonstrate that the dynamic programming algorithms outperform Gurobi in time efficiency.

Abstract: We study strategyproof mechanisms for building a pathway between two regions of a line segment separated by an obstacle. Each of the $n$ agents has a private location within its region and may use either its original route to a facility or the new pathway, whose traversal cost is a fraction $k\in[0,1)$ of its length. We seek strategyproof (SP) and group-strategyproof (GSP) mechanisms that approximately minimize maximum cost or social cost. After characterizing optimal pathways for both objectives, we establish a tight deterministic maximum-cost approximation ratio of $\frac{2}{1+k}$ and a deterministic social-cost upper bound of $\frac{n}{1+k(n-1)}$, together with complementary lower bounds. Both upper bounds are achieved by GSP mechanisms. We then study randomized mechanisms under strategyproofness in expectation. A power-proportional mechanism achieves a social-cost approximation ratio at most $5$, independent of $n$ and $k$, with a tight guarantee of $3$ for this mechanism when $k=0$. We prove randomized lower bounds of $\frac{3+2k}{2+3k}$ for maximum cost and $\max\big\{1,\frac{285}{263+385k}\big\}$ for social cost, the latter for $n\ge7$. Finally, we improve several bounds for the real-line pathway model of [Chan and Wang, AAMAS 2023]. Our deterministic maximum-cost lower bound of $2$ matches the upper bound obtainable from [Qin, Fang, and Liu, COCOA 2024]. We strengthen the deterministic social-cost lower bound from $\frac32$ to $2$ under SP and to $\max\{2,n-1\}$ under GSP. For randomized social cost, we sharpen the guarantee of Chan and Wang's proportional mechanism from $6$ to $3$ and raise their lower bound from $1.02$ to $\frac{285}{263}\approx1.08365$ for $n\ge7$.