🤖 AI Summary
This study addresses the lack of precise characterization regarding the efficiency gap between local and global optima in task allocation. Focusing on the matchable semi-matching problem, this work proposes an analytical framework combining explicit rational potential functions on comparison graphs with extremal construction techniques to derive closed-form expressions for the worst-case ratios of local search under varying move step sizes and load constraints. The primary contribution is the first proof that worst-case behavior can be realized even under simple tree-structured constraints, thereby surpassing traditional approximation bounds. Specifically, it establishes exact worst-case ratios of 3/2 for single-step moves and approximately 1.2945 for double-step moves in the absence of load limits, while providing rigorous theoretical guarantees under degree-constrained settings.
📝 Abstract
An assignment of tasks to servers can resist every small improvement and still make tasks wait longer than necessary. We determine exactly how inefficient such an assignment can be when each task requires one unit of service and the eligibility constraints permit all tasks to use distinct servers. For every move size $r$ and maximum current server load $K$, we give a closed formula for the worst ratio between locally optimal and globally optimal total completion time. Local optimality here allows every feasible reassignment changing at most $r$ tasks. Every finite-cap bound is attained on a tree where each task has at most two eligible servers. Thus the worst behavior already occurs under simple eligibility constraints. At load cap two, the exact ratio is $1+1/(r+2)$, attained on a path with $r+2$ tasks. Without a load cap, the worst-case supremum is $3/2$ for single-task moves and approximately $1.294503159$ for two-task moves; its excess above one is $1/(r+2)+O(2^{-r}/r)$ as $r$ grows. The proof uses an explicit rational potential on a comparison graph and matching extremal constructions. These results give sharp guarantees for bounded-size local search on matchable semi-matchings, including exact guarantees under degree bounds.