Semi-Streaming Estimation of the Held--Karp Bound and Graph-Metric TSP Cost

The Held--Karp bound is a linear-programming lower bound for the metric traveling salesperson problem (TSP). We approximate this bound in the multi-pass insertion-only streaming model for metrics induced by connected positive-weight graphs. For an $n$-vertex, $m$-edge graph and $0<\eps\le1/2$, the algorithm returns a $(1+\eps)$ upper approximation with high probability, using $\wtO(\eps^{-4})$ passes, $\wtO(n\eps^{-2})$ words, and $\wtO(m\eps^{-4})$ arithmetic operations per pass. We replace the changing edge-weight vector by a short history of multiplicative-weights (MWU) updates. After initialization, updates repeat until a column is removed. We store each repeated update once, with its repetition count. The record length depends on the number of columns, not updates; the argument applies to any fixed nonnegative matrix. For Held--Karp, tree-path queries recover edge weights from these records. Separate search and update thresholds allow fresh sparsifiers at successive search stages. Controlled rounding gives a bounded-precision implementation. The estimate yields a $(3/2+\eps)$ graph-metric TSP value estimate and a $(7/5+\eps)$ estimate when the input graph is unweighted. Conversely, in weighted graph streams, every one-pass $(1+η)$ upper approximation to Held--Karp, for fixed $0\leη<1$, requires quadratic space. The algorithms estimate values; they do not construct tours within these bounds.