Concentration for random Euclidean combinatorial optimization

Abstract

We prove concentration bounds for random Euclidean combinatorial optimization problems with $p$--costs. For bipartite matching and for the (mono- and bi-partite) traveling salesperson problem in dimension $d\ge 3$, we obtain concentration at the natural energy scale $n^{1-p/d}$ for $1\le p<d^2/2$. Our method combines a Poincaré inequality with a robust geometric mechanism providing uniform bounds on the edges of optimizers. We also formulate a conjectural $p\!\to\!q$ transfer principle for the $p$--optimal matching which, if true, would extend the concentration range to all $p\ge 1$.

Figures (6)

• Figure 1: Local geometry used in Lemma \ref{['lem:l_infty_to_loc']}.
• Figure 2: TSP: a standard $2$--opt move cutting $(x_{\tau(a)},x_{\tau(a{+}1)})$ and $(x_{\tau(b)},x_{\tau(b{+}1)})$ and reconnecting $(x_{\tau(a)},x_{\tau(b)})$, $(x_{\tau(a{+}1)},x_{\tau(b{+}1)})$.
• Figure 3: Two possible reconnections. One of the two reconnections always yields a single alternating cycle.
• Figure 4: Monopartite TSP: normalized $q$--costs on $\tau^*_p$. In both regimes, the curves remain approximately flat after normalization by $n^{1-q/d}$.
• Figure 5: Bipartite matching: normalized $q$--costs on $\sigma^*_p$. The curves remain stable under normalization by $n^{1-q/d}$.

Theorems & Definitions (15)

• Conjecture 1.1: $p\!\to\!q$ transfer for the $p$--optimal matching
• Theorem 2.1: Concentration for bipartite matching
• Lemma 2.2: $2$--opt inequality
• proof
• Lemma 2.3
• proof : Proof of Lemma \ref{['lem:l_infty_to_loc']}
• proof
• Theorem 3.1: Concentration for Euclidean TSP
• Lemma 3.2: 2-opt inequality for an optimal tour
• proof