Hidden Hamiltonian Cycle Recovery via Linear Programming

• 成果类型: Article
• 署名作者: Bagaria, Vivek; Ding, Jian; Tse, David; Wu, Yihong; Xu, Jiaming
• 署名单位: Stanford University; University of Pennsylvania; Yale University; Duke University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2019.1886
• 发表日期: 2020
• 页码: 53-70
• 关键词: b-matchings; seriation; matrices; clique

摘要

We introduce the problem of hidden Hamiltonian cycle recovery, where there is an unknown Hamiltonian cycle in an n-vertex complete graph that needs to be inferred from noisy edge measurements. The measurements are independent and distributed according to P-n for edges in the cycle and Q(n) otherwise. This formulation is motivated by a problem in genome assembly, where the goal is to order a set of contigs (genome subsequences) according to their positions on the genome using long-range linking measurements between the contigs. Computing the maximum likelihood estimate in this model reduces to a traveling salesman problem (TSP). Despite the NP-hardness of TSP, we show that a simple linear programming (LP) relaxation-namely, the fractional 2-factor (F2F) LP-recovers the hidden Hamiltonian cycle with high probability as n -> infinity provided that alpha(n) - logn -> infinity, where alpha(n) (sic) -2 log integral root dP(n)dQ(n) is the Renyi divergence of order I. This condition is information-theoretically optimal in the sense that, under mild distributional assumptions, alpha(n) >= (1 + 0(1))log n is necessary for any algorithm to succeed regardless of the computational cost. Departing from the usual proof techniques based on dual witness construction, the analysis relies on the combinatorial characterization (in particular, the half-integrality) of the extreme points of the F2F polytope. Represented as bicolored multigraphs, these extreme points are further decomposed into simpler blossom-type structures for the large deviation analysis and counting arguments. Evaluation of the algorithm on real data shows improvements over existing approaches.