Hidden Hamiltonian Cycle Recovery via Linear Programming

Abstract

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 → ∞ provided that α n − log n → ∞, where α n −2 log √ dP n dQ n is the Rényi divergence of order 1 2 . This condition is information-theoretically optimal in the sense that, under mild distributional assumptions, α n ≥ (1 + o(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 multi-graphs, 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.

Reconstruction of line-embeddings of graphons

“…We then take many small random subgraphs of G (2) α and attempt to order them using a deterministic algorithm. Since G (2) α is "almost" Robinsonian, most of these subgraphs will be exactly Robinsonian. 3.…”

“…where d = min{0.5, 2d}. This shows a sketch of w(0, •) and w (2) (0, •) for a graphon of type (1.4) that satisfies Assumption 1.5. Note that w (2) violates the diagonally increasing condition, but w…”

“…Remark 1.6. Denote by w (2) the usual "square" of a graphon (see Definition 2.1). We note that the left hand side of Equation (1.7) equals inf s∈[0,d] w (2) (0, s), and the right hand side equals w (2)…”

“…Very similar random graphs have been extensively studied (see e.g. [2,5,14,36]). Special cases of this noisy seriation problem (with points sometimes embedded on a circle instead of an interval) appear in many areas.…”

“…However, generally such results only apply to a narrow class of random graphs, and the restriction of the problem to this specific class allows for the use of highly specialized methods. A few examples beyond archaeology include genomics [2,35], ranking [17], data visualization [32], and ecology and sociology [24].…”