[Paper Review] The rank of a divisor on a finite graph: geometry and computation

Madhusudan Manjunath|arXiv (Cornell University)|Nov 30, 2011

Limits and Structures in Graph Theory11 references3 citations

TL;DR

This paper presents a polynomial-time algorithm for computing the rank of a divisor on a finite graph with a fixed number of vertices, leveraging geometric insights from the geometry of numbers and a novel graph invariant called the critical automorphism group. The key contribution is an algorithm with time complexity $O(2^{nar{\log} n})\text{poly}(\text{size}(G))$, where $n+1$ is the number of vertices, and a new proof that testing non-negative rank lies in $NP \cap co-NP$, leading to the introduction of the critical automorphism group as a structural invariant.

ABSTRACT

We study the problem of computing the rank of a divisor on a finite graph, a quantity that arises in the Riemann-Roch theory on a finite graph developed by Baker and Norine (Advances of Mathematics, 215(2): 766-788, 2007). Our work consists of two parts: the first part is an algorithm whose running time is polynomial for a multigraph with a fixed number of vertices. More precisely, our algorithm has running time O(2^{n \log n})poly(size(G)), where n+1 is the number of vertices of the graph G. The second part consists of a new proof of the fact that testing if rank of a divisor is non-negative or not is in the complexity class NP intersection co-NP and motivated by this proof and its generalisations, we construct a new graph invariant that we call the critical automorphism group of the graph.

Motivation & Objective

• To develop a polynomial-time algorithm for computing the rank of a divisor on a finite multigraph with a fixed number of vertices.
• To provide a geometric interpretation of the rank of a divisor using lattice geometry and polyhedral structures.
• To establish that testing whether the rank of a divisor is non-negative lies in $NP \cap co-NP$ via a duality theorem.
• To introduce and characterize a new graph invariant, the critical automorphism group, based on duality and symmetry in divisor configurations.
• To explore structural properties of the critical automorphism group, particularly in complete graphs, and provide a constructive characterization.

Proposed method

• The algorithm uses the Riemann-Roch theorem and a rank formula from Baker and Norine’s work as foundational tools.
• It applies techniques from the geometry of numbers, particularly the enumeration of lattice points and projections onto hyperplanes.
• The method involves computing orthogonal projections of extremal points of the Laplacian polytope onto the hyperplane $H_0 = \{(1,\dots,1)^\perp\}$ to identify critical configurations.
• A key component is the use of a duality theorem that characterizes divisors with non-negative rank, enabling the $NP \cap co-NP$ proof.
• The critical automorphism group is constructed as the group of automorphisms preserving the set of critical configurations under the distance function $d_S$.
• For complete graphs, the method uses permutation symmetries and lattice basis transformations to derive a general form of critical automorphisms involving integer shifts and permutations.

Experimental results

Research questions

• RQ1 Can the rank of a divisor on a finite graph be computed in polynomial time when the number of vertices is fixed?
• RQ2 What is the geometric structure underlying the rank of a divisor, and how can it be leveraged algorithmically?
• RQ3 Is the problem of testing whether $r(D) \geq 0$ in $NP \cap co-NP$, and what structural duality underlies this complexity class membership?
• RQ4 What new invariants can be derived from the duality structure of divisor configurations, and how do they relate to graph symmetries?
• RQ5 Can the critical automorphism group be fully characterized, especially for symmetric graphs like complete graphs?

Key findings

• The paper presents an algorithm with time complexity $O(2^{n\log n})\text{poly}(\text{size}(G))$ for computing the rank of a divisor on a multigraph with $n+1$ vertices, which is polynomial when $n$ is fixed.
• The rank computation is based on a geometric interpretation of divisor configurations as projections of extremal points of the Laplacian polytope onto the hyperplane $H_0$.
• A duality theorem is proven that characterizes divisors with non-negative rank, which establishes the membership of the decision problem in $NP \cap co-NP$.
• The critical automorphism group of a graph is introduced as the group of automorphisms preserving the set of critical configurations, and it is shown to be isomorphic to the automorphism group of the critical polytope.
• For the complete graph $K_{n+1}$, the critical automorphism group contains all permutations of vertices and more general maps involving integer shifts $H \cdot q_i$ where $H$ divides the height $n$.
• The paper constructs a general form of critical automorphisms for $K_{n+1}$: maps sending $b_i$ to $b_{\pi(i)} + H \cdot q_i$, where $q_i$ lies in the span of previous basis vectors and $H$ divides $n$.