arXiv subjects

Farbod Shokrieh

Publications and source records attributed to Farbod Shokrieh.

24 records · Page 2Linked to original sources

arXiv2015

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $\mathbb{Z}$-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.

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arXiv2014

Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem

Let $Γ$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $Γ$. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an "integral" version of this result which is of independent interest. As an application, we provide a "geometric proof" of (a dual version of) Kirchhoff's celebrated Matrix-Tree Theorem. Indeed, we show that each weighted graph model $G$ for $Γ$ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus ${\rm Pic}^g(Γ)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${\rm Pic}^g(Γ)$ is the sum of the volumes of the cells in the decomposition.

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arXiv2013

Divisors on graphs, Connected flags, and Syzygies

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner bases for each higher syzygy module. In each case the given minimal Gröbner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied by Postnikov and Shapiro and by Manjunath and Sturmfels. The case of a general graph was stated as an open problem.

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arXiv2012

Chip-Firing Games, $G$-Parking Functions, and an Efficient Bijective Proof of the Matrix-Tree Theorem

Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of $G$. We outline an efficient bijective proof of this theorem, by studying a canonical finite abelian group attached to $G$ whose order is equal to the value of same matrix determinant. More specifically, we show how one can efficiently compute a bijection between the group elements and the spanning trees of the graph. The main ingredient for computing the bijection is an efficient algorithm for finding the unique $G$-parking function (reduced divisor) in a linear equivalence class defined by a chip-firing game. We also give applications, including a new and completely algebraic algorithm for generating random spanning trees. Other applications include algorithms related to chip-firing games and sandpile group law, as well as certain algorithmic problems about the Riemann-Roch theory on graphs.

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arXiv2012

Chip-firing games, potential theory on graphs, and spanning trees

We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an "efficient bijective" proof of Kirchhoff's matrix-tree theorem and a new algorithm for finding random spanning trees. The running times of our algorithms are analyzed using potential theory, and we show that the bounds thus obtained generalize and improve upon several previous results in the literature. We also extend some of these considerations to metric graphs.

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arXiv2011

The monodromy pairing and discrete logarithm on the Jacobian of finite graphs

Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.

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