[WIP] Add partial orientations and rank-related optimizations

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<issue_title>Partial Orientations and Rank Related Optimizations [Spencer Backman] for Heuristics</issue_title>
<issue_description>Reference: https://arxiv.org/pdf/1401.3309</issue_description>

<agent_instructions>RIEMANN-ROCH THEORY FOR GRAPH ORIENTATIONS
SPENCER BACKMAN
Abstract. We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan’s cycle–cocycle reversal system for
partial orientations. An oriented version of Dhar’s burning algorithm is introduced
and employed in the study of acyclicity for partial orientations. We then show that
the Baker–Norine rank of a partially orientable divisor is one less than the minimum
number of directed paths which need to be reversed in the generalized cycle–cocycle
reversal system to produce an acyclic partial orientation. These results are applied in
providing new proofs of the Riemann–Roch theorem for graphs as well as Luo’s topological characterization of rank-determining sets. We prove that the max-flow mincut theorem is equivalent to the Euler characteristic description of orientable divisors
and extend this characterization to the setting of partial orientations. Furthermore,
we demonstrate that P icg−1
(G) is canonically isomorphic as a P ic0
(G)-torsor to the
equivalence classes of full orientations in the cycle–cocycle reversal system acted on
by directed path reversals. Efficient algorithms for computing break divisors and
constructing partial orientations are presented.
Contents

1. Introduction 1
2. Notation and terminology 5
3. Generalized cycle, cocycle, and cycle–cocycle reversal systems 7
4. Oriented Dhar’s algorithm 10
5. Directed path reversals and the Riemann–Roch formula 18
6. Luo’s theorem on rank-determining sets 24
7. Max-flow min-cut and divisor theory 27
8. Acknowledgements 33
   References 34
9. Introduction
   Baker and Norine [5] introduced a combinatorial Riemann–Roch theorem for graphs
   analogous to the classical statement for Riemann surfaces. For proving the theorem,
   they employed chip-firing, a deceptively simple game on graphs with connections to
   various areas of mathematics. Given a graph G, we define a configuration of chips D
   on G as a function from the vertices to the integers. A vertex v fires by sending a chip
   Key words and phrases. Chip-firing, partial graph orientation, cycle-cocycle reversal system, Dhar’s
   algorithm, Riemann-Roch theorem for graphs, rank-determining set, max-flow min-cut theorem.
   1
   arXiv:1401.3309v4 [math.CO] 14 Apr 2017
   2 SPENCER BACKMAN
   to each of its neighbors, losing its degree number of chips in the process. If we take D
   to be a vector, firing the vertex vi precisely corresponds to subtracting the ith column
   of the Laplacian matrix from D. In this way we may view chip-firing as a combinatorial
   language for describing the integer translates of the lattice generated by the columns
   of the Laplacian matrix, e.g. [2, 4].
   Reinterpreting chip configurations as divisors, we say that two divisors are linearly
   equivalent if one can be obtained from the other by a sequence of chip-firing moves,
   and a divisor is effective if each vertex has a nonnegative number of chips. Baker and
   Norine define the rank of a divisor, denoted r(D), to be one less than the minimum
   number of chips which need to be removed so that D is no longer equivalent to an
   effective divisor. Defining the canonical divisor K to have values K(v) = deg(v) − 2,
   the genus of G to be g = |E(G)| − |V (G)| + 1, and the degree deg(D) of a divisor D to
   be the total number of chips in D, they prove the Riemann–Roch formula for graphs:
   Theorem 1.1 (Baker–Norine [5]).
   r(D) − r(K − D) = deg(D) − g + 1.
   Baker and Norine’s proof depends in a crucial way on the theory of q-reduced divisors,
   known elsewhere as G-parking functions or superstable configurations [13, 35]. A divisor
   D is said to be q-reduced if (i) D(v) ≥ 0 for all v 6= q, and (ii) for any non-empty subset
   A ⊂ V (G) \ {q}, firing the set A causes some vertex in A to go into debt, i.e., to have
   a negative number of chips. They show that every divisor D is linearly equivalent to
   a unique q-reduced divisor D0
   , and r(D) ≥ 0 if and only if D0
   is effective. We note
   that q-reduced divisors are dual, in a precise sense, to the recurrent configurations (also
   known as q-critical configurations), which play a prominent role in the abelian sandpile
   model [5, Lemma 5.6]
   There is a second story, which runs parallel to that of chip-firing, describing certain
   constrained reorientations of graphs first introduced by Mosesian [31] in the context
   of Hasse diagrams for posets. Given an acyclic orientation of a graph O and a sink
   vertex q, we can perform a sink reversal, reorienting all of the edges incident to q.
   This operation is directly connected to the theory of chip-firing: we can associate to
   O a divisor DO with entries DO(v) = indegO(v) − 1, and performing a sink reversal
   at vi we obtain the orientation O0 with associated divisor DO0 given by the firing of
   vi
   . Mosesian observed that, provided an acyclic orientation O and a vertex q, there
   exists a unique acyclic orientation O0 having q as the unique sink, which is obtained
   from O by sink reversals. The divisors associated to (the reverse of) these q-rooted
   acyclic orientations are the maximal noneffective q-reduced divisors. This connection
   between acyclic orientations and chip-firing dates back at least to Bj¨orner, Lov´asz, and
   Shor’s seminal paper on the topic [10], and has been utilized in recent proofs of the
   Riemann–Roch formula [1, 11, 30].
   Gioan [21] generalized this setup to arbitrary (not necessarily acyclic) orientations
   by introducing the cocycle reversal, wherein all of the edges in a consistently oriented
   cut can be reversed, and a cycle reversal, in which the edges in a consistently oriented
   cycle can be reversed. Using these two operations, he defined the cycle–cocycle reversal
   system as the collection of full orientations modulo cycle and cocycle reversals, and
   RIEMANN-ROCH THEORY FOR GRAPH ORIENTATIONS 3
   proved that the number of equivalence classes in this system is equal to the number
   of spanning trees of the underlying graph. He also showed that each orientation is
   equivalent in the cocycle reversal system to a unique q-connected orientation. These
   are the orientations in which every vertex is reachable from q by a directed path.
   Gioan and Las Vergnas [22], and Bernardi [9], combined these results, presenting an
   explicit bijections between the q-connected orientations with a standardized choice of
   the orientation’s cyclic part and spanning trees of a graph. Bernardi’s bijection is
   determined by a choice of “combinatorial map”, which is essentially a combinatorial
   embedding of a graph in a surface. Recently, An, Baker, Kuperberg, and Shokrieh [3]
   showed that the divisors associated to the q-connected orientations are precisely the
   break divisors of Mikhalkin and Zharkov [30] offset by a chip at q. They then applied
   this observation to give a tropical “volume proof” of Kirchoff’s matrix-tree theorem via
   a canonical polyhedral decomposition of Picg
   (G), the collection of divisors of degree g
   modulo linear equivalence.
   A limitation of the orientation-based perspective is that the divisor associated to an
   orientation will always have degree g − 1. In this work, we introduce a generalization
   of the cycle–cocycle reversal system for investigating partial orientations, thus allowing
   for a discussion of divisors with degrees less than g − 1. The generalized cycle–cocycle
   reversal system is defined by the introduction of edge pivots, whereby an edge (u, v)
   oriented towards v is unoriented and an unoriented edge (w, v) is oriented towards v
   (see Fig. 1). Note that edge pivots, as with cycle reversals, leave the divisor associated
   to a partial orientation unchanged. We demonstrate that this additional operation is
   dynamic enough to allow for a characterization of linear equivalence.
   Theorem 1.2. Two partial orientations are equivalent in the generalized cycle–cocycle
   reversal system if and only if their associated divisors are linearly equivalent.
   Moreover, we use edge pivots and cocycle reversals to show that a divisor with degree
   at most g − 1 is linearly equivalent to a divisor associated to a partial orientation or it
   is linearly equivalent a divisor dominated by a divisor associated to an acyclic partial
   orientation. These results allow us to reduce the study of linear equivalence of divisors
   of degree at most g − 1 on graphs to the study of partial orientations.
   Dhar’s burning algorithm is one of the key tools in the study of chip-firing. Originally
   discovered in the context of the abelian sandpile model, Dhar’s algorithm provides a
   quadratic-time test for determining whether a given configuration is q-reduced. There
   are variants of Dhar’s algorithm which produce bijections between q-reduced divisors
   and spanning trees, some of which respect important tree statistics such as external
   activity [12] or tree inversion number [32]. In the work of Baker and Norine, this algorithm was implicitly employed in the proof of their core lemma RR1, which states that
   if a divisor has negative rank then it is dominated by a divisor of degree g − 1 which
   also has negative rank. We present an oriented version of Dhar’s algorithm whose iterated application provides a method for determining whether a partial orientation is
   equivalent in the generalized cocycle reversal system to an acyclic partial orientation
   or a sourceless partial orientation. We combine these results to obtain the following
   theorem.
   4 SPENCER BACKMAN
   (a)
   (b)
   (c)
   Figure 1. A partial orientation with (a) an edge pivot, (b) a cocycle
   reversal, and (c) a cycle reversal.
   Theorem 1.3. Let D be a divisor with deg(D) ≤ g − 1, then
   (i) r(D) = −1 if and only if D ∼ D0 ≤ DO with O an acyclic partial orientation.
   (ii) r(D) ≥ 0 if and only if D ∼ DO with O a sourceless partial orientation.
   This implies that for understanding whether the rank of a divisor is negative or
   nonnegative, it suffices to investigate partial orientations. We introduce q-connected
   partial orientations and use them to prove the following explicit description of ranks of
   divisors associated</agent_instructions>

Comments on the Issue (you are @copilot in this section)

@DhyeyMavani2003 Need to talk to Prof. Pflueger about what optimizations this can lead to!