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Optimal Homologous Cycles, Total Unimodularity, and Linear Programming

Tamal K. Dey, Anil N. Hirani, Bala Krishnamoorthy

arXiv:1001.0338v3math.ATcs.CGcs.DSmath.OC

TL;DR

The paper studies how to find a minimum-weight cycle homologous to a given cycle, where integer coefficients make the optimization inherently integral. It uses total unimodularity to solve the optimization by linear programming in polynomial time, including for triangulations of compact orientable manifolds.

  • Problem

    Given a cycle in a simplicial complex, the problem is to find a minimum-weight cycle in the same homology class.

    Source: §1 · p. 2

  • Method

    The paper formulates weighted homologous-chain optimization and uses linear programming when the boundary matrix is totally unimodular, recovering an integral optimum.

    Source: §2.2 · p. 4, §3 · p. 5, §3.1 · p. 6

  • Results

    A totally unimodular boundary matrix gives polynomial-time solvability, and such matrices occur for finite simplicial complexes triangulating compact orientable manifolds.

    Source: §3.1 · p. 6, §4.1 · p. 9

  • Takeaways & Limitations

    For finite simplicial complexes triangulating compact orientable manifolds, optimal homologous chains can be computed in polynomial time regardless of simplex orientations.

    Source: §3.1 · p. 6, §4.1 · p. 9

    The paper leaves open whether the problem is NP-hard in cases with relative torsion.

    Source: §7 · p. 16

Abstract

from arXiv · show

Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer coefficients, we show the following : For a finite simplicial complex $K$ of dimension greater than $p$, the boundary matrix $[\partial_{p+1}]$ is totally unimodular if and only if $H_p(L, L_0)$ is torsion-free, for all pure subcomplexes $L_0, L$ in $K$ of dimensions $p$ and $p+1$ respectively, where $L_0$ is a subset of $L$. Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain integer solution. Thus the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under $\mathbb{Z}_2$ coefficients which, being a field, is in general easier to deal with. One consequence of our result, among others, is that one can compute in polynomial time an optimal 2-cycle in a given homology class for any finite simplicial complex embedded in $\mathbb{R}^3$. Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles.

1 Introduction

The paper asks how to find a minimum-weight cycle homologous to a given cycle, a problem with applications ranging from shape analysis to higher-dimensional spaces. It shows that total unimodularity makes this inherently integer optimization solvable by linear programming with integer solutions, and describes related optimization variants.

1 Introduction

  • For a given p-cycle, the goal is to find a minimum-weight p-cycle in its homology class, with weights inducing a weighted 1-norm.The homology coefficients may be chosen from groups such as Q, R, Z, or Z2.Source: §1 · p. 2
  • Under Z2 coefficients, optimal p-cycle computation is NP-hard for p ≥ 1, while the paper reports polynomial-time solvability for a large class using integer coefficients.The cited hardness result also implies that constant-factor approximation is NP-hard.Source: §1 · p. 2
  • Real-coefficient linear programming may yield fractional simplex coefficients, whereas integer coefficients produce integral chains but otherwise require integer programming.The authors report that experiments suggest optimal solutions may use only coefficients in {−1, 0, 1}, and say constraints can enforce that range.Source: §1 · p. 2
  • When the boundary matrix is totally unimodular, the inherently integer optimization can be solved by linear programming with integer solutions in polynomial time.The paper says it gives a topological characterization of the complexes satisfying this condition.Source: §1 · p. 2
  • The optimization formulation also supports intermediate solutions, bounding chains, and optimal chains homologous to a given chain even when it is not a cycle.The authors frame these variants as leveraging the flexibility of optimization to connect algebraic topology and optimization theory.Source: §1 · p. 2

2 Background

The background introduces integer chains, their boundaries and homology, then explains how total unimodularity connects integer optimization to linear programming. These definitions provide the algebraic and optimization concepts used in the paper.

2.1 Basic definitions from algebraic topology

  • An integer p-chain is a formal sum of oriented p-simplices; the boundary operator maps chains to lower-dimensional chains, and its matrix records the resulting coefficients.The chain group is free abelian, and reversing simplex orientation changes the sign of its coefficient.Source: §2.1 · p. 3
  • Cycles are chains with zero boundary, and homology identifies cycles that differ by a boundary; relative homology is defined using chains modulo a subcomplex.Finite-complex homology groups are finitely generated abelian groups, which may have torsion or be torsion-free.Source: §2.1 · pp. 3–4

2.2 Total unimodularity and optimization

  • Total unimodularity means every square submatrix has determinant 0, 1, or −1, and it makes the associated polyhedron integral when the right-hand side is integral.An integral polyhedron is the convex hull of its integral vectors.Source: §2.2 · p. 4
  • For a totally unimodular constraint matrix, the integer linear program can be solved in polynomial time by relaxing its integrality constraint to a linear program and finding an integral optimum.The discussion notes that an optimal vertex exists when the optimal cost is finite, enabling recovery of an integral optimal solution.Source: §2.2 · p. 4

3 Problem formulation

The section formulates optimal homologous chains as weighted ℓ1 minimization and shows that, when the boundary matrix is totally unimodular, linear programming yields an integral optimum in polynomial time. It also develops a constrained ℓ0 variant and extensions to weighted bounding chains.

3 Problem formulation

  • OHCP minimizes the weighted ℓ1 norm over chains homologous to the input, with cycles as a special case and simplex weights able to encode volume or sparsity.The cycle case requires no changes to the formulation, equations, algorithm, or theorems.Source: §3 · pp. 4–5

3.1 Optimal homologous chains and linear programming

  • The formulation converts the ℓ1 objective to a linear one with auxiliary variables, then drops integrality constraints to obtain an LP.The absolute-value terms are piecewise linear; the resulting constraint matrix includes [∂p+1].Source: §3 · p. 5, §3.1 · p. 6
  • A solution to the integer OHCP exists for every given p-chain and weight matrix.The proof bounds attention to feasible integral chains whose weighted norm is no greater than that of the input.Source: §3.1 · pp. 5–6
  • When [∂p+1] is totally unimodular, optimal homologous chains can be found in polynomial time with an integral solution.The LP relaxation has a totally unimodular equality-constraint matrix, so its optimum is integral.Source: §3.1 · pp. 6–7
  • Simulating Z2 chains with integers introduces a coefficient 2 that destroys total unimodularity, so the LP relaxation no longer guarantees integer solutions.This obstruction applies even when [∂p+1] itself is totally unimodular.Source: §3.1 · p. 7
  • Weights on both the homologous p-chain and its bounding (p+1)-chain can minimize their combined weight; for a null-homologous input, the resulting bounding chain is optimal.The (p+1)-chain has boundary equal to the difference between the optimal chain and the input.Source: §3.1 · p. 7

3.2 Minimizing the number of simplices

  • Without the added coefficient constraint on x, an optimal ℓ1 chain may contain values outside {−1, 0, 1}, and its support-minimality is not established.The paper’s example can produce edge values of 2 or −2 even when the input values lie in {−1, 0, 1}.Source: §3.2 · pp. 7–8
  • With x constrained to {−1, 0, 1}, the ℓ1 objective finds a homologous chain with the fewest nonzero entries in polynomial time when [∂p+1] is totally unimodular.The input chain must also take values in {−1, 0, 1}.Source: §3.2 · pp. 7–8

4 Manifolds

For orientable manifold triangulations, the top boundary matrix is totally unimodular regardless of simplex orientations; non-orientable examples show this property can fail alongside relative torsion.

4 Manifolds

  • The manifold results contrast the orientable case, where total unimodularity holds, with non-orientable examples where it may fail.The paper examines relative torsion in those examples in preparation for its general characterization.Source: §4 · p. 8, §4.2 · p. 9

4.1 Orientable manifolds

  • The top boundary matrix of a finite triangulation of a compact orientable manifold is totally unimodular, regardless of simplex orientations.Consequently, the optimal homologous-chain problem for p-dimensional chains is solvable in polynomial time in this setting.Source: §4.1 · p. 9

4.2 Non-orientable manifolds

  • For the Möbius strip, the boundary matrix has a submatrix with determinant −2, while the absolute H1 group is torsion-free and the relative homology has torsion.The submatrix is a relative boundary matrix, and the example is stated to have torsion in relative homology but not absolute homology.Source: §4.2 · p. 10
  • A projective-plane triangulation also contains a Möbius strip whose relative boundary matrix has determinant −2, showing the boundary matrix is not totally unimodular.The triangles and edges of the strip specify the columns and rows of the selected submatrix.Source: §4.2 · pp. 10–11

5 Simplicial complexes

For arbitrary simplicial complexes, the paper characterizes when boundary matrices are totally unimodular through relative torsion, yielding polynomial-time optimization under that condition. It also gives an embedding-based special case and, for p ≤ 1, a forbidden-subcomplex characterization.

5.1 Total unimodularity and relative torsion

  • Theorem 5.2 states that [∂p+1] is totally unimodular if and only if Hp(L, L0) is torsion-free for all pure subcomplexes L0 ⊂ L of K, of dimensions p and p + 1, respectively.Source: §5.1 · p. 11
  • A polynomial-time algorithm decides whether all such relative homology groups are torsion-free, equivalently whether some pair has torsion.The algorithm tests total unimodularity of [∂p+1]; the naive check over every pair L, L0 is exponential, and the characterization is no easier to check than total unimodularity.Source: §5.1 · p. 12

5.2 A special case

  • For a finite complex embedded in Rd+1, Hd(L, L0) is torsion-free for every such pure pair, so an optimal homologous d-chain can be computed by linear programming.This includes optimal 2-chains in complexes embedded in R3.Source: §5.2 · p. 13

5.3 Total unimodularity and Möbius complexes

  • For p ≤ 1, [∂p+1] is totally unimodular exactly when the complex contains no (p + 1)-dimensional Möbius subcomplex.For 2-complexes, this is equivalent to having no relative torsion; the absence condition remains necessary for higher p, but is not sufficient there.Source: §5.3 · pp. 13, 15
  • A 3-complex with no 3-dimensional Möbius subcomplex can still have a non-totally-unimodular [∂3], as a 7-tetrahedron example demonstrates.Its boundary matrix has a 7 × 7 submatrix with determinant −2.Source: §5.3 · p. 15

6 Experimental Results

Preliminary experiments apply the linear program to homologous chains on a torus and a solid annulus, and illustrate the resulting chains.

6 Experimental Results

  • The torus examples show the algorithm tightening initial 1-chains to shorter homologous chains.The top row of Figure 3 shows the initial and computed optimal chains.Source: §6 · p. 15, Figure 3 · p. 16
  • On a tetrahedral triangulation of a solid annulus, the computed optimal 2-chain is the inner surface, with the outer surface as the initial chain.The figure includes two cut-away views.Source: §6 · pp. 15–16, Figure 3 · p. 16
  • The experiments produced oriented curves with edge values ±1, but the theoretical guarantee is integrality rather than values restricted to {−1, 0, 1}.When that restriction is essential, the authors say to impose it using linear program (8) or its Euclidean variant.Source: §6 · p. 16

7 Discussion

The discussion considers alternative objectives, open questions about relative torsion and general weights, and cases where linear programming may still return integer solutions.

7 Discussion

  • Theorem 3.6 remains valid for minimizing the signed sum of weighted coefficients, while the formulation no longer needs x_i^+ variables.The authors also ask whether replacing diagonal weights with a general matrix W permits linear programming; they do not know.Source: §7 · p. 16
  • The authors leave open whether OHCP under Z coefficients is NP-hard when relative torsion is present.They also ask whether the broader weighted objective using a general matrix W can be solved in polynomial time for interesting complex classes.Source: §7 · p. 16
  • The known NP-hardness of OHCP under Z2 coefficients applies even to surfaces in R^3 that are torsion-free under Z.This rules out polynomial solvability for that Z2 case based solely on the absence of relative torsion under Z.Source: §7 · p. 17
  • For some 2-complexes with relative torsion, experiments found integer LP solutions for some input chains, motivating a search for which chains have this property.The authors note that relative torsion does not rule out integer vertices of the constraint polyhedron.Source: §7 · p. 17
  • The authors raise optimal homology-basis computation under Z as a related open question, following positive low-dimensional results and high-dimensional Z2 hardness.The cited positive cases concern surfaces and one-dimensional homology of simplicial complexes.Source: §7 · p. 17

Boundary matrices for non-orientable surfaces

The section presents boundary matrices for triangulations of the Möbius strip and projective plane, with rows indexed by edges and columns by triangles.

Boundary matrices for non-orientable surfaces

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