An official website of the United States government
We introduce the conormal fan of a matroid M \operatorname {M} , which is a Lagrangian analog of the Bergman fan of M \operatorname {M} . We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} . This allows us to express the h h -vector of the broken circuit complex of M \operatorname {M} in terms of the intersection theory of the conormal fan of M \operatorname {M} . We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} , when combined with the Hodge–Riemann relations for the conormal fan of M \operatorname {M} , implies Brylawski’s and Dawson’s conjectures that the h h -vectors of the broken circuit complex and the independence complex of M \operatorname {M} are log-concave sequences.
more »
« less
On Optimization and Counting of Non-Broken Bases of Matroids
Given a matroid M = (E,I), and a total ordering over the elements E, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in I with no broken circuit. The set of NBC independent sets of any matroid M define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.
more »
« less
- Award ID(s):
- 2203541
- PAR ID:
- 10561417
- Editor(s):
- Megow, Nicole; Smith, Adam
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 275
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-296-9
- Page Range / eLocation ID:
- 275-275
- Subject(s) / Keyword(s):
- Complexity Hardness Optimization Counting Random walk Local to Global Matroids Theory of computation → Random walks and Markov chains Theory of computation → Computational complexity and cryptography
- Format(s):
- Medium: X Size: 14 pages; 748501 bytes Other: application/pdf
- Size(s):
- 14 pages 748501 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.more » « less
-
The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions, and is provably not hard under "local" reductions computable in TIME(n^0.49) . The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC^0) is closely related to many of the longstanding open questions in complexity theory. All prior hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function. Some of these results were proved by exploiting a connection to a notion of time-bounded Kolmogorov complexity (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed, but this approach establishes only hardness of extremely good approximations of the form 1+o(1), and these improved hardness results are not yet known to hold for MCSP. In particular, it is known that MKTP is hard for the complexity class DET under nonuniform AC^0 m-reductions, implying MKTP is not in AC^0[p] for any prime p. It was still open if similar circuit lower bounds hold for MCSP. One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond 1 + o(1) to omega(1), as KT-complexity and circuit size are polynomially-related. In this paper, we show that this approach cannot succeed. More speci cally, we prove that PARITY does not reduce to the problem of computing superlinear approximations to KT-complexity or circuit size via AC^0-Turing reductions that make O(1) queries. This is signi cant, since approximating any set in P/poly AC^0-reduces to just one query of a much worse approximation of circuit size or KT-complexity. For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in earlier work (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC^0 reductions. It may also be a step toward con rming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform AC0 m-reductions.more » « less
-
We prove that the maximum likelihood degree of a matroid M equals its beta invariant β(M). For an element e of M that is neither a loop nor a coloop, this is defined to be the degree of the intersection of the Bergman fan of (M,e) and the inverted Bergman fan of N = (M/e)^⊥. Equivalently, for a generic vector w ∈ R^E−e, this is the number of ways to find weights (0, x) on M and y on N with x + y = w such that on each circuit of M (resp. N), the minimum x-weight (resp. y-weight) occurs at least twice.more » « less
-
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has the same cardinality. In this paper, we extend these notions and define $$\mathcal{M}$$-separated collections for any oriented matroid $$\mathcal{M}$$. We show that maximal by size $$\mathcal{M}$$-separated collections are in bijection with fine zonotopal tilings (if $$\mathcal{M}$$ is a realizable oriented matroid), or with one-element liftings of $$\mathcal{M}$$ in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid $$\mathcal{M}$$ is pure if $$\mathcal{M}$$-separated collections form a pure simplicial complex, i.e., any maximal by inclusion $$\mathcal{M}$$-separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank $$3$$, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank $$3$$ is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an $$n$$-gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank $$3$$, graphical, uniform).more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.40
