Two-Dimensional Partial Cubes

Abstract

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

Labeled sample compression schemes for complexes of oriented matroids

“…To achieve improper labeled compression schemes our results provide a new approach, extending the one of [16,17] presented in the introduction. Is it possible to extend a given set system or a partial cube to a COM without increasing the VC-dimension too much?…”

“…Apart from the above, COMs have spiked research and appear as the next natural class to attack in different areas such as combinatorial semigroup theory [43], algebraic combinatorics in relation to the Varchenko determinant [35], with respect to neural codes [36], poset cones [22], as well as sweeping sequences [48]. In particular, relations to COMs have already been established within sample compression, see [9,16,17]. A central feature of COMs, is that they can be studied via their tope graphs.…”

“…For instance, using the result of Moran and Warmuth [44] that ample set systems of VC-dimension d admit labeled sample compression schemes of size d, one can try to extend a given set system to an ample set system without increasing the VC-dimension too much and then apply their result. Two recent works of the authors go into this direction: In [16] it is shown that partial cubes of VC-dimension 2 can be extended to ample set systems of VC-dimension 2. Furthermore, in [17] it is shown that OMs and complexes of uniform oriented matroids can be extended to ample set systems without increasing the VC-dimension.…”

“…On the other hand, there exist partial cubes of VC-dimension 3 that cannot be extended to ample set systems of VC-dimension 3, see [17], as well as set systems of VC-dimension 2, that cannot be extended to partial cubes of VC-dimension 2, see [16]. In [17] it is conjectured that every COM of VC-dimension d can be extended to an ample set system of VC-dimension d. This would yield improper labeled sample compression schemes for COMs of size d.…”

Corners and simpliciality in oriented matroids and partial cubes

“…, n}, r(G) coincides with the VC-dimension of , see [49]. The latter correspondence has led to some recent interest in partial cubes of bounded rank, see [12]. See Figure 2 for an antipodal partial cube of rank 3.…”