Distance labeling schemes are schemes that label the vertices of a graph with short labels in suc... more Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural classes of graphs admitting distance and/or routing labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance and routing labeling schemes with labels of O(log 3 n) bits.
This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample... more This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture -one of the central open questions of learning theory. We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension.
The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of... more The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer p ≥ 2, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power G p of G. This extends some characterizations of graphs with connected medians (case p = 1) provided by . The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have G 2 -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.
HAL (Le Centre pour la Communication Scientifique Directe), 2019
This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample... more This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture -one of the central open questions of learning theory. We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension. OMs, COMs, and AMPs (viewed as isometric subgraphs of hypercubes) give rise to set families for which the VC-dimension has a particular significance: the VC-dimension of an AMP is the largest dimension of a cube of its cube complex, the VC-dimension of an OM is its rank, and the VC-dimension of a COM is the largest VC-dimension of its faces. Littlestone and Warmuth introduced the sample compression technique for deriving generalization bounds in machine learning. Floyd and Warmuth [19] asked whether any set family of VC-dimension d has a sample compression scheme of size O(d). This question, known as the sample compression conjecture, remains one of the oldest open problems in machine learning. It was shown in [28] that labeled compression schemes of size O(2 d ) exist. Moran and Warmuth [27] designed labeled compression schemes of size d for ample set systems. Chalopin et al. [9] designed (stronger) unlabeled compression schemes of size d for maximum families and characterized such schemes for ample set systems. In view of the above, it was noticed in [30] and [27] that the sample compression conjecture would be solved if any set family of VC-dimension d can be completed to an ample (or maximum) set system of VC-dimension O(d) or covered by O(2 d ) ample set systems of VC-dimension O(d). This opens a perspective that apart from its application to sample compression, is interesting in its own right: ample completions of structured set families. This is extending a given set system to an ample system by adding sets. A natural problem is ample completions of set families defined by partial cubes (i.e., isometric subgraphs of hypercubes). In [14], we prove that any partial cube of VC-dimension 2 admits an ample completion of VC-dimension 2. Moreover, we give a set family of VC-dimension 2 which has no ample completion of the same VC-dimension. In the present paper, we give an example of a partial cube of VC-dimension 3 which cannot be completed to an ample set system of VC-dimension 3. Hence, in higher dimension we cannot complete all partial cubes without increasing the VC-dimension. In the light of the above perspective, one may ask if there exists a constant c such that every partial cube of VC-dimension d admits an ample completion of VC-dimension ≤ cd? Even stronger, we wonder if partial cubes of VC-dimension d admit an ample completion of VC-dimension d + c. Note that no such additive constant c exists for general set families . In [14], we perform the ample completion of a partial cube of VC-dimension 2 in two steps. First, we show that they can be completed to tope graphs of COMs and next we complete the resulting graphs into ample ones, in both cases, without increasing the VC-dimension. In the present article, we are interested in the completion of this intermediate class of partial cubes. For COMs, we are inclined to believe that the following stronger result holds: Conjecture 1. The tope graph of any COM of VC-dimension d has an ample completion of VC-dimension d. COMs of rank 2 have the nice property that their faces are uniform OMs. This is not longer true in higher dimensions: COMs whose faces are uniform OMs constitute a proper subclass of COMs (that we call CUOMs). In this paper, we prove that Conjecture 1 holds for all tope graphs of OMs and CUOMs. This proves that set families arising from topes of OMs and CUOMs satisfy the sample compression conjecture. In Fig. , we present an example of the tope graph of a CUOM of VC-dimension 3, which we further use as a running example. 2.1. VC-dimension. Let S be a family of subsets of an m-element set U . A subset X of U is shattered by S if for all Y ⊆ X there exists S ∈ S such that S ∩ X = Y . The Vapnik-Chervonenkis dimension [35] (the VC-dimension for short) VC-dim(S) of S is the cardinality of the largest subset of U shattered by S. Any set family S ⊆ 2 U can be viewed as a subset of vertices of the m-dimensional hypercube Q m = Q(U ). Denote by G(S) the subgraph of Q m
HAL (Le Centre pour la Communication Scientifique Directe), Feb 6, 2017
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do ... more We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex Q - 3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -a property naturally generalizing the notion of median graphs.
We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension d admi... more We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension d admit a proper labeled sample compression scheme of size d. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture -one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.
k-Approximate distance labeling schemes are schemes that label the vertices of a graph with short... more k-Approximate distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the k-approximation of the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. One of the important problems is finding natural classes of graphs admitting exact or approximate distance labeling schemes with labels of polylogarithmic size. In this paper, we describe a 4-approximate distance labeling scheme for the class of K4-free bridged graphs. This scheme uses labels of poly-logarithmic length O(log n 3 ) allowing a constant decoding time. Given the labels of two vertices u and v, the decoding function returns a value between the exact distance dG(u, v) and its quadruple 4dG(u, v).
The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly... more The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups. The fixed point theorem is proved by using a graph-theoretical tool-dismantlability. In particular we show that 1-skeleta of weakly systolic complexes, i.e. weakly bridged graphs, are dismantlable. On the way we show numerous characterizations of weakly bridged graphs and weakly systolic complexes.
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do ... more We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex Q - 3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -a property naturally generalizing the notion of median graphs.
In this paper, we investigate the graphs in which all balls are convex and the groups acting on t... more In this paper, we investigate the graphs in which all balls are convex and the groups acting on them geometrically (which we call CB-graphs and CB-groups).These graphs have been introduced andcharacterized by Soltan and Chepoi (1983) and Farber and Jamison (1987). CB-graphs and CB-groups generalize systolic (alias bridged) and weakly systolic graphs and groups,which play an important role in geometric group theory. We present metric and local-to-global characterizations of CB-graphs. Namely, we characterize CB-graphs G as graphs whose triangle-pentagonal complexes are simply connected and balls of radius at most 3 are convex. Similarly to systolic and weakly systolic graphs, we provea dismantlability result for CB-graphs G: we show that their squares G^2 are dismantlable. This implies that the Rips complexes of CB-graphsare contractible. Finally, we adapt and extend the approach of Januszkiewicz and Swiatkowski (2006) for systolic groups and of Chalopin et al. (2020) for Helly group...
The median function is a location/consensus function that maps any profile π (a finite multiset o... more The median function is a location/consensus function that maps any profile π (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from π. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function. In this paper, we show that modular graphs with G 2 -connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T 2 )) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even ∆-matroids) are ABCT-graphs and that benzenoid graphs are ABCT 2 -graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs.
25 pages, 5 figuresInternational audienceWe investigate the structure of two-dimensional partial ... more 25 pages, 5 figuresInternational audienceWe 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). Furthermore we point out relations to tope graphs of COMs of low rank and r...
This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample... more This paper considers completions of tope graphs of COMs (complexes of oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that tope graphs of OMs and CUOMs satisfy the sample compression conjecture -one of the central open questions of learning theory. We conjecture that the tope graph of every COM can be completed to an ample partial cube without increasing the VC-dimension. OMs, COMs, and AMPs (viewed as isometric subgraphs of hypercubes) give rise to set families for which the VC-dimension has a particular significance: the VC-dimension of an AMP is the largest dimension of a cube of its cube complex, the VC-dimension of an OM is its rank, and the VC-dimension of a COM is the largest VC-dimension of its faces. Littlestone and Warmuth introduced the sample compression technique for deriving generalization bounds in machine learning. Floyd and Warmuth [19] asked whether any set family of VC-dimension d has a sample compression scheme of size O(d). This question, known as the sample compression conjecture, remains one of the oldest open problems in machine learning. It was shown in [28] that labeled compression schemes of size O(2 d ) exist. Moran and Warmuth [27] designed labeled compression schemes of size d for ample set systems. Chalopin et al. [9] designed (stronger) unlabeled compression schemes of size d for maximum families and characterized such schemes for ample set systems. In view of the above, it was noticed in [30] and [27] that the sample compression conjecture would be solved if any set family of VC-dimension d can be completed to an ample (or maximum) set system of VC-dimension O(d) or covered by O(2 d ) ample set systems of VC-dimension O(d). This opens a perspective that apart from its application to sample compression, is interesting in its own right: ample completions of structured set families. This is extending a given set system to an ample system by adding sets. A natural problem is ample completions of set families defined by partial cubes (i.e., isometric subgraphs of hypercubes). In [14], we prove that any partial cube of VC-dimension 2 admits an ample completion of VC-dimension 2. Moreover, we give a set family of VC-dimension 2 which has no ample completion of the same VC-dimension. In the present paper, we give an example of a partial cube of VC-dimension 3 which cannot be completed to an ample set system of VC-dimension 3. Hence, in higher dimension we cannot complete all partial cubes without increasing the VC-dimension. In the light of the above perspective, one may ask if there exists a constant c such that every partial cube of VC-dimension d admits an ample completion of VC-dimension ≤ cd? Even stronger, we wonder if partial cubes of VC-dimension d admit an ample completion of VC-dimension d + c. Note that no such additive constant c exists for general set families . In [14], we perform the ample completion of a partial cube of VC-dimension 2 in two steps. First, we show that they can be completed to tope graphs of COMs and next we complete the resulting graphs into ample ones, in both cases, without increasing the VC-dimension. In the present article, we are interested in the completion of this intermediate class of partial cubes. For COMs, we are inclined to believe that the following stronger result holds: Conjecture 1. The tope graph of any COM of VC-dimension d has an ample completion of VC-dimension d. COMs of rank 2 have the nice property that their faces are uniform OMs. This is not longer true in higher dimensions: COMs whose faces are uniform OMs constitute a proper subclass of COMs (that we call CUOMs). In this paper, we prove that Conjecture 1 holds for all tope graphs of OMs and CUOMs. This proves that set families arising from topes of OMs and CUOMs satisfy the sample compression conjecture. In Fig. , we present an example of the tope graph of a CUOM of VC-dimension 3, which we further use as a running example. 2.1. VC-dimension. Let S be a family of subsets of an m-element set U . A subset X of U is shattered by S if for all Y ⊆ X there exists S ∈ S such that S ∩ X = Y . The Vapnik-Chervonenkis dimension [35] (the VC-dimension for short) VC-dim(S) of S is the cardinality of the largest subset of U shattered by S. Any set family S ⊆ 2 U can be viewed as a subset of vertices of the m-dimensional hypercube Q m = Q(U ). Denote by G(S) the subgraph of Q m
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017
In case of δ = 0 (trees) and = r = 0, (subtrees of a tree) we recover the result of Alon ( ) abou... more In case of δ = 0 (trees) and = r = 0, (subtrees of a tree) we recover the result of Alon ( ) about the transversal and packing numbers of a set-family in which each set is a union of at most κ subtrees of a tree.
In this note, we present first linear time algorithms for computing the center and the diameter o... more In this note, we present first linear time algorithms for computing the center and the diameter of several classes of face regular plane graphs: triangulations with inner vertices of degree 6, quadrangulations with inner vertices of degree 4 and the subgraphs of the regular hexagonal grid bounded by a simple circuit of this grid.
Transactions of the American Mathematical Society, 2014
The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly... more The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups. The fixed point theorem is proved by using a graph-theoretical tool — dismantlability. In particular we show that 1 1 –skeleta of weakly systolic complexes, i.e., weakly bridged graphs, are dismantlable. On the way we show numerous characterizations of weakly bridged graphs and weakly systolic complexes.
In this article, we characterize the graphs G that are the retracts of Cartesian products of chor... more In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K 2,3 , the 4-wheel minus one spoke W - 4 , and the k-wheels W k (for k ≥ 4) as induced subgraphs. We also show that these graphs G are exactly the cage-amalgamation graphs as introduced by Bre šar and Tepeh Horvat (Cage-amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071-1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of G by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed
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Papers by Victor Chepoi