arXiv subjects

Victor Chepoi

Publications and source records attributed to Victor Chepoi.

At least 55 records · Page 3Linked to original sources

arXiv2018

COMs: Complexes of Oriented Matroids

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of "conditional oriented matroid" (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs.

math.CO↗

arXiv2017

On density of subgraphs of halved cubes

Let $\mathcal S$ be a family of subsets of a set $X$ of cardinality $m$ and $\text{VC-dim}(\mathcal S)$ be the Vapnik-Chervonenkis dimension of $\mathcal S$. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if $G_1(\mathcal S)=(V,E)$ is the subgraph of the hypercube $Q_m$ induced by $\mathcal S$ (called the 1-inclusion graph of $\mathcal S$), then $\frac{|E|}{|V|}\le \text{VC-dim}({\mathcal S})$. Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if $\mathcal S$ is an arbitrary set family and $G_{1,2}(\mathcal S)=(V,E)$ is the 1,2-inclusion graph of $\mathcal S$ (i.e., the subgraph of the square $Q^2_m$ of the hypercube $Q_m$ induced by $\mathcal S$), then $\frac{|E|}{|V|}\le \binom{d}{2}$, where $d:=\text{cVC-dim}^*(\mathcal S)$ is the clique-VC-dimension of $\mathcal S$ (which we introduce in this paper). The 1,2-inclusion graphs are exactly the subgraphs of halved cubes and comprise subgraphs of Johnson graphs as a subclass.

cs.DM↗

arXiv2017

Weakly modular graphs and nonpositive curvature

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various `nonpositive curvature' and `local-to-global' properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises in several seemingly-unrelated fields of mathematics: Metric graph theory, Geometric group theory, Incidence geometries and buildings, Theoretical computer science and combinatorial optimization. We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With $l_1$-embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even $\triangle$-matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs.

math.MG↗

arXiv2017

Packing and covering with balls on Busemann surfaces

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $δ$, the minimum number of closed balls of radius $δ$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $δ$ centered at points of $S$: $ν(S) \le ρ(S) \le 19ν(S)$, where $ρ(S)$ and $ν(S)$ are the covering and the packing numbers of $S$ by $δ$-balls.

math.MG↗

arXiv2016

Core congestion is inherent in hyperbolic networks

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset $X$ of vertices of a $δ$-hyperbolic graph $G$ there exists a vertex $m$ of $G$ such that the disk $D(m,4 δ)$ of radius $4 δ$ centered at $m$ intercepts at least one half of the total flow between all pairs of vertices of $X$, where the flow between two vertices $x,y\in X$ is carried by geodesic (or quasi-geodesic) $(x,y)$-paths. A set $S$ intercepts the flow between two nodes $x$ and $y$ if $S$ intersect every shortest path between $x$ and $y$. Differently from what was conjectured by Jonckheere et al., we show that $m$ is not (and cannot be) the center of mass of $X$ but is a node close to the median of $X$ in the so-called injective hull of $X$. In case of non-uniform traffic between nodes of $X$ (in this case, the unit flow exists only between certain pairs of nodes of $X$ defined by a commodity graph $R$), we prove a primal-dual result showing that for any $ρ>5δ$ the size of a $ρ$-multi-core (i.e., the number of disks of radius $ρ$) intercepting all pairs of $R$ is upper bounded by the maximum number of pairwise $(ρ-3δ)$-apart pairs of $R$.

cs.DS↗

arXiv2015

Distance-preserving subgraphs of Johnson graphs

We give a characterization of distance--preserving subgraphs of Johnson graphs, i.e. of graphs which are isometrically embeddable into Johnson graphs (the Johnson graph $J(m,Λ)$ has the subsets of cardinality $m$ of a set $Λ$ as the vertex--set and two such sets $A,B$ are adjacent iff $|A\triangle B|=2$). Our characterization is similar to the characterization of D. Ž. Djoković (J. Combin. Th. Ser. B 14 (1973), 263--267) of distance--preserving subgraphs of hypercubes and provides an explicit description of the wallspace (split system) defining the embedding.

math.CO↗

arXiv2015

On two conjectures of Maurer concerning basis graphs of matroids

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14 (1973), 216-240). We also establish Conjecture 1 from the same paper about the redundancy of the conditions in the characterization of basis graphs. We indicate positive-curvature-like aspects of the local properties of the studied complexes. We characterize similarly the corresponding 2-dimensional complexes of even $Δ$-matroids.

math.CO↗

arXiv2014

Cop and robber game and hyperbolicity

In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s' with s' 0, this establishes a new - game-theoretical - characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between δand s is linear for any s'<s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity δof a graph on n vertices in O(n^2) time when the graph is given by its distance-matrix.

math.CO↗

arXiv2013

Isometric embedding of Busemann surfaces into $L_1$

In this paper, we prove that any non-positively curved 2-dimensional surface (alias, Busemann surface) is isometrically embeddable into $L_1$. As a corollary, we obtain that all planar graphs which are 1-skeletons of planar non-positively curved complexes with regular Euclidean polygons as cells are $L_1$-embeddable with distortion at most $2+π/2<4$. Our results significantly improve and simplify the results of the recent paper {\it A. Sidiropoulos, Non-positive curvature, and the planar embedding conjecture, FOCS 2013.}}

cs.CG↗

arXiv2013

Bucolic Complexes

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.

math.CO↗

arXiv2012

Dismantlability of weakly systolic complexes and applications

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.

math.GR↗

arXiv2012

Shortest path problem in rectangular complexes of global nonpositive curvature

CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x,y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees \geq4). Namely, we show that for a CAT(0) rectangular complex K with n vertices, one can construct a data structure D of size $O(n^2)$ so that, given any two points x,y in K, the shortest path γ(x,y) between x and y can be computed in O(d(p,q)) time, where p and q are vertices of two faces of K containing the points x and y, respectively, such that γ(x,y) is contained in K(I(p,q)) and d(p,q) is the distance between p and q in the underlying graph of K. If K is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p,q)logΔ) time, where Δ is the maximal degree of a vertex of G(K). Finally, if K is a squaregraph, then one can construct a data structure D of size O(nlogn) and answer two-point shortest path queries in O(d(p,q)) time.

cs.CG↗

arXiv2011

Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) are axis-parallel and oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this paper, we present a polynomial factor 2 approximation algorithm for the bidirected minimum Manhattan network problem.

cs.CG↗

arXiv2011

Nice labeling problem for event structures: a counterexample

In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ admits a labeling with at most $f(n)$ labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barthélemy and Constantin in 1993.

math.CO↗

arXiv2010

Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu, Indyk, and Sidiropoulos (2007) and Bǎdoiu, Demaine, Hajiaghayi, Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of metric relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an outerplanar metric.

math.MG↗

arXiv2010

Minimum Manhattan network problem in normed planes with polygonal balls: a factor 2.5 approximation algorithm

Let B be a centrally symmetric convex polygon of R^2 and || p - q || be the distance between two points p,q in R^2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R^2, a B-Manhattan network on T is a network N(T) = (V,E) with the property that its edges are parallel to the directions of B and for every pair of terminals t_i and t_j, the network N(T) contains a shortest B-path between them, i.e., a path of length || t_i - t_j ||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum possible length. The problem of finding minimum B-Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance || p - q || is the l_1 or the l_infty-distance between p and q) and it has been shown recently by Chin, Guo, and Sun (SoCG'09) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4 ,3 , and 2) for minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for minimum B-Manhattan network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (APPROX'05) and subsequently used in other factor 2 approximation algorithms for minimum Manhattan problem.

cs.CG↗

arXiv2010

Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.

math.CO↗

arXiv2010

Cop and robber games when the robber can hide and ride

In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G. The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski, Winkler (1983) and Quilliot (1983) characterized the cop-win graphs as graphs admitting a dismantling scheme. In this paper, we characterize in a similar way the class CW(s,s') of cop-win graphs in the game in which the cop and the robber move at different speeds s' and s, s'<= s. We also establish some connections between cop-win graphs for this game with s' =3, coincide and we provide a structural characterization of these graphs. We also investigate several dismantling schemes necessary or sufficient for the cop-win graphs in the game in which the robber is visible only every k moves for a fixed integer k>1. We characterize the graphs which are cop-win for any value of k. Finally, we consider the game where the cop wins if he is at distance at most 1 from the robber and we characterize via a specific dismantling scheme the bipartite graphs where a single cop wins in this game.

cs.DM↗