arXiv subjects

Jean Cardinal

Publications and source records attributed to Jean Cardinal.

At least 19 recordsLinked to original sources

arXiv2026

A simple algorithm for computing Hamilton paths on independent set polytopes

The independent set polytope, or stable set polytope, of a graph $G$ is the 0/1-polytope defined by the convex hull of the characteristic vectors of all independent sets of $G$. We present a simple algorithm for computing a Hamilton path on the independent set polytope of a given $n$-vertex graph $G$ with amortized delay $\mathcal{O}(n)$. The independent sets are listed such that two consecutive sets differ either in removing a vertex, or adding a vertex and removing its neighbors from the independent set, i.e., the symmetric difference between two consecutive independent sets induces a star in $G$. As applications of this result, we obtain an algorithm to compute a Hamilton path on the matching polytope of an $m$-edge graph $G$ with worst-case delay $\mathcal{O}(m)$, which lists all matchings of $G$ in such a way that the symmetric difference between two consecutive matchings is a path on at most three edges. Furthermore, we obtain an algorithm to compute a Hamilton path on the chain polytope and order polytope of an $n$-element poset $P$ with amortized delay $\mathcal{O}(n)$, which lists all antichains of $P$ or all ideals of $P$, respectively, by star exchanges. Our algorithms are derived from the generic framework proposed by Merino and M\"utze (FOCS'23+SICOMP'24) for computing Hamilton paths on arbitrary 0/1-polytopes, which uses a linear optimization procedure as a black box. Our algorithms bypass solving the computationally intractable maximum weight independent set problem by a simple and purely combinatorial greedy rule.

cs.DM↗

arXiv2026

Biclique decompositions from Welzl orders

A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in ``well-structured'' instances.

math.CO↗

arXiv2026

Finding Shortest Reconfiguration Sequences on Independent Set Polytopes

We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the $1$-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in $\mathbb{R}^n$ described by $O(n)$ linear inequalities is hard to approximate within a factor of $(1-\varepsilon)\ln n$ for any constant $\epsilon >0$, unless $P=NP$. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

cs.DS↗

arXiv2026

Implicit representations via the polynomial method

Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on $n$ vertices in this family, we can assign a label consisting of $O(n^{1-2/(d+1) + \varepsilon})$ bits to each vertex (where $\varepsilon > 0$ can be made arbitrarily small and the constant of proportionality depends on $\varepsilon$ and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of $O(n^{1/3 + \varepsilon})$ bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of $O(n^{1-1/d}\log n)$ for $d$-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-$d$ polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size $O(\log n)$. We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size $O(\log^3 n)$.

cs.CG↗

arXiv2025

Combinatorial generation via permutation languages. VII. Supersolvable hyperplane arrangements

For an arrangement $\mathcal{H}$ of hyperplanes in $\mathbb{R}^n$ through the origin, a region is a connected subset of $\mathbb{R}^n\setminus\mathcal{H}$. The graph of regions $G(\mathcal{H})$ has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from $\mathcal{H}$. We aim to compute a Hamiltonian path or cycle in the graph $G(\mathcal{H})$, i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if $\mathcal{H}$ is a supersolvable arrangement, then the graph of regions $G(\mathcal{H})$ has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions $P(\mathcal{H},R_0)$, obtained by orienting the graph $G(\mathcal{H})$ away from a particular base region $R_0$. Our second main result is that if $\mathcal{H}$ is supersolvable and $R_0$ is a canonical base region, then for any lattice congruence $\equiv$ on $P(\mathcal{H},R_0)=:L$, the cover graph of the quotient lattice $L/\equiv$ has a Hamiltonian path. [...]

math.CO↗

arXiv2025

Compact Representation of Semilinear and Terrain-like Graphs

We consider the existence and construction of \textit{biclique covers} of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The \textit{size} of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size $O(n\polylog n)$. This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz's problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (\textit{Forum Math. Sigma}, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size $O(n\log^3 n)$. This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (\textit{Discrete Comput. Geom.} 1994). Finally, we prove that there exists families of unit disk graphs on $n$ vertices that do not admit biclique coverings of size $o(n^{4/3})$, showing that we are unlikely to improve on Szemer\'edi-Trotter type incidence bounds for higher-degree semialgebraic graphs.

math.CO↗

arXiv2025

Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

cs.CG↗

arXiv2024

Facet-Hamiltonicity

We consider facet-Hamiltonian cycles of polytopes, defined as cycles in their skeleton such that every facet is visited exactly once. These cycles can be understood as optimal watchman routes that guard the facets of a polytope. We consider the existence of such cycles for a variety of polytopes, the facets of which have a natural combinatorial interpretation. In particular, we prove the following results: - Every permutahedron has a facet-Hamiltonian cycle. These cycles consist of circular sequences of permutations of $n$ elements, where two successive permutations differ by a single adjacent transposition, and such that every subset of $[n]$ appears as a prefix in a contiguous subsequence. With these cycles we associate what we call rhombic strips which encode interleaved Gray codes of the Boolean lattice, one Gray code for each rank. These rhombic strips correspond to simple Venn diagrams. - Every generalized associahedron has a facet-Hamiltonian cycle. This generalizes the so-called rainbow cycles of Felsner, Kleist, M\"utze, and Sering (SIDMA 2020) to associahedra of any finite type. We relate the constructions to the Conway-Coxeter friezes and the bipartite belts of finite type cluster algebras. - Graph associahedra of wheels, fans, and complete split graphs have facet-Hamiltonian cycles. For associahedra of complete bipartite graphs and caterpillars, we construct facet-Hamiltonian paths. The construction involves new insights on the combinatorics of graph tubings. We also consider the computational complexity of deciding whether a given polytope has a facet-Hamiltonian cycle and show that the problem is NP-complete, even when restricted to simple 3-dimensional polytopes.

math.CO↗

arXiv2024

The Existential Theory of the Reals as a Complexity Class: A Compendium

We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model of real computation, and the other following work by Mn\"{e}v and Shor on the universality of realization spaces of oriented matroids. Over the years the number of problems for which $\exists \mathbb{R}$ rather than NP has turned out to be the proper way of measuring their complexity has grown, particularly in the fields of computational geometry, graph drawing, game theory, and some areas in logic and algebra. $\exists \mathbb{R}$ has also started appearing in the context of machine learning, Markov decision processes, and probabilistic reasoning. We have aimed at collecting a comprehensive compendium of problems complete and hard for $\exists \mathbb{R}$, as well as a long list of open problems. The compendium is presented in the third part of our survey; a tour through the compendium and the areas it touches on makes up the second part. The first part introduces the reader to the existential theory of the reals as a complexity class, discussing its history, motivation and prospects as well as some technical aspects.

cs.CC↗

arXiv2024

The Complexity of Intersection Graphs of Lines in Space and Circle Orders

We consider the complexity of the recognition problem for two families of combinatorial structures. A graph $G=(V,E)$ is said to be an intersection graph of lines in space if every $v\in V$ can be mapped to a straight line $\ell (v)$ in $\mathbb{R}^3$ so that $vw$ is an edge in $E$ if and only if $\ell(v)$ and $\ell(w)$ intersect. A partially ordered set $(X,\prec)$ is said to be a circle order, or a 2-space-time order, if every $x\in X$ can be mapped to a closed circular disk $C(x)$ so that $y\prec x$ if and only if $C(y)$ is contained in $C(x)$. We prove that the recognition problems for intersection graphs of lines and circle orders are both $\exists\mathbb{R}$-complete, hence polynomial-time equivalent to deciding whether a system of polynomial equalities and inequalities has a solution over the reals. The second result addresses an open problem posed by Brightwell and Luczak.

cs.CG↗

arXiv2024

Rectangulotopes

Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles. They are defined as quotientopes of natural lattice congruences on the weak Bruhat order on permutations in $\mathfrak{S}_n$, and their skeleta are flip graphs on rectangulations. We give simple vertex and facet descriptions of these polytopes, in particular elementary formulas for computing the coordinates of the vertex corresponding to each rectangulation, in the spirit of J.-L. Loday's realization of the associahedron.

math.CO↗

arXiv2024

The expansion of half-integral polytopes

The expansion of a polytope is an important parameter for the analysis of the random walks on its graph. A conjecture of Mihai and Vazirani states that all $0/1$-polytopes have expansion at least 1. We show that the generalization to half-integral polytopes does not hold by constructing $d$-dimensional half-integral polytopes whose expansion decreases exponentially fast with $d$. We also prove that the expansion of half-integral zonotopes is uniformly bounded away from $0$. As an intermediate result, we show that half-integral zonotopes are always graphical.

math.CO↗

arXiv2024

Combinatorics of rectangulations: Old and new bijections

A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.

math.CO↗

arXiv2023

A General Technique for Searching in Implicit Sets via Function Inversion

In recent years, the Fiat-Naor function inversion scheme has been used to disprove conjectures in fine-grained complexity theory and design state of the art data structures for a number of combinatorial problems. We pursue this line of research by considering its application to data structures for searching in implicit sets, defined as the image of a function. We show that, if $f$ is of the form $[N]\to [2^{w}]^d$ for some $w=polylog(N)$ and is computable in constant time, then, for any $0<\alpha <1$, we can obtain a data structure using $\~O(N^{1-\alpha/3})$ space such that, for a given $d$-dimensional axis-aligned box $B$, we can search for some $x\in [N]$ such that $f(x) \in B$ in time $\~O(N^{\alpha})$. (Here the $\~O(.)$ notation omits polylogarithmic factors.) Using similar techniques, we further obtain - data structures for range counting and reporting, predecessor, selection, ranking queries, and combinations thereof, on the set $f([N])$, - data structures for preimage size and preimage selection queries for a given value of $f$, and - data structures for selection and ranking queries on geometric quantities computed from tuples of points in $d$-space. These results unify and generalize previously known results on 3SUM-indexing and string searching, and are widely applicable as a black box to a variety of problems. In particular, we give a data structure for a generalized version of gapped string indexing, and show how to preprocess a set of points on an integer grid in order to efficiently compute (in sublinear time), for points contained in a given axis-aligned box, their Theil-Sen estimator, the $k$th largest area triangle, or the induced hyperplane that is the $k$th furthest from the origin.

cs.DS↗

arXiv2023

Shortest paths on polymatroids and hypergraphic polytopes

Base polytopes of polymatroids, also known as generalized permutohedra, are polytopes whose edges are parallel to a vector of the form $\mathbf{e}_i - \mathbf{e}_j$. We consider the following computational problem: Given two vertices of a generalized permutohedron $P$, find a shortest path between them on the skeleton of $P$. This captures many known flip distance problems, such as computing the minimum number of exchanges between two spanning trees of a graph, the rotation distance between binary search trees, the flip distance between acyclic orientations of a graph, or rectangulations of a square. We prove that this problem is $NP$-hard, even when restricted to very simple polymatroids in $\mathbb{R}^n$ defined by $O(n)$ inequalities. Assuming $P\not= NP$, this rules out the existence of an efficient simplex pivoting rule that performs a minimum number of nondegenerate pivoting steps to an optimal solution of a linear program, even when the latter defines a polymatroid. We also prove that the shortest path problem is inapproximable when the polymatroid is specified via an evaluation oracle for a corresponding submodular function, strengthening a recent result by Ito et al. (ICALP'23). More precisely, we prove the $APX$-hardness of the shortest path problem when the polymatroid is a hypergraphic polytope, whose vertices are in bijection with acyclic orientations of a given hypergraph. The shortest path problem then amounts to computing the flip distance between two acyclic orientations of a hypergraph. On the positive side, we provide a polynomial-time approximation algorithm for the problem of computing the flip distance between two acyclic orientations of a hypergraph, where the approximation factor is the maximum codegree of the hypergraph. Our result implies an exact polynomial-time algorithm for the flip distance between two acyclic orientations of any linear hypergraph.

cs.DS↗

arXiv2022

Combinatorial generation via permutation languages. V. Acyclic orientations

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs. Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung-Hoang-M\"utze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.

math.CO↗

arXiv2022

Improved Algebraic Degeneracy Testing

In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that $F(a_1,\ldots,a_k) = 0$. We consider a generalization of this problem in which $F$ is an arbitrary constant-degree polynomial, we are given $k$ sets of $n$ numbers, and have to determine whether there exist a $k$-tuple of numbers, one in each set, on which $F$ vanishes. We give the first improvement over the na\"ive $O^*(n^{k-1})$ algorithm for this problem (where the $O^*(\cdot)$ notation omits subpolynomial factors). We show that the problem can be solved in time $O^*\left( n^{k - 2 + \frac 4{k+2}}\right)$ for even $k$ and in time $O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right)$ for odd $k$ in the real RAM model of computation. We also prove that for $k=4$, the problem can be solved in time $O^*(n^{2.625})$ in the algebraic decision tree model, and for $k=5$ it can be solved in time $O^*(n^{3.56})$ in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for $k$-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft's point-line incidence detection problem in any dimension.

cs.CG↗

arXiv2022

The rotation distance of brooms

The associahedron $\mathcal{A}(G)$ of a graph $G$ has the property that its vertices can be thought of as the search trees on $G$ and its edges as the rotations between two search trees. If $G$ is a simple path, then $\mathcal{A}(G)$ is the usual associahedron and the search trees on $G$ are binary search trees. Computing distances in the graph of $\mathcal{A}(G)$, or equivalently, the rotation distance between two binary search trees, is a major open problem. Here, we consider the different case when $G$ is a complete split graph. In that case, $\mathcal{A}(G)$ interpolates between the stellohedron and the permutohedron, and all the search trees on $G$ are brooms. We show that the rotation distance between any two such brooms and therefore the distance between any two vertices in the graph of the associahedron of $G$ can be computed in quasi-quadratic time in the number of vertices of $G$.

math.CO↗