Jean Cardinal

SODA Conference 2025 Conference Paper

Facet-Hamiltonicity

• Hugo A. Akitaya
• Jean Cardinal
• Stefan Felsner
• Linda Kleist
• Robert Lauff

MFCS Conference 2025 Conference Paper

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

• Jean Cardinal
• Xavier Goaoc
• Sarah Wajsbrot

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.

SODA Conference 2023 Conference Paper

Zigzagging through acyclic orientations of chordal graphs and hypergraphs

• Jean Cardinal
• Hung Phuc Hoang
• Arturo Merino
• Torsten Mütze

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 (SODA 2022). 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 (FPSAC 2022). This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group (SODA 2020). Our algorithms are derived from the Hartung-Hoang-Mutze-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. * Arturo Merino was supported by ANID Becas Chile 2019-72200522. Torsten Mütze was supported by Czech Science Foundation grant GA 22-15272S. Arturo Merino and Torsten Mütze were also supported by German Science Foundation grant 413902284.

SODA Conference 2022 Conference Paper

Efficient generation of elimination trees and graph associahedra

• Jean Cardinal
• Arturo Merino
• Torsten Mütze

An elimination tree for a connected graph G is a rooted tree on the vertices of G obtained by choosing a root x and recursing on the connected components of G – x to produce the subtrees of x. Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph G can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph G can be implemented in time (m + n ) per generated elimination tree, where m and n are the number of edges and vertices of G, respectively. If G is a tree, we improve this to a loopless algorithm running in time (1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of G, rather than just Hamilton path, if the graph G is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i. e. , it computes a Hamilton path on the graph associahedron of G if and only if G is chordal.

SODA Conference 2020 Conference Paper

Competitive Online Search Trees on Trees

• Prosenjit Bose
• Jean Cardinal
• John Iacono
• Grigorios Koumoutsos
• Stefan Langerman

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O (log log n )-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.

TCS Journal 2020 Journal Article

Reconfiguration of satisfying assignments and subset sums: Easy to find, hard to connect

• Jean Cardinal
• Erik D. Demaine
• David Eppstein
• Robert A. Hearn
• Andrew Winslow

We consider the computational complexity of reconfiguration problems, in which one is given two combinatorial configurations satisfying some constraints, and is asked to transform one into the other using elementary operations, while satisfying the constraints at all times. Such problems appear naturally in many contexts, such as model checking, motion planning, enumeration, sampling, and recreational mathematics. We provide hardness results for problems in this family, in which the constraints and operations are particularly simple. More precisely, we prove the PSPACE-completeness of the following decision problems: • Given two satisfying assignments of a planar monotone instance of NAE 3-SAT, can one assignment be transformed into the other by a sequence of variable flips such that the formula remains satisfied at every step? • Given two subsets of a set S of integers with the same sum, can one subset be transformed into the other by adding or removing at most three elements of S at a time, such that the intermediate subsets also have the same sum? • Given two points in { 0, 1 } n contained in a polytope P specified by a constant number of linear inequalities, is there a path in the n-hypercube connecting the two points and contained in P? These problems can be interpreted as reconfiguration analogues of standard problems in NP. Interestingly, the sets of instances that appear as input to the reconfiguration problems in our reductions lie in P. In particular, the elements of S and the coefficients of the inequalities defining P can be restricted to have logarithmic bit-length.

SODA Conference 2014 Conference Paper

Making Octants Colorful and Related Covering Decomposition Problems

• Jean Cardinal
• Kolja B. Knauer
• Piotr Micek
• Torsten Ueckerdt

TCS Journal 2010 Journal Article

Connected vertex covers in dense graphs

• Jean Cardinal
• Eythan Levy

We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worst-case ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2 / ( 1 + ε ) in graphs with average degree ε n, and give a family of near-tight examples.

TCS Journal 2010 Journal Article

Non-cooperative facility location and covering games

• Jean Cardinal
• Martin Hoefer

We consider a general class of non-cooperative games related to combinatorial covering and facility location problems. A game is based on an integer programming formulation of the corresponding optimization problem, and each of the k players wants to satisfy a subset of the constraints. For that purpose, resources available in integer units must be bought, and their cost can be shared arbitrarily between players. We consider the existence and cost of exact and approximate pure-strategy Nash equilibria. In general, the prices of anarchy and stability are in Θ ( k ) and deciding the existence of a pure Nash equilibrium is NP-hard. Under certain conditions, however, cheap Nash equilibria exist, in particular if the integrality gap of the underlying integer program is 1, or in the case of single constraint players. We also present algorithms that compute simultaneously near-stable and near-optimal approximate Nash equilibria in polynomial time.

STOC Conference 2010 Conference Paper

Sorting under partial information (without the ellipsoid algorithm)

• Jean Cardinal
• Samuel Fiorini
• Gwenaël Joret
• Raphaël M. Jungers
• J. Ian Munro

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set $P$, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of $P$. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical. We develop efficient algorithms for sorting under partial information. Like Kahn and Kim, our approach relies on graph entropy. However, our algorithms differ in essential ways from theirs. Rather than resorting to convex programming for computing the entropy, we approximate the entropy, or make sure it is computed only once in a restricted class of graphs, permitting the use of a simpler algorithm. Specifically, we present: an O(n 2 ) algorithm performing O(log n ⋅ log e(P)) comparisons; an O(n 2.5 ) algorithm performing at most (1+ε) log e(P) + O ε(n) comparisons; an O(n 2.5 ) algorithm performing O(log e(P)) comparisons. All our algorithms are simple to implement.

STOC Conference 2009 Conference Paper

An efficient algorithm for partial order production

• Jean Cardinal
• Samuel Fiorini
• Gwenaël Joret
• Raphaël M. Jungers
• J. Ian Munro

We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S, by comparing a minimum number of pairs in T. Special cases of this problem include sorting by comparisons, selection, multiple selection, and heap construction.

SODA Conference 2009 Conference Paper

Decomposition of multiple coverings into more parts

• Greg Aloupis
• Jean Cardinal
• Sébastien Collette
• Stefan Langerman
• David Orden
• Pedro Ramos 0001

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk -fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.

TCS Journal 2009 Journal Article

Improved approximation bounds for edge dominating set in dense graphs

• Jean Cardinal
• Stefan Langerman
• Eythan Levy

We analyze the simple greedy algorithm that iteratively removes the endpoints of a maximum-degree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2-approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less than 2 in the cases where the input graph has n vertices and at least ϵ ( n 2 ) edges, for ϵ > 1 / 2. This ratio is shown to be asymptotically tight for ϵ > 1 / 2.

TCS Journal 2005 Journal Article

Designing small keyboards is hard

• Jean Cardinal
• Stefan Langerman

We study the problem of placing symbols of an alphabet onto the minimum number of keys of a small keyboard so that any word of a given dictionary can be recognized univoquely only by looking at the corresponding sequence of keys. This problem is motivated by the design of small keyboards for mobile devices. We show that the problem is hard in general, and NP-complete even if we only wish to decide whether two keys are sufficient. We also consider two variants of the problem. In the first one, symbols on a key must be contiguous in an ordered alphabet. In the second variant, a well-chosen measure of ambiguity in the recognition of the words is minimized given the number of keys. Hardness and approximability results are given.