arXiv subjects

Sophie Toulouse

Publications and source records attributed to Sophie Toulouse.

6 recordsLinked to original sources

arXiv2026

Optimizing alphabet reduction pairs of arrays

In our earlier paper, "2 CSPs all are approximable within a constant differential factor" (ISCO 2018, LNCS 10856), we introduced a family of combinatorial designs called 'alphabet reduction pairs of arrays' (ARPAs). These designs are parameterized by three integers $q,p,k$, with $p\leq q$ and $k\leq p$: $q$ is the size of the alphabet from which the arrays draw their entries; $p$ is the maximum number of distinct symbols allowed in a row of the second array; $k$ is the largest integer for which the two arrays coincide -- up to row permutations -- on any $k$-element subset of their columns. The first array must contain at least one occurrence of the word $0\ 1 \cdots\ q-1$ as a row. The idea is to cover as many occurrences of this word as possible using as few words as possible, each containing at most $p$ distinct symbols. ARPAs are related to the approximability of constraint satisfaction problems with bounded constraint arity ($k$-CSPs). In this context, we are particularly interested in ARPAs that maximize the frequency of the word $0\ 1 \cdots\ q-1$. We call such ARPAs 'optimal' and study them in this paper. To this end, we introduce a simpler family of combinatorial designs called 'Cover pairs of arrays' (CPAs), which can be viewed as partially defined ARPAs with Boolean entries. We prove that ARPAs and CPAs are equivalent with respect to maximizing the frequency of their target word. As a corollary of our proof, computing the frequency of the target word in optimal ARPAs reduces to solving a linear program in $q + p + 1$ continuous variables and $k + 1$ constraints. We also prove the optimality of previously known ARPAs for $p=k$ and provide optimal ARPAs for $k=1$ and $k=2$.

math.CO↗

arXiv2025

Deriving differential approximation results for $k\,$CSPs from combinatorial designs

Inapproximability results for $\mathsf{Max\,k\,CSP\!-\!q}$ have been traditionally established using balanced $t$-wise independent distributions, which are closely related to orthogonal arrays, a famous family of combinatorial designs. In this work, we investigate the role of these combinatorial structures in the context of the differential approximability of $\mathsf{k\,CSP\!-\!q}$, providing new structural insights and approximation bounds. We first establish a direct connection between the average differential ratio on $\mathsf{k\,CSP\!-\!q}$ instances and orthogonal arrays. This allows us to derive the new differential approximability bounds of $1/q^k$ for $(k +1)$-partite instances, $Ω(1/n^{\lfloor k/2\rfloor})$ for Boolean instances, $Ω(1/n)$ when $k =2$, and $Ω(1/n^{k -\lceil\log_{Θ(q)}k\rceil})$ when $k, q\geq 3$. We then introduce families of array pairs, called {\em alphabet reduction pairs of arrays}, that are still related to balanced $k$-wise independence. Using these pairs of arrays, we establish a reduction from $\mathsf{k\,CSP\!-\!q}$ to $\mathsf{k\,CSP\!-\!k}$ (where $q >k$), with an expansion factor of $1/(q -k/2)^k$ on the differential approximation guarantee. Combining this with a 1998 result by Yuri Nesterov, we conclude that $\mathsf{2\,CSP\!-\!q}$ is approximable within a differential factor of $0.429/(q -1)^2$. Finally, using similar Boolean array pairs, {\em called cover pairs of arrays}, we prove that every Hamming ball of radius $k$ provides a $Ω(1/n^k)$-approximation of the instance diameter. Thus, our work highlights the relevance of combinatorial designs for establishing structural differential approximation guarantees for CSPs.

math.CO↗

arXiv2020

Approximation of the Double Travelling Salesman Problem with Multiple Stacks

The Double Travelling Salesman Problem with Multiple Stacks, DTSPMS, deals with the collect and delivery of n commodities in two distinct cities, where the pickup and the delivery tours are related by LIFO constraints. During the pickup tour, commodities are loaded into a container of k rows, or stacks, with capacity c. This paper focuses on computational aspects of the DTSPMS, which is NP-hard. We first review the complexity of two critical subproblems: deciding whether a given pair of pickup and delivery tours is feasible and, given a loading plan, finding an optimal pair of pickup and delivery tours, are both polynomial under some conditions on k and c. We then prove a (3k)/2 standard approximation for the MinMetrickDTSPMS, where k is a universal constant, and other approximation results for various versions of the problem. We finally present a matching-based heuristic for the 2DTSPMS, which is a special case with k=2 rows, when the distances are symmetric. This yields a 1/2-o(1), 3/4-o(1) and 3/2+o(1) standard approximation for respectively Max2DTSPMS, its restriction Max2DTSPMS-(1,2) with distances 1 and 2, and Min2DTSPMS-(1,2), and a 1/2-o(1) differential approximation for Min2DTSPMS and Max2DTSPMS.

cs.DM↗

arXiv2010

On the complexity of the multiple stack TSP, kSTSP

The multiple Stack Travelling Salesman Problem, STSP, deals with the collect and the deliverance of n commodities in two distinct cities. The two cities are represented by means of two edge-valued graphs (G1,d2) and (G2,d2). During the pick-up tour, the commodities are stored into a container whose rows are subject to LIFO constraints. As a generalisation of standard TSP, the problem obviously is NP-hard; nevertheless, one could wonder about what combinatorial structure of STSP does the most impact its complexity: the arrangement of the commodities into the container, or the tours themselves? The answer is not clear. First, given a pair (T1,T2) of pick-up and delivery tours, it is polynomial to decide whether these tours are or not compatible. Second, for a given arrangement of the commodities into the k rows of the container, the optimum pick-up and delivery tours w.r.t. this arrangement can be computed within a time that is polynomial in n, but exponential in k. Finally, we provide instances on which a tour that is optimum for one of three distances d1, d2 or d1+d2 lead to solutions of STSP that are arbitrarily far to the optimum STSP.

cs.CC↗

arXiv2010

Approximability of the Multiple Stack TSP

STSP seeks a pair of pickup and delivery tours in two distinct networks, where the two tours are related by LIFO contraints. We address here the problem approximability. We notably establish that asymmetric MaxSTSP and MinSTSP12 are APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12.

cs.CC↗

arXiv2008

Rapport de recherche sur le problème du plus court chemin contraint

This article provides an overview of the performance and the theoretical complexity of approximate and exact methods for various versions of the shortest path problem. The proposed study aims to improve the resolution of a more general covering problem within a column generation scheme in which the shortest path problem is the sub-problem.

cs.DS↗