arXiv subjects

Lydia Mirabel Mendoza-Cadena

Publications and source records attributed to Lydia Mirabel Mendoza-Cadena.

10 recordsLinked to original sources

arXiv2025

Diameter reduction via arc reversal

The diameter of a directed graph is the maximum distance between any pair of vertices. We study a problem that generalizes \textsc{Oriented Diameter}: For a given directed graph and a positive integer $d$, what is the minimum number of arc reversals required to obtain a graph with diameter at most $d$? We investigate variants of this problem, considering the number of arc reversals and the target diameter as parameters. We show hardness results under certain parameter restrictions, and give polynomial time algorithms for planar and cactus graphs. This work is partly motivated by the relation between oriented diameter and the volume of directed edge polytopes, which we show to be independent.

math.CO↗

arXiv2025

New upper bounds on the order of mixed cages of girth 6

A $[z,r;g]$-mixed cage is a mixed graph of minimum order such that each vertex has $z$ in-arcs, $z$ out-arcs, $r$ edges, and it has girth $g$, and the minimum order for $[z,r;g]$-mixed graphs is denoted by $n[z,r;g]$. In this paper, we present an infinite family of mixed graphs with girth $6$, that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. \textit{On Mixed Cages of girth 6}, arXiv:2401.14768v2. In particular, if $q$ is an even prime power we construct a family of graphs that satisfies $n[\frac{q}{4},q;6]\leq 4q^2-4$, and if $q$ is an odd prime power, and $\frac{q-3}{2}$ is odd then our family satisfies that $n[\frac{q-1}{4},q;6]\leq 4q^2-4$, otherwise $n[\frac{q-3}{4},q;6]\leq 4q^2-4$.

math.CO↗

arXiv2025

A Note on Mixed Cages of Girth 5

A mixed regular graph is a graph where every vertex has $z$ incoming arcs, $z$ outgoing arcs, and $r$ edges; furthermore, if it has girth $g$, we say that the graph is a \emph{$[z,r;g]$-mixed graph}. A \emph{$[z,r;g]$-mixed cage} is a $[z,r;g]$-mixed graph with the smallest possible order. In this note, we give a family of $[z,q;5]$-mixed graphs for $q\geq 7$ power of prime and $q-1\leq 4z+R$ with $z\geq 1$ and $R \in \{1,\ldots,5\}$. This provides better upper bounds on the order of mixed cages until this moment.

math.CO↗

arXiv2025

The Strong Core of Housing Markets with Partial Order Preferences

We study the strong core of housing markets when agents' preferences over houses are expressed as partial orders. We provide a structural characterization of the strong core, and propose an efficient algorithm that finds an allocation in the strong core or decides that it is empty, even in the presence of forced and forbidden arcs. The algorithm satisfies the property of group-strategyproofness. Additionally, we show that certain results known for the strong core in the case when agents' preferences are weak orders can be extended to the setting with partial order preferences; among others, we show that the strong core in such housing markets satisfies the property of respecting improvements.

cs.GT↗

arXiv2024

On the Complexity of Inverse Bivariate Multi-unit Assignment Valuation Problems

Inverse and bilevel optimization problems play a central role in both theory and applications. These two classes are known to be closely related due to the pioneering work of Dempe and Lohse (2006), and thus have often been discussed together ever since. In this paper, we consider inverse problems for multi-unit assignment valuations. Multi-unit assignment valuations form a subclass of strong-substitutes valuations that can be represented by edge-weighted complete bipartite graphs. These valuations play a key role in auction theory as the strong substitutes condition implies the existence of a Walrasian equilibrium. A recent line of research concentrated on the problem of deciding whether a bivariate valuation function is an assignment valuation or not. In this paper, we consider an \emph{inverse} variant of the problem: we are given a bivariate function $g$, and our goal is to find a bivariate multi-unit assignment valuation function $f$ that is as close to $g$ as possible. The difference between $f$ and $g$ can be measured either in $\ell_1$- or $\ell_\infty$-norm. Using tools from discrete convex analysis, we show that the problem is strongly NP-hard. On the other hand, we derive linear programming formulations that solve relaxed versions of the problem.

math.OC↗

arXiv2023

Shortest Odd Paths in Undirected Graphs with Conservative Weight Functions

We consider the Shortest Odd Path problem, where given an undirected graph $G$, a weight function on its edges, and two vertices $s$ and $t$ in $G$, the aim is to find an $(s,t)$-path with odd length and, among all such paths, of minimum weight. For the case when the weight function is conservative, i.e., when every cycle has non-negative total weight, the complexity of the Shortest Odd Path problem had been open for 20 years, and was recently shown to be NP-hard. We give a polynomial-time algorithm for the special case when the weight function is conservative and the set $E^-$ of negative-weight edges forms a single tree. Our algorithm exploits the strong connection between Shortest Odd Path and the problem of finding two internally vertex-disjoint paths between two terminals in an undirected edge-weighted graph. It also relies on solving an intermediary problem variant called Shortest Parity-Constrained Odd Path where for certain edges we have parity constraints on their position along the path. Also, we exhibit two FPT algorithms for solving Shortest Odd Path in graphs with conservative weight functions. The first FPT algorithm is parameterized by $|E^-|$, the number of negative edges, or more generally, by the maximum size of a matching in the subgraph of $G$ spanned by $E^-$. Our second FPT algorithm is parameterized by the treewidth of $G$.

cs.DS↗

arXiv2023

Newton-type algorithms for inverse optimization I: weighted bottleneck Hamming distance and $\ell_\infty$-norm objectives

In minimum-cost inverse optimization problems, we are given a feasible solution to an underlying optimization problem together with a linear cost function, and the goal is to modify the costs by a small deviation vector so that the input solution becomes optimal. The difference between the new and the original cost functions can be measured in several ways. In this paper, we focus on two objectives: the weighted bottleneck Hamming distance and the weighted $\ell_\infty$-norm. We consider a general model in which the coordinates of the deviation vector are required to fall within given lower and upper bounds. For the weighted bottleneck Hamming distance objective, we present a simple, purely combinatorial algorithm that determines an optimal deviation vector in strongly polynomial time. For the weighted $\ell_\infty$-norm objective, we give a min-max characterization for the optimal solution, and provide a pseudo-polynomial algorithm for finding an optimal deviation vector that runs in strongly polynomial time in the case of unit weights. For both objectives, we assume that an algorithm with the same time complexity for solving the underlying combinatorial optimization problem is available. For both objectives, we also show how to extend the results to inverse optimization problems with multiple cost functions.

math.OC↗

arXiv2023

Newton-type algorithms for inverse optimization II: weighted span objective

In inverse optimization problems, the goal is to modify the costs in an underlying optimization problem in such a way that a given solution becomes optimal, while the difference between the new and the original cost functions, called the deviation vector, is minimized with respect to some objective function. The $\ell_1$- and $\ell_\infty$-norms are standard objectives used to measure the size of the deviation. Minimizing the $\ell_1$-norm is a natural way of keeping the total change of the cost function low, while the $\ell_\infty$-norm achieves the same goal coordinate-wise. Nevertheless, none of these objectives is suitable to provide a balanced or fair change of the costs. In this paper, we initiate the study of a new objective that measures the difference between the largest and the smallest weighted coordinates of the deviation vector, called the weighted span. We give a min-max characterization for the minimum weighted span of a feasible deviation vector, and provide a Newton-type algorithm for finding one that runs in strongly polynomial time in the case of unit weights.

math.OC↗

arXiv2022

Resolving Infeasibility of Linear Systems: A Parameterized Approach

Deciding feasibility of large systems of linear equations and inequalities is one of the most fundamental algorithmic tasks. However, due to data inaccuracies or modeling errors, in practical applications one often faces linear systems that are infeasible. Extensive theoretical and practical methods have been proposed for post-infeasibility analysis of linear systems. This generally amounts to detecting a feasibility blocker of small size $k$, which is a set of equations and inequalities whose removal or perturbation from the large system of size $m$ yields a feasible system. This motivates a parameterized approach towards post-infeasibility analysis, where we aim to find a feasibility blocker of size at most $k$ in fixed-parameter time $f(k) \cdot m^{O(1)}$. We establish parameterized intractability ($W[1]$- and $NP$-hardness) results already in very restricted settings for different choices of the parameters maximum size of a deletion set, number of positive/negative right-hand sides, treewidth, pathwidth and treedepth. Additionally, we rule out a polynomial compression for MinFB parameterized by the size of a deletion set and the number of negative right-hand sides. Furthermore, we develop fixed-parameter algorithms parameterized by various combinations of these parameters when every row of the system corresponds to a difference constraint. Our algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose fixed-parameter tractability was shown by Chen et al. (STOC 2008).

cs.DS↗

arXiv2022

Inverse optimization problems with multiple weight functions

We introduce a new class of inverse optimization problems in which an input solution is given together with $k$ linear weight functions, and the goal is to modify the weights by the same deviation vector $p$ so that the input solution becomes optimal with respect to each of them, while minimizing $\|p\|_1$. In particular, we concentrate on three problems with multiple weight functions: the inverse shortest $s$-$t$ path, the inverse bipartite perfect matching, and the inverse arborescence problems. Using LP duality, we give min-max characterizations for the $\ell_1$-norm of an optimal deviation vector. Furthermore, we show that the optimal $p$ is not necessarily integral even when the weight functions are so, therefore computing an optimal solution is significantly more difficult than for the single-weighted case. We also give a necessary and sufficient condition for the existence of an optimal deviation vector that changes the values only on the elements of the input solution, thus giving a unified understanding of previous results on arborescences and matchings.

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