arXiv subjects

José A. Soto

Publications and source records attributed to José A. Soto.

At least 19 recordsLinked to original sources

arXiv2026

The Strong Secretary Conjecture is True for Linear Matroids

We prove a $1/e$ guarantee for the matroid secretary problem on linear matroids, therefore settling the strong secretary conjecture in this class of matroids. The result holds both when the matroid is known in advance and when a linear representation over a finite field is given online. In the known-matroid model, the result extends more generally to matroids admitting a finitary modular extension. Each element of a fixed optimal basis is selected with probability at least $1/e$. $\mathbf{\text{Concurrent Discovery Disclosure:}}$ The proof of the main result in this manuscript was obtained in a conversation with ChatGPT-6 Astra on Tuesday, September 15, 2026 at 1:02 AM PDT. We then prepared this manuscript for public release, with the intent of uploading it on the morning of Thursday, September 17, 2026. In the early morning hours of September 17, while finalizing the submission, we discovered the manuscript https://arxiv.org/abs/2609.19118 of Abdi, Banihashem, Hajiaghayi, and Mittal, uploaded on September 16, 2026, which contains the same result via an essentially identical approach. We are sharing our manuscript nonetheless in case our exposition is of independent utility to the community, and we hope this experience stimulates broader discussion about concurrent discovery in the AI era.

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arXiv2026

Secretary Problems with Interactive Ordinal Queries

We study secretary problems in which the online algorithm can ask a reliable oracle a limited number of ordinal queries about unseen elements. Values are revealed only when elements arrive, and all accept/reject decisions are irrevocable. The oracle never reveals numerical values; it only answers questions defined relative to the information already observed by the algorithm. We first consider the single-choice secretary problem. A query returns the number of unseen elements whose value exceeds the current record. We analyze random order, adversarial order, and two sample-based adversarial orders, with no queries, one query, and unlimited queries. We obtain tight bounds for all variants except one. In adversarial order, queries do not improve over the trivial $1/n$ guarantee. In random order, one query improves the optimal success probability from $1/e$ to $0.4477\ldots$, and unlimited queries achieve the optimal value $1/2$. In the sample-based models, one query already yields the tight value $1/e$ against an online adversary, while unlimited queries give the exact value $(20\sqrt{10}-29)/81$ against an offline adversary. The remaining one-query offline-adversarial case is left open, and we give two partial upper bounds for it. We also extend the model to the matroid secretary problem, where we study two ordinal query interfaces. A simple query identifies the unseen elements that would individually improve the current optimum on the observed set, while a complete query gives a finer ordinal partition of the unseen elements by weight relative to that optimum. Under sample-based online adversarial order, a single simple query yields a $1/4$ probability-competitive algorithm, and a complete query improves this to the optimal $1/e$. Finally, with unlimited complete queries in the random-order model, we achieve the optimal $1/2$ probability-competitive ratio.

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arXiv2026

Deterministic online matching under short augmenting paths and restricted vertex reassignments

We study deterministic online bipartite matching with local recourse. Online vertices arrive one by one and reveal edges to a fixed offline set. After each arrival, the algorithm maintains a matching and may update it only by an augmenting path of length $1$ or $3$ starting at the new vertex. In the budgeted model $\operatorname{OMP}_3(s,t)$, each offline vertex can be reassigned at most $s$ times and each online vertex at most $t$ times. Our main result is a deterministic algorithm, Lowest-Cost-Path ($\operatorname{LCP}$), that achieves the optimal competitive ratio for every choice of budgets. If $s=0$ or $t=0$, the best possible ratio is $1/2$. For every $s\ge 1$ and finite $t$, the optimal ratio is $$ γ_t=\frac{2\cdot 2^t-1}{3\cdot 2^t-1}. $$ This value equals $3/5$ for $t=1$ and converges to $2/3$ as $t$ increases. For $t=\infty$, the optimal ratio is exactly $2/3$. In particular, allowing a single reassignment per offline vertex already matches the best guarantee achievable with any larger offline budget. We also prove matching upper bounds for all deterministic algorithms. Our analysis is primal--dual and uses a structural description of the history graph of $\operatorname{LCP}$, which explains how exhausted reassignment budgets can block length-$3$ augmentations. We also study offline-weighted variants. In particular, we determine the exact optimal deterministic ratios for two unit-offline-budget cases: $2-\sqrt2$ for $\operatorname{WOMP}_3(1,1)$, and $(\sqrt5-1)/2$ for $\operatorname{WOMP}_3(1,\infty)$. Both are strictly below their unweighted counterparts. If weights are placed on online vertices instead, no deterministic algorithm has a positive competitive ratio.

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arXiv2026

The Multiple-Choice Matroid Secretary Problem

We introduce and study the multiple-choice matroid secretary problem, denoted $(J,κ)$-MSP. For rank-one matroids and $κ=\infty$, it reduces to the classical secretary problem with $J$ choices. Elements arrive in uniformly random order. Algorithms may keep a candidate pool $\mathrm{AUX}$ feasible in the $J$-fold union matroid $\mathcal{M}^{(J)}$ satisfying $|\mathrm{AUX}|\le κ\cdot\mathrm{rank}(\mathcal{M})$. Finally, one extracts the maximum-weight independent subset of $\mathrm{AUX}$ in $\mathcal{M}$. This model separates online storage from the final feasible solution. We study two multiple-choice implementations: multi-track algorithms (maintaining $J$ independent sets of $\mathcal{M}$) and union-based algorithms (maintaining the pool directly in $\mathcal{M}^{(J)}$). Our main result is an exact optimal algorithm for transversal matroids in the uncapacitated $(J,\infty)$ setting. For fixed $J$, its probability-competitive ratio equals the optimal success probability of the classical $J$-choice secretary problem. Thus, rank-one instances are the worst case for the whole transversal class, and the optimal guarantee converges exponentially fast to $1$ as $J$ grows. We also analyze a simple single-threshold routing algorithm for capacitated transversal matroids with local capacities $b$ and global capacity $κ\cdot\mathrm{rank}(\mathcal{M})$. Its analysis provides explicit finite-parameter bounds and asymptotic formulas, showing how finite-rank loss caused by global capacity decays, and how $b$, $J$, and $κ$ interact. Finally, we instantiate the multi-track approach for $k$-column-sparse matroids (guarantee $1-O(e^{-J/(ke)})$) and the union-based approach for laminar matroids (guarantee $1-O(e^{-J/e})$).

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arXiv2026

The k-Sum Lateness Problem on a Single Machine

We study a single-machine scheduling problem in which each job $j$ has a nonnegative processing time $p_j\ge 0$ and a due date $d_j\in\mathbb{R}$. For a non-idling schedule $S$, let $C_j(S)$ be the completion time and let $L_j(S)=C_j(S)-d_j$ be the (possibly negative) lateness. The objective is to minimize the sum of the $k$ largest lateness values, interpolating between maximum lateness ($k=1$) and total lateness ($k=n$). We prove that the decision version is weakly NP-complete when $k$ is part of the input. For fixed $k$, we give an $O(k^2 n^{k+2})$ algorithm. As a consequence, we resolve a conjecture of Woeginger on the top-$k$ tardiness problem and obtain an $O(n^{k+2})$ algorithm for every fixed $k$. Our main structural result shows that there exists an optimal schedule that admits a block-island decomposition. Outside a suitable top-$k$ set, jobs form due-date blocks ordered by due date. Within each due-date class, the top-$k$ jobs form a suffix in lexicographic shortest-processing-time (SPT) order. This structure also yields an FPT algorithm parameterized by $D+k$, where $D$ is the number of distinct due dates. Independently, a standard dual representation of the top-$k$ objective reduces the problem to a family of total-tardiness instances with uniformly shifted due dates. For integral data, this gives a pseudopolynomial algorithm and a fully polynomial additive approximation scheme with error at most $\varepsilon M$, where $M=\max\{1,\max_j p_j,\max_j |d_j|\}$. The same route also gives XP algorithms for fixed $P$ and fixed $D$, where $P$ is the number of distinct processing times.

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arXiv2026

Free-Order Online Selection for k-Systems

The Matroid Secretary Problem is a central question in online optimization, modeling sequential decision-making under combinatorial constraints. We introduce a bipartite graph framework that unifies and extends several known formulations, including bipartite matching, matroid intersection, and matroid secretary problems. In this model, agents and items form a bipartite graph, and the objective is to select a matching that satisfies independence constraints on both sides. We first study the free-order setting under edge-arrivals. For $k$-matroid intersection, we leverage a core lemma by [FSZ, 2022] to design an $Ω(1/k^2)$-competitive algorithm, extending known results for single matroids. Building on this, we introduce $k$-growth systems -- a new class of independence systems that lie properly between $k$-matchoids and $k$-extendible systems and may be of independent combinatorial interest. We establish a generalized core lemma for $k$-growth systems, showing that a suitably defined set of critical elements retains a $Ω(1/k^2)$ fraction of the optimal weight. Using this lemma, we extend our $Ω(1/k^2)$-competitive algorithm to $k$-growth systems. We then study the agent-arrival model, which presents unique challenges to our framework. We extend the core lemma to this model and then apply it to obtain an $Ω(β/k^2)$-competitive algorithm for $k$-growth systems, where $β$ denotes the competitiveness of an appropriate type of order-oblivious algorithm for the item-side constraint. Finally, we extend our results to the case of multiple item selection, and obtain constant-competitive algorithms for fundamental cases such as partition matroids and $k$-matching constraints. We also study the closure properties and structural role and of $k$-growth systems within the hierarchy of $k$-systems.

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arXiv2024

Set Selection with Uncertain Weights: Non-Adaptive Queries and Thresholds

We study set selection problems where the weights are uncertain. Instead of its exact weight, only an uncertainty interval containing its true weight is available for each element. In some cases, some solutions are universally optimal; i.e., they are optimal for every weight that lies within the uncertainty intervals. However, it may be that no universal optimal solution exists, unless we are revealed additional information on the precise values of some elements. In the minimum cost admissible query problem, we are tasked to (non-adaptively) find a minimum-cost subset of elements that, no matter how they are revealed, guarantee the existence of a universally optimal solution. We introduce thresholds under uncertainty to analyze problems of minimum cost admissible queries. Roughly speaking, for every element e, there is a threshold for its weight, below which e is included in all optimal solutions and a second threshold above which e is excluded from all optimal solutions. We show that computing thresholds and finding minimum cost admissible queries are essentially equivalent problems. Thus, the analysis of the minimum admissible query problem reduces to the problem of computing thresholds. We provide efficient algorithms for computing thresholds in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching. By making use of the equivalence between the two problems these results translate into efficient algorithms for minimum cost admissible queries in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching.

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arXiv2023

Online Combinatorial Assignment in Independence Systems

We consider an online multi-weighted generalization of several classic online optimization problems, called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matchings of a hypergraph we recover the combinatorial auction problem, where every node represents an item to be sold, and every edge represents a bundle of items. For combinatorial auctions, Kesselheim et al. showed upper bounds of O(loglog(k)/log(k)) and $O(\log \log(n)/\log(n))$ on the competitiveness of any online algorithm, even in the random order model, where $k$ is the maximum bundle size and $n$ is the number of items. We provide an exponential improvement on these upper bounds to show that the competitiveness of any online algorithm in the prophet IID setting is upper bounded by $O(\log(k)/k)$, and $O(\log(n)/\sqrt{n})$. Furthermore, using linear programming, we provide new and improved guarantees for the $k$-bounded online combinatorial auction problem (i.e., bundles of size at most $k$). We show a $(1-e^{-k})/k$-competitive algorithm in the prophet IID model, a $1/(k+1)$-competitive algorithm in the prophet-secretary model using a single sample per agent, and a $k^{-k/(k-1)}$-competitive algorithm in the secretary model. Our algorithms run in polynomial time and work in more general independence systems where the offline combinatorial assignment problem admits the existence of a polynomial-time randomized algorithm that we call certificate sampler. We show that certificate samplers have a nice interplay with random order models, and we also provide new polynomial-time competitive algorithms for some classes of matroids, matroid intersections, and matchoids.

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arXiv2021

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ of minimum size such that adding $S'$ to $G$ makes it $(k+1)$-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse. In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for $2$-vertex-connectivity augmentation, for the case in which $G$ is a cycle. This is the first step for attacking the more general problem of augmenting a $2$-connected graph. Our algorithm is based on local search and attains an approximation ratio of $1.8704$. To derive it, we prove novel results on the structure of minimal solutions.

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arXiv2019

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. We consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given $n$ cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on $n$ consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, as in the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least $0.45292$. In the second one, similar to the prophet inequality, the player maximizes the expectation of the selected hidden value. We show a guarantee of at least $0.63518$ with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial $n$ visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently $n$ visible numbers in the table. And the third one is similar to the latter but it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. Our guarantee improves upon $1-1/e$ for this problem, which is the currently best known guarantee and only works for the i.i.d. case.

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arXiv2019

The minimum cost query problem on matroids with uncertainty areas

We study the minimum weight basis problem on matroid when elements' weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base.

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arXiv2018

Symmetry exploitation for Online Machine Covering with Bounded Migration

Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple $(1.7 + \varepsilon)$-competitive algorithm using a migration factor of $O(1/\varepsilon)$ which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved $(4/3+\varepsilon)$-competitive algorithm using a migration factor of $\tilde{O}(1/\varepsilon^3)$. At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.

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arXiv2018

Strong Algorithms for the Ordinal Matroid Secretary Problem

In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is $α$ probability-competitive if every element from the optimum appears with probability $1/α$ in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on $2e$ by Korula and Pál [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of $k$ column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a $1+O(\sqrt{\log ρ/ρ})$ probability-competitive algorithm for uniform matroids of rank $ρ$ based on Kleinberg's $1+O(\sqrt{1/ρ})$ utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank $ρ$. We devise an $O(\log ρ)$ probability-competitive algorithm and an $O(\log\log ρ)$ ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the $O(\log\log ρ)$ utility-competitive algorithm by Feldman et al.~[SODA 2015].

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arXiv2017

Robust randomized matchings

The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the $k$ heaviest edges of $M$ to the maximum weight of a matching of size at most $k$. If $M$ guarantees a payoff of at least $α$ then it is called $α$-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a $1/\sqrt{2}$-robust matching, which is best possible. We show that Alice can improve her payoff to $1/\ln(4)$ by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.

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arXiv2014

Independent sets and hitting sets of bicolored rectangular families

A bicolored rectangular family BRF is a collection of all axis-parallel rectangles contained in a given region Z of the plane formed by selecting a bottom-left corner from a set A and an upper-right corner from a set B. We prove that the maximum independent set and the minimum hitting set of a BRF have the same cardinality and devise polynomial time algorithms to compute both. As a direct consequence, we obtain the first polynomial time algorithm to compute minimum biclique covers, maximum cross-free matchings and jump numbers in a class of bipartite graphs that significantly extends convex bipartite graphs and interval bigraphs. We also establish several connections between our work and other seemingly unrelated problems. Furthermore, when the bicolored rectangular family is weighted, we show that the problem of finding the maximum weight of an independent set is NP-hard, and provide efficient algorithms to solve it on certain subclasses.

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arXiv2014

Advances on Matroid Secretary Problems: Free Order Model and Laminar Case

The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and Vondrák [ESA 2011]). However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constant-factor approximation was found. We address this point by presenting a 9-approximation for the \emph{free order model}, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constant-factor approximation was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, a constant-factor approximation has been found for this case, using a clever but rather involved method and analysis (Im and Wang, [SODA 2011]) that leads to a 16000/3-approximation. This is arguably the most involved special case of the matroid secretary problem for which a constant-factor approximation is known. We present a considerably simpler and stronger $3\sqrt{3}e\approx 14.12$-approximation, based on reducing the problem to a matroid secretary problem on a partition matroid.

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arXiv2014

Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line : Algorithms and Complexity

Finding a maximum independent set (MIS) of a given fam- ily of axis-parallel rectangles is a basic problem in computational geom- etry and combinatorics. This problem has attracted significant atten- tion since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set (MHS), is bounded by a universal constant. An interesting special case, that may prove use- ful to tackling the general problem, is the diagonal-intersecting case, in which the given family of rectangles is intersected by a diagonal. Indeed, Chepoi and Felsner recently gave a factor 6 approximation algorithm for MHS in this setting, and showed that the duality gap is between 3/2 and 6. In this paper we improve upon these results. First we show that MIS in diagonal-intersecting families is NP-complete, providing one smallest subclass for which MIS is provably hard. Then, we derive an $O(n^2)$-time algorithm for the maximum weight independent set when, in addition the rectangles intersect below the diagonal. This improves and extends a classic result of Lubiw, and amounts to obtain a 2-approximation algo- rithm for the maximum weight independent set of rectangles intersecting a diagonal. Finally, we prove that for diagonal-intersecting families the duality gap is between 2 and 4. The upper bound, which implies an approximation algorithm of the same factor, follows from a simple com- binatorial argument, while the lower bound represents the best known lower bound on the duality gap, even in the general case.

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arXiv2013

TSP Tours in Cubic Graphs: Beyond 4/3

After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic 2-connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2-connected graphs and matching the conjectured value of the famous 4/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4/3 - 1/61236)n, implying that cubic 2-connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4/3. With the previous result, and by considering an even smaller epsilon, we show that the integrality gap of the TSP relaxation is at most 4/3 - epsilon, even if the graph is not 2-connected (i.e. for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4/3 - 1/18)n.

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