arXiv subjects

Thiago Oliveira

Publications and source records attributed to Thiago Oliveira.

4 recordsLinked to original sources

arXiv2026

On some k-fold generalizations of Lovász theta and their sandwich theorems

We study several $k$-fold generalizations of the Lovász theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan--Manber parameter $\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $φ_k$ obtained from orthonormal representations of graphs and show the inequality $φ_k \leq \vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\vartheta_k$, $φ_k$, $\vartheta(G\square K_k)$, and related convexifications.

math.CO↗

arXiv2026

If it is Good Then Drop it -- a Spiteful Poisson Process for Submodular Maximization

We study the problem of maximizing a general and not necessarily monotone submodular function subject to a matroid independence constraint. This problem has a rich history, with multiple algorithms using both discrete and continuous methods. Recently, [Ganz-Rozenman, Kulik, Schwartz and Singh STOC `26] presented a novel hybrid approach based on a Poisson process that aims to combine the strengths of both discrete and continuous methods for the special case of the problem where the submodular function is monotone. Our main result is a new Poisson process based hybrid algorithm that works for both non-monotone and monotone submodular functions, achieving an approximation of $ \frac{1}{e}$ for the former and $1-\frac{1}{e}$ for the latter. The algorithm always maintains a feasible set and at random times governed by the Poisson process it performs a single element swap based on a best response set. The new idea is that our algorithm is spiteful as it can purposefully discard an element that is in both the current set and the best response set. Surprisingly, this spiteful step does not harm the approximation our algorithm achieves for monotone submodular functions but is necessary for the non-monotone case. As applications, we obtain fast approximation algorithms for maximizing non-monotone submodular function subject to a general matroid independence constraint as well as faster algorithms for a partition matroid.

cs.DS↗

arXiv2026

Philosopher and Prophet Inequalities for Divisible Items

We study online welfare maximization with divisible resources. A sequence of $n$ players arrive one by one; upon arrival, each player draws a valuation function over $m$ divisible items from a known distribution, reveals this valuation, and must be allocated an irrevocable fractional bundle subject to unit supply constraints. While online welfare maximization has been extensively studied for indivisible items and combinatorial valuations, much less is known when the resources are divisible and players have multi-dimensional concave valuations. We give approximation algorithms for monotone concave valuations satisfying diminishing returns. Our main result is a $2/3$-approximation to the optimal online policy, also known as the philosopher benchmark. The algorithm is guided by a low-dimensional concave relaxation of the online benchmark and rounds it via a new single-item capped online contention resolution scheme. This Capped-OCRS problem allocates to each realized type no more than its prescribed fractional bundle while preserving a $2/3$-fraction of that bundle in expectation. Its analysis uses a submartingale potential for the remaining side, we show that computing the optimal online policy is #P-hard even for a single divisible item. We also obtain a tight prophet inequality against the offline hindsight optimum. We show that a fixed-price auction with one linear per-unit price for each original divisible item achieves a $1/2$-approximation to the offline/prophet benchmark. The prices are obtained by aggregating Aumann--Shapley supporting prices, a continuous analogue of supporting prices for submodular/XOS set functions, and yield simple item prices rather than copy-dependent prices arising from discretization. The factor $1/2$ for the prophet benchmark is information-theoretically tight even for one item with linear valuations.

cs.DS↗

arXiv2025

A connection between the $A_α$-spectrum and the Lovász theta number

We show that the smallest $α$ so that $αD + (1-α)A \succcurlyeq 0$ is at least $1/\vartheta(\overline{G})$, significantly improving upon a result due to Nikiforov and Rojo (2017). In fact, we display an even stronger connection: if the nonzero entries of $A$ are allowed to vary and those of $D$ vary accordingly, then we show that this smallest $α$ is in fact equal to $1/\vartheta(\overline{G})$. We also show other results obtained as an application of this optimization framework, including a connection to the well-known quadratic formulation for $ω(G)$ due to Motzkin and Straus (1964).

math.CO↗