Welcome to Combinatorics and Optimization

Title:Combinatorial rules for the geometry of Hessenberg varieties

progressions

 Abstract:

Hessenberg varieties were introduced by De Mari, Procesi, and Shayman in the early 1990s and lie at the intersection of geometry, representation theory, and combinatorics.  In 2012, Insko and Yong studied a class of Hessenberg varieties using patch ideals, a technique dating back to at least the 1970s from the study of Schubert varieties. In this talk, we will derive patch ideals and use them to study two classes of Hessenberg varieties.  We will see the combinatorics that govern the behaviour of these patch ideals and translate these results to the geometric setting. Based in part on work with Sergio Da Silva, Megumi Harada, and Jenna Rajchgot.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: A constant factor approximation for Nash social welfare with subadditive valuations

Abstract:Social welfare refers to a class of optimization problems where the goal is to allocate subsets of resources $\mathcal{I}$ among agents $\mathcal{A}$ (or people) such that maximizes the overall "happiness" of society in a fair and efficient manner. More specifically, each agent $i \in \mathcal{I}$ has an intrinsic \emph{valuation} function $v_i: 2^{\mathcal{I}}\rightarrow \mathbb{R}$, which is a monotone function with $v_i(\emptyset)=0$, and $v_i$ quantifies the intrinsic value for subsets of items $S\subseteq \mathcal{I}$.

Variations of allocation with different valuation and objective functions has been studied in different areas of computer science, economies, and game theory. In this talk we focus on the Nash social welfare welfare (NSW) problem; given an allocation $\mathcal{S}= (S_i)_{i\in \mathcal{A}}$ the goal is to maximize the geometric mean of agents valuations.

Unfortunately, Nash social welfare problem is NP-hard already in the case of two agents with identical additive valuations, and it is NP-hard to approximate within a factor better than 0.936 for additive valuations and $(1-\frac{1}{e})$ for submodular valuation. 

Moreover, the current best approximation factors of $\simeq 0.992$ for additive valuations and $(\frac{1}{4}-\epsilon)$ for submodular valuations were found by Barman et al (2018) and Garg et al. (2023), respectively.

In this talk, we present a sketch of the algorithm in recent work by Dobzinski et al., which proves the first constant-factor approximation algorithm (with a fairly large constant $\sim \frac{1}{375,000}$) for NSW problem with subadditive valuations accessible via demand queries.

Title: Inapproximability of Sparsest Vector in a Real Subspace

Abstract:We establish strong inapproximability for finding the sparsest nonzero vector in a real subspace (where sparsity refers to the number of nonzero entries). Formally we show that it is NP-Hard (under randomized reductions) to approximate the sparsest vector in a subspace within any constant factor. We recover as a corollary state of the art inapproximability factors for the shortest vector problem (SVP), a foundational problem in lattice based cryptography. Our proof is surprisingly simple, bypassing even the PCP theorem.

Our main motivation in this work is the development of inapproximability techniques for problems over the reals. Analytic variants of sparsest vector have connections to small set expansion, quantum separability and polynomial maximization over convex sets, all of which cause similar barriers to inapproximability. The approach we develop could lead to progress on the hardness of some of these problems.

Joint work with Euiwoong Lee.