Search

Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning. The current best results include an $e^{2k}$-estimation for any matroid of rank $k$ and a $(1+\epsilon)^d$-approximation for a uniform matroid of rank $k\ge d+\frac d\epsilon$, where the rank $k\ge d$ denotes the desired size of the optimal set. Our main result is a new approximation algorithm with an approximation guarantee that depends only on the dimension $d$ of the vectors and not on the size $k$ of the output set. In particular, we show an $(O(d))^{d}$-estimation and an $(O(d))^{d^3}$-approximation for any matroid, giving a significant improvement over prior work when $k\gg d$. Our result relies on the existence of an optimal solution to a convex programming relaxation for the problem which has sparse support; in particular, no more than $O(d^2)$ variables of the solution have fractional values. The sparsity results rely on the interplay between the first-order optimality conditions for the convex program and matroid theory. We believe that the techniques introduced to show sparsity of optimal solutions to convex programs will be of independent interest. We also give a randomized algorithm that rounds a sparse fractional solution to a feasible integral solution to the original problem. To show the approximation guarantee, we utilize recent works on strongly log-concave polynomials and show new relationships between different convex programs studied for the problem. Finally, we use the estimation algorithm and sparsity results to give an efficient deterministic approximation algorithm with an approximation guarantee that depends solely on the dimension $d$.

We give a characterization result for the integrality gap of the natural linear programming relaxation for the vertex cover problem. We show that integrality gap of the standard linear programming relaxation for any graph G equals $\left(2-\frac{2}{\chi^f(G)}\right)$ where $\chi^f(G)$ denotes the fractional chromatic number of G., Comment: 6 pages

Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the "multi-criteria dimensionality reduction" problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as our novel Fair-PCA problem and the Nash Social Welfare (NSW) problem. In Fair-PCA, the input data is divided into $k$ groups, and the goal is to find a single $d$-dimensional representation for all groups for which the minimum variance of any one group is maximized. In NSW, the goal is to maximize the product of the individual variances of the groups achieved by the common low-dimensional space. Our main result is an exact polynomial-time algorithm for the two-criterion dimensionality reduction problem when the two criteria are increasing concave functions. As an application of this result, we obtain a polynomial time algorithm for Fair-PCA for $k=2$ groups and a polynomial time algorithm for NSW objective for $k=2$ groups. We also give approximation algorithms for $k>2$. Our technical contribution in the above results is to prove new low-rank properties of extreme point solutions to semi-definite programs. We conclude with experiments indicating the effectiveness of algorithms based on extreme point solutions of semi-definite programs on several real-world data sets., Comment: The preliminary version appeared in NeurIPS2019. This version combines the motivation from "The Price of Fair PCA: One Extra Dimension" (NeurIPS 2018) by the same set of author, adds new a motivation, and introduces new heuristics and more experimental results

In this work, we consider robust submodular maximization with matroid constraints. We give an efficient bi-criteria approximation algorithm that outputs a small family of feasible sets whose union has (nearly) optimal objective value. This algorithm theoretically performs less function calls than previous works at cost of adding more elements to the final solution. We also provide significant implementation improvements showing that our algorithm outperforms the algorithms in the existing literature. We finally assess the performance of our contributions in three real-world applications.

We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a $1/e$ approximation factor of the objective. Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree $m$-homogeneous stable polynomial on $m$ variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo.

Tur\'an problems in extremal combinatorics ask to find asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful methods based on semidefinite programming to find sums of squares that establish edge density inequalities in Tur\'an problems. Working with polynomial analogs of the flag algebra entities, we prove that such sums of squares created by flag algebras can be retrieved from a restricted version of the symmetry-adapted semidefinite program proposed by Gatermann and Parrilo. This involves using the representation theory of the symmetric group for finding succinct sums of squares expressions for invariant polynomials. The connection reveals several combinatorial and structural properties of flag algebra sums of squares, and offers new tools for Tur\'an and other related problems., Comment: 28 pages; changed some notation and definitions to better match the literature; added a conclusion

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$.

In this paper we study the problem of computing max-entropy distributions over a discrete set of objects subject to observed marginals. Interest in such distributions arises due to their applicability in areas such as statistical physics, economics, biology, information theory, machine learning, combinatorics and, more recently, approximation algorithms. A key difficulty in computing max-entropy distributions has been to show that they have polynomially-sized descriptions. We show that such descriptions exist under general conditions. Subsequently, we show how algorithms for (approximately) counting the underlying discrete set can be translated into efficient algorithms to (approximately) compute max-entropy distributions. In the reverse direction, we show how access to algorithms that compute max-entropy distributions can be used to count, which establishes an equivalence between counting and computing max-entropy distributions.