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Determinant maximization provides an elegant generalization of problems in many areas, including convex geometry, statistics, machine learning, fair allocation of goods, and network design. In an instance of the determinant maximization problem, we are given a collection of vectors $v_1,\ldots, v_n \in \mathbb{R}^d$, and the goal is to pick a subset $S\subseteq [n]$ of given vectors to maximize the determinant of the matrix $\sum_{i \in S} v_iv_i^\top$, where the picked set of vectors $S$ must satisfy some combinatorial constraint such as cardinality constraint ($|S| \leq k$) or matroid constraint ($S$ is a basis of a matroid defined on $[n]$). In this work, we give a combinatorial algorithm for the determinant maximization problem under a matroid constraint that achieves $O(d^{O(d)})$-approximation for any matroid of rank $r\geq d$. This complements the recent result of~\cite{BrownLPST22} that achieves a similar bound for matroids of rank $r\leq d$, relying on a geometric interpretation of the determinant. Our result matches the best-known estimation algorithms~\cite{madan2020maximizing} for the problem, which could estimate the objective value but could not give an approximate solution with a similar guarantee. Our work follows the framework developed by~\cite{BrownLPST22} of using matroid intersection based algorithms for determinant maximization. To overcome the lack of a simple geometric interpretation of the objective when $r \geq d$, our approach combines ideas from combinatorial optimization with algebraic properties of the determinant. We also critically use the properties of a convex programming relaxation of the problem introduced by~\cite{madan2020maximizing}.

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 $d$.

A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a well-studied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its "combinatorial shadow". We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famous Blakley-Roy inequalities for paths of odd length. As a consequence, these simple inequalities cannot be written as a rational sum of squares of graph densities., Comment: 29 pages, corrected small typos and changed the exposition of Lemma 2.2

Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. A standard tool to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this paper, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares. These results answer two questions raised by Lov\'asz. Our main tool is used again to show that the smallest open case of Sidorenko's conjectured inequality cannot be certified by a sum of squares. Finally, we show that our setup is equivalent to existing frameworks by Razborov and Lov\'asz-Szegedy, and thus our results hold in these settings too., Comment: 12 pages

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.

We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the $k$-element subsets of $[n]$. For simplicity, we focus on the case $k=2$, but our results extend naturally to all values of $k \geq 2$. We develop a variant of the Gatermann-Parrilo symmetry-reduction method tailored to our setting that allows for several simplifications and a connection to flag algebras. We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. Our method bypasses much of the technical difficulties needed to apply the Gatermann-Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by Razborov in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting., Comment: 32 pages; corrected typo in acknowledgments

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.

In the k-arc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least k-arc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)-approximation to the unweighted variant of the problem, where all arcs of G have the same cost. This improves on the 1 + 2/k approximation of Gabow et al. [GGTW09]. Similar to the 2-approximation algorithm for this problem [FJ81], our algorithm simply takes the union of a k in-arborescence and a k out-arborescence. The main difference is in the selection of the two arborescences. Here, inspired by the recent applications of the rounding by sampling method (see e.g. [AGM+ 10, MOS11, OSS11, AKS12]), we select the arborescences randomly by sampling from a distribution on unions of k arborescences that is defined based on an extreme point solution of the linear programming relaxation of the problem. In the analysis, we crucially utilize the sparsity property of the extreme point solution to upper-bound the size of the union of the sampled arborescences. To complement the algorithm, we also show that the integrality gap of the minimum cost strongly connected subgraph problem (i.e., when k = 1) is at least 3/2 - c, for any c > 0. Our integrality gap instance is inspired by the integrality gap example of the asymmetric traveling salesman problem [CGK06], hence providing further evidence of connections between the approximability of the two problems., Comment: 10 pages, 2 figures, ICALP2012