arXiv subjects

Alexandra Kolla

Publications and source records attributed to Alexandra Kolla.

At least 19 recordsLinked to original sources

arXiv2024

Monogamy of Entanglement Bounds and Improved Approximation Algorithms for Qudit Hamiltonians

We prove new monogamy of entanglement bounds for two-local qudit Hamiltonians of rank-one projectors without one-local terms. In particular, we certify the maximum energy in terms of the maximum matching of the underlying interaction graph via low-degree sum-of-squares proofs. Algorithmically, we show that a simple matching-based algorithm approximates the maximum energy to at least $1/d$ for general graphs and to at least $1/d + \Theta(1/D)$ for graphs with bounded degree, $D$. This outperforms random assignment, which, in expectation, achieves energy of only $1/d^2$ of the maximum energy for general graphs. Notably, on $D$-regular graphs with degree, $D \leq 5$, and for any local dimension, $d$, we show that this simple matching-based algorithm has an approximation guarantee of $1/2$. Lastly, when $d=2$, we present an algorithm achieving an approximation guarantee of $0.595$, beating that of [PT22, arXiv:2206.08342], which gave an approximation ratio of $1/2$.

quant-ph↗

arXiv2023

Approximation Algorithms for Quantum Max-$d$-Cut

We initiate the algorithmic study of the Quantum Max-$d$-Cut problem, a quantum generalization of the well-known Max-$d$-Cut problem. The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, $d$-dimensional qudits over all local interactions. Equivalently, this problem is physically motivated by the $SU(d)$-Heisenberg model, a spin glass model that generalized the well-known Heisenberg model over qudits. We develop a polynomial-time randomized approximation algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees. Moreover, we prove the tightness of our analysis by presenting an algorithmic gap instance for Quantum Max-d-Cut problem with $d \geq 3$.

quant-ph↗

arXiv2023

Approximately counting independent sets in dense bipartite graphs via subspace enumeration

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph -- in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to ``high-temperature'' problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. The proof exploits the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e. bounded threshold rank), which via subspace enumeration lets us enumerate small cuts efficiently.

cs.DS↗

arXiv2023

A quantum advantage over classical for local max cut

We compare the performance of a quantum local algorithm to a similar classical counterpart on a well-established combinatorial optimization problem LocalMaxCut. We show that a popular quantum algorithm first discovered by Farhi, Goldstone, and Gutmannn [1] called the quantum optimization approximation algorithm (QAOA) has a computational advantage over comparable local classical techniques on degree-3 graphs. These results hint that even small-scale quantum computation, which is relevant to the current state-of the art quantum hardware, could have significant advantages over comparably simple classical computation.

quant-ph↗

arXiv2022

Algorithms for the ferromagnetic Potts model on expanders

We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger's algorithm to count cuts that may be of independent interest. It is \#BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of \#BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high temperature proof follows more standard polymer model analysis, our result holds in the largest known range of parameters $d$ and $q$.

cs.DS↗

arXiv2021

Computational thresholds for the fixed-magnetization Ising model

The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair. Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins). In particular, letting $\beta_c(\Delta)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $\Delta$-regular tree, and $\eta_{\Delta,\beta,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $\Delta$-regular tree at inverse temperature $\beta$, we prove, for the class of graphs of maximum degree $\Delta$: 1. For $\beta < \beta_c(\Delta)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $\eta$. 2. For $\beta > \beta_c(\Delta)$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $\eta$ such that $|\eta| >\eta_{\Delta,\beta,1}^+ $. 3. For $\beta > \beta_c(\Delta)$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $\eta$ such that $|\eta| <\eta_{\Delta,\beta,1}^+ $ unless NP=RP\@.

cs.DS↗

arXiv2020

Efficient algorithms for the Potts model on small-set expanders

An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.

cs.DS↗

arXiv2019

Statistical physics approaches to Unique Games

We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would refute the Unique Games Conjecture.

cs.DS↗

arXiv2018

Lower bounds for Max-Cut in $H$-free graphs via semidefinite programming

For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming (SDP), to prove lower bounds on $f(G)$. We use this approach to find large cuts in graphs with few triangles and in $K_r$-free graphs.

cs.DS↗

arXiv2018

Spectrally Robust Graph Isomorphism

We initiate the study of spectral generalizations of the graph isomorphism problem. (a)The Spectral Graph Dominance (SGD) problem: On input of two graphs $G$ and $H$ does there exist a permutation $\pi$ such that $G\preceq \pi(H)$? (b) The Spectrally Robust Graph Isomorphism (SRGI) problem: On input of two graphs $G$ and $H$, find the smallest number $\kappa$ over all permutations $\pi$ such that $ \pi(H) \preceq G\preceq \kappa c \pi(H)$ for some $c$. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology. Here $G\preceq c H$ means that for all vectors $x$ we have $x^T L_G x \leq c x^T L_H x$, where $L_G$ is the Laplacian $G$. We prove NP-hardness for SGD. We also present a $\kappa$-approximation algorithm for SRGI for the case when both $G$ and $H$ are bounded-degree trees. The algorithm runs in polynomial time when $\kappa$ is a constant.

cs.DS↗

arXiv2017

Optimal Lower Bounds for Sketching Graph Cuts

We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must use $\Omega(n\log n/\epsilon^2)$ bits of space in the worst case, improving the $\Omega(n/\epsilon^2)$ bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two $d-$regular graphs which approximate each other's cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.

cs.DS↗

arXiv2016

Invertibility and Largest Eigenvalue of Symmetric Matrix Signings

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral properties. Our results are twofold: 1. We show NP-completeness for the following three problems: verifying whether a given matrix has a symmetric signing that is positive semi-definite/singular/has bounded eigenvalues. However, we also illustrate that the complexity could substantially differ for input matrices that are adjacency matrices of graphs. 2. We exhibit a stark contrast between invertibility and the above-mentioned spectral properties: we show a combinatorial characterization of matrices with invertible symmetric signings and design an efficient algorithm using this characterization to verify whether a given matrix has an invertible symmetric signing. Next, we give an efficient algorithm to solve the search problem of finding an invertible symmetric signing for matrices whose support graph is bipartite. We also provide a lower bound on the number of invertible symmetric signed adjacency matrices. Finally, we give an efficient algorithm to find a minimum increase in support of a given symmetric matrix so that it has an invertible symmetric signing. We use combinatorial and spectral techniques in addition to classic results from matching theory. Our combinatorial characterization of matrices with invertible symmetric signings might be of independent interest.

cs.DM↗

arXiv2015

Multisection in the Stochastic Block Model using Semidefinite Programming

We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on $k$-clusters: a random model on $n=km$ vertices, partitioned in $k$ equal sized clusters, with edges sampled independently across clusters with probability $q$ and within clusters with probability $p$, $p>q$. The goal is to recover the initial "hidden" partition of $[n]$. We study semidefinite programming (SDP) based algorithms in this context. In the regime $p = \frac{α\log(m)}{m}$ and $q = \frac{β\log(m)}{m}$ we show that a certain natural SDP based algorithm solves the problem of {\em exact recovery} in the $k$-community SBM, with high probability, whenever $\sqrtα - \sqrtβ > \sqrt{1}$, as long as $k=o(\log n)$. This threshold is known to be the information theoretically optimal. We also study the case when $k=θ(\log(n))$. In this case however we achieve recovery guarantees that no longer match the optimal condition $\sqrtα - \sqrtβ > \sqrt{1}$, thus leaving achieving optimality for this range an open question.

cs.DS↗

arXiv2015

Approximation of non-boolean 2CSP

We develop a polynomial time $Ω\left ( \frac 1R \log R \right)$ approximate algorithm for Max 2CSP-$R$, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size $R$, and we want to find an assignment to the variables that maximizes the number of satisfied constraints. Assuming the Unique Games Conjecture, this is the best possible approximation up to constant factors. Previously, a $1/R$-approximate algorithm was known, based on linear programming. Our algorithm is based on semidefinite programming (SDP) and on a novel rounding technique. The SDP that we use has an almost-matching integrality gap.

cs.DS↗

arXiv2014

Dimension-Free $L^p$-Maximal Inequalities in $\mathbb{Z}_{m+1}^N$

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the $L^1$-normalized indicator of the $r$-sphere, \[ σ_r := \frac{1}{|\{|x| = r\}|} 1_{\{|x| =r\}}.\] We study the $L^p \to L^p$ mapping properties of the maximal operator \[ M^{N} f (x) := \sup_{r \leq N} | σ_r*f| \] acting on functions defined on $\mathbb{Z}_{m+1}^N$. Specifically, we prove that for all $p>1$, there exist absolute constants $C_{m,p}$ so that \[ \| M^{N} f \|_{L^p(\mathbb{Z}_{m+1}^N)} \leq C_{m,p} \| f \|_{L^p(\mathbb{Z}_{m+1}^N)} \] for all $N$.

math.CA↗

arXiv2014

Unique Games on the Hypercube

In this paper, we investigate the validity of the Unique Games Conjecture when the constraint graph is the boolean hypercube. We construct an almost optimal integrality gap instance on the Hypercube for the Goemans-Williamson semidefinite program (SDP) for Max-2-LIN$(\mathbb{Z}_2)$. We conjecture that adding triangle inequalities to the SDP provides a polynomial time algorithm to solve Unique Games on the hypercube.

cs.CC↗

arXiv2014

High Throughput Data Center Topology Design

With high throughput networks acquiring a crucial role in supporting data-intensive applications, a variety of data center network topologies have been proposed to achieve high capacity at low cost. While this literature explores a large number of design points, even in the limited case of a network of identical switches, no proposal has been able to claim any notion of optimality. The case of heterogeneous networks, incorporating multiple line-speeds and port-counts as data centers grow over time, introduces even greater complexity. In this paper, we present the first non-trivial upper-bound on network throughput under uniform traffic patterns for any topology with identical switches. We then show that random graphs achieve throughput surprisingly close to this bound, within a few percent at the scale of a few thousand servers. Apart from demonstrating that homogeneous topology design may be reaching its limits, this result also motivates our use of random graphs as building blocks to explore the design of heterogeneous networks. Given a heterogeneous pool of network switches, through experiments and analysis, we explore how the distribution of servers across switches and the interconnection of switches affect network throughput. We apply these insights to a real-world heterogeneous data center topology, VL2, demonstrating as much as 43% higher throughput with the same equipment.

cs.NI↗

arXiv2014

Measuring and Understanding Throughput of Network Topologies

High throughput is of particular interest in data center and HPC networks. Although myriad network topologies have been proposed, a broad head-to-head comparison across topologies and across traffic patterns is absent, and the right way to compare worst-case throughput performance is a subtle problem. In this paper, we develop a framework to benchmark the throughput of network topologies, using a two-pronged approach. First, we study performance on a variety of synthetic and experimentally-measured traffic matrices (TMs). Second, we show how to measure worst-case throughput by generating a near-worst-case TM for any given topology. We apply the framework to study the performance of these TMs in a wide range of network topologies, revealing insights into the performance of topologies with scaling, robustness of performance across TMs, and the effect of scattered workload placement. Our evaluation code is freely available.

cs.NI↗