arXiv subjects

Prahladh Harsha

Publications and source records attributed to Prahladh Harsha.

At least 37 records · Page 2Linked to original sources

arXiv2021

A Characterization of hard-to-cover CSPs

We continue the study of the covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, Håstad and Sudan [SIAM J. Comp. 2002] and Dinur and Kol [CCC'13]. The covering number of a CSP instance $Φ$ is the smallest number of assignments to the variables of $Φ$, such that each constraint of $Φ$ is satisfied by at least one of the assignments. We show the following results: 1. Assuming a covering variant of the Unique Games Conjecture, introduced by Dinur and Kol, we show that for every non-odd predicate $P$ over any constant-size alphabet and every integer $K$, it is NP-hard to approximate the covering number within a factor of $K$. This yields a complete characterization of CSPs over constant-size alphabets that are hard to cover. 2. For a large class of predicates that are contained in the 2k-LIN predicate, we show that it is quasi-NP-hard to distinguish between instances with covering number at most $2$ and those with covering number at least $Ω(\log\log n)$. This generalizes and improves the 4-LIN covering hardness result of Dinur and Kol.

cs.CC↗

arXiv2020

Agreement tests on graphs and hypergraphs

Agreement tests are a generalization of low degree tests that capture a local-to-global phenomenon, which forms the combinatorial backbone of most PCP constructions. In an agreement test, a function is given by an ensemble of local restrictions. The agreement test checks that the restrictions agree when they overlap, and the main question is whether average agreement of the local pieces implies that there exists a global function that agrees with most local restrictions. There are very few structures that support agreement tests, essentially either coming from algebraic low degree tests or from direct product tests (and recently also from high-dimensional expanders). In this work, we prove a new agreement theorem which extends direct product tests to higher dimensions, analogous to how low degree tests extend linearity testing. As a corollary of our main theorem, it follows that an ensemble of small graphs on overlapping sets of vertices can be glued together to one global graph assuming they agree with each other on average. We prove the agreement theorem by (re)proving the agreement theorem for dimension 1, and then generalizing it to higher dimensions (with the dimension 1 case being the direct product test, and dimension 2 being the graph case). A key technical step in our proof is the reverse union bound, which allows us to treat dependent events as if they are disjoint, and may be of independent interest. An added benefit of the reverse union bound is that it can be used to show that the "majority decoded" function also serves as a global function that explains the local consistency of the agreement theorem, a fact that was not known even in the direct product setting (dimension 1) prior to our work.

cs.CC↗

arXiv2020

Decoding Multivariate Multiplicity Codes on Product Sets

The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma to construct high-rate locally-decodable codes. However, the algorithmic results about these codes crucially rely on the fact that the polynomials are evaluated on a vector space and not an arbitrary product set. In this work, we show how to decode multivariate multiplicity codes of large multiplicities in polynomial time over finite product sets (over fields of large characteristic and zero characteristic). Previously such decoding algorithms were not known even for a positive fraction of errors. In contrast, our work goes all the way to the distance of the code and in particular exceeds both the unique decoding bound and the Johnson bound. For errors exceeding the Johnson bound, even combinatorial list-decodablity of these codes was not known. Our algorithm is an application of the classical polynomial method directly to the multivariate setting. In particular, we do not rely on a reduction from the multivariate to the univariate case as is typical of many of the existing results on decoding codes based on multivariate polynomials. However, a vanilla application of the polynomial method in the multivariate setting does not yield a polynomial upper bound on the list size. We obtain a polynomial bound on the list size by taking an alternative view of multivariate multiplicity codes. In this view, we glue all the partial derivatives of the same order together using a fresh set $z$ of variables. We then apply the polynomial method by viewing this as a problem over the field $\mathbb{F}(z)$ of rational functions in $z$.

cs.IT↗

arXiv2020

Rigid Matrices From Rectangular PCPs

We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the other determining the column. We construct PCPs that are efficient, short, smooth and (almost-)rectangular. As a key application, we show that proofs for hard languages in $NTIME(2^n)$, when viewed as matrices, are rigid infinitely often. This strengthens and simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit rigid matrices in FNP. Namely, we prove the following theorem: - There is a constant $δ\in (0,1)$ such that there is an FNP-machine that, for infinitely many $N$, on input $1^N$ outputs $N \times N$ matrices with entries in $\mathbb{F}_2$ that are $δN^2$-far (in Hamming distance) from matrices of rank at most $2^{\log N/Ω(\log \log N)}$. Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed--Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothness-inducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.

cs.CC↗

arXiv2020

COVID-19 Epidemic in Mumbai: Projections, full economic opening, and containment zones versus contact tracing and testing: An Update

Mumbai, amongst the most densely populated cities in the world, has witnessed the fourth largest number of cases and the largest number of deaths among all the cities in India (as of 28th October 2020). Along with the rest of India, lockdowns (of varying degrees) have been in effect in Mumbai since March 25, 2020. Given the large economic toll on the country from the lockdown and the related restrictions on mobility of people and goods, swift opening of the economy especially in a financial hub such as Mumbai becomes critical. In this report, we use the IISc-TIFR agent based simulator to develop long term projections for Mumbai under realistic scenarios related to Mumbai's opening of the workplaces, or equivalently, the economy, and the associated public transportation through local trains and buses. These projections were developed taking into account a possible second wave if the economy and the local trains are fully opened either on November 1, 2020 or on January 1, 2021. The impact on infection spread in Mumbai if the schools and colleges open on January first week 2021 is also considered. We also try to account for the increased intermingling amongst the population during the Ganeshotsav festival as well as around the Navratri/Dussehra and Diwali festival. Our conclusion, based on our simulations, is that the impact of fully opening up the economy on November 1 is manageable provided reasonable medical infrastructure is in place. Further, schools and colleges opening in January do not lead to excessive increase in infections. The report also explores the relative effectiveness of contact tracing vs containment zones, and also includes very rudimentary results of the effect of vaccinating the elderly population in February 2021.

physics.soc-ph↗

arXiv2020

On the Probabilistic Degree of OR over the Reals

We study the probabilistic degree over reals of the OR function on $n$ variables. For an error parameter $ε$ in (0,1/3), the $ε$-error probabilistic degree of any Boolean function $f$ over reals is the smallest non-negative integer $d$ such that the following holds: there exists a distribution $D$ of polynomials entirely supported on polynomials of degree at most $d$ such that for all $z \in \{0,1\}^n$, we have $Pr_{P \sim D} [P(z) = f(z) ] \geq 1- ε$. It is known from the works of Tarui ({Theoret. Comput. Sci.} 1993) and Beigel, Reingold, and Spielman ({ Proc. 6th CCC} 1991), that the $ε$-error probabilistic degree of the OR function is at most $O(\log n.\log 1/ε)$. Our first observation is that this can be improved to $O{\log {{n}\choose{\leq \log 1/ε}}}$, which is better for small values of $ε$. In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials $P$ in the support of the distribution $D$ have the following special structure:$P = 1 - (1-L_1).(1-L_2)...(1-L_t)$, where each $L_i(x_1,..., x_n)$ is a linear form in the variables $x_1,...,x_n$, i.e., the polynomial $1-P(x_1,...,x_n)$ is a product of affine forms. We show that the $ε$-error probabilistic degree of OR when restricted to polynomials of the above form is $Ω( \log a/\log^2 a )$ where $a = \log {{n}\choose{\leq \log 1/ε}}$. Thus matching the above upper bound (up to poly-logarithmic factors).

cs.CC↗

arXiv2020

Explicit SoS lower bounds from high-dimensional expanders

We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

cs.CC↗

arXiv2020

City-Scale Agent-Based Simulators for the Study of Non-Pharmaceutical Interventions in the Context of the COVID-19 Epidemic

We highlight the usefulness of city-scale agent-based simulators in studying various non-pharmaceutical interventions to manage an evolving pandemic. We ground our studies in the context of the COVID-19 pandemic and demonstrate the power of the simulator via several exploratory case studies in two metropolises, Bengaluru and Mumbai. Such tools become common-place in any city administration's tool kit in our march towards digital health.

q-bio.PE↗

arXiv2020

COVID-19 Epidemic Study II: Phased Emergence From the Lockdown in Mumbai

The nation-wide lockdown starting 25 March 2020, aimed at suppressing the spread of the COVID-19 disease, was extended until 31 May 2020 in three subsequent orders by the Government of India. The extended lockdown has had significant social and economic consequences and `lockdown fatigue' has likely set in. Phased reopening began from 01 June 2020 onwards. Mumbai, one of the most crowded cities in the world, has witnessed both the largest number of cases and deaths among all the cities in India (41986 positive cases and 1368 deaths as of 02 June 2020). Many tough decisions are going to be made on re-opening in the next few days. In an earlier IISc-TIFR Report, we presented an agent-based city-scale simulator(ABCS) to model the progression and spread of the infection in large metropolises like Mumbai and Bengaluru. As discussed in IISc-TIFR Report 1, ABCS is a useful tool to model interactions of city residents at an individual level and to capture the impact of non-pharmaceutical interventions on the infection spread. In this report we focus on Mumbai. Using our simulator, we consider some plausible scenarios for phased emergence of Mumbai from the lockdown, 01 June 2020 onwards. These include phased and gradual opening of the industry, partial opening of public transportation (modelling of infection spread in suburban trains), impact of containment zones on controlling infections, and the role of compliance with respect to various intervention measures including use of masks, case isolation, home quarantine, etc. The main takeaway of our simulation results is that a phased opening of workplaces, say at a conservative attendance level of 20 to 33\%, is a good way to restart economic activity while ensuring that the city's medical care capacity remains adequate to handle the possible rise in the number of COVID-19 patients in June and July.

q-bio.PE↗

arXiv2020

Locally testable codes via high-dimensional expanders

Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are "far" from all codewords by probing a given word only at a very few (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. A major open problem is whether there exist LTCs with positive rate, constant relative distance and testable with a constant number of queries. In this paper, we present a new approach towards constructing such LTCs using the machinery of high-dimensional expanders. To this end, we consider the Tanner representation of a code, which is specified by a graph and a base code. Informally, our result states that if this graph is part of a high-dimensional expander then the local testability of the code follows from the local testability of the base code. This work unifies and generalizes the known results on testability of the Hadamard, Reed-Muller and lifted codes on the Subspace Complex, all of which are proved via local self correction. However, unlike previous results, constant rounds of self correction do not suffice as the diameter of the underlying test graph can be logarithmically large in a high-dimensional expander and not constant as in all known earlier results. We overcome this technical hurdle by performing iterative self correction with logarithmically many rounds and tightly controlling the error in each iteration using properties of the high-dimensional expander. Given this result, the missing ingredient towards constructing a constant-query LTC with positive rate and constant relative distance is an instantiation of a base code that interacts well with a constant-degree high-dimensional expander.

cs.CC↗

arXiv2020

A note on the explicit constructions of tree codes over polylogarithmic-sized alphabet

Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudlák's construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular matrices. In this short note, we give a unified and simpler presentation of Pudlák and Cohen-Haeupler-Schulman's constructions.

cs.CC↗

arXiv2018

On Polynomial Approximations to ${AC}^0$

We make progress on some questions related to polynomial approximations of ${\rm AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC, 1991), that any ${\rm AC}^0$ circuit of size $s$ and depth $d$ has an $\varepsilon$-error probabilistic polynomial over the reals of degree $(\log (s/\varepsilon))^{O(d)}$. We improve this upper bound to $(\log s)^{O(d)}\cdot \log(1/\varepsilon)$, which is much better for small values of $\varepsilon$. We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that $(\log s)^{O(d)}\cdot \log(1/\varepsilon)$-wise independence fools ${\rm AC}^0$, improving on Tal's strengthening of Braverman's theorem (J. ACM, 2010) that $(\log (s/\varepsilon))^{O(d)}$-wise independence fools ${\rm AC}^0$. Up to the constant implicit in the $O(d)$, our result is tight. As far as we know, this is the first PRG construction for ${\rm AC}^0$ that achieves optimal dependence on the error $\varepsilon$. We also prove lower bounds on the best polynomial approximations to ${\rm AC}^0$. We show that any polynomial approximating the ${\rm OR}$ function on $n$ bits to a small constant error must have degree at least $\widetildeΩ(\sqrt{\log n})$. This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015).

cs.CC↗

arXiv2018

Robust Multiplication-based Tests for Reed-Muller Codes

We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is a codeword of the Reed-Muller code of dimension $n$ and order $d$ over the finite field $\mathbb{F}_q$ for prime $q$ (i.e., $f$ is the evaluation of a degree-$d$ polynomial over $\mathbb{F}_q$ for $q$ prime). * $\mathrm{Test}_{e,k}$: Pick $P_1,\ldots,P_k$ independent random degree-$e$ polynomials and accept iff the function $fP_1\cdots P_k$ is the evaluation of a degree-$(d+ek)$ polynomial (i.e., is a codeword of the Reed-Muller code of dimension $n$ and order $(d+ek)$). We prove the robust soundness of the above tests for large values of $e$, answering a question of Dinur and Guruswami [Israel Journal of Mathematics, 209:611-649, 2015]. Previous soundness analyses of these tests were known only for the case when either $e=1$ or $k=1$. Even for the case $k=1$ and $e>1$, earlier soundness analyses were not robust. We also analyze a derandomized version of this test, where (for example) the polynomials $P_1,\dots,P_k$ can be the same random polynomial $P$. This generalizes a result of Guruswami et al. [SIAM J. Comput., 46(1):132-159, 2017]. One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields $\mathbb{F}_q$, which may be of independent interest.

cs.CC↗

arXiv2018

On Multilinear Forms: Bias, Correlation, and Tensor Rank

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors. 1. Correlation bounds : We show that a random $d$-linear form has exponentially low correlation with low-degree polynomials. More precisely, for $d \ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \dots, X_d) : \left(GF(2)^{k}\right)^d \rightarrow GF(2)$ has correlation $2^{-k(1-o(1))}$ with any polynomial of degree at most $d/10$. This result is proved by giving near-optimal bounds on the bias of random $d$-linear form, which is in turn proved by giving near-optimal bounds on the probability that a random rank-$t$ $d$-linear form is identically zero. 2. Tensor-rank vs Bias : We show that if a $d$-dimensional tensor has small rank, then the bias of the associated $d$-linear form is large. More precisely, given any $d$-dimensional tensor $$T :\underbrace{[k]\times \ldots [k]}_{\text{$d$ times}}\to GF(2)$$ of rank at most $t$, the bias of the associated $d$-linear form $$f_T(X_1,\ldots,X_d) := \sum_{(i_1,\dots,i_d) \in [k]^d} T(i_1,i_2,\ldots, i_d) X_{1,i_1}\cdot X_{1,i_2}\cdots X_{d,i_d}$$ is at least $\left(1-\frac1{2^{d-1}}\right)^t$. The above bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds for $d=3$. In particular, we use this approach to prove that the finite field multiplication tensor has tensor rank at least $3.52 k$ matching the best known lower bound for any explicit tensor in three dimensions over $GF(2)$.

cs.CC↗

arXiv2017

Multiplayer parallel repetition for expander games

We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are connected in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.

cs.CC↗

arXiv2017

On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)} \pmod {2^k}$. We consider the fractional agreement (which we refer to as $γ_{d,k}(F)$) of $F_k$ with degree $d$ polynomials from $\mathbb{Z}/2^k\mathbb{Z}[x_1,\ldots,x_n]$. Our results are the following: - Increasing $k$ can help: We observe that as $k$ increases, $γ_{d,k}(F)$ cannot decrease. We give two kinds of examples where $γ_{d,k}(F)$ actually increases. The first is an infinite family of functions $F$ such that $γ_{2d,2}(F) - γ_{3d-1,1}(F) \geq Ω(1)$. The second is an infinite family of functions $F$ such that $γ_{d,1}(F)\leq\frac{1}{2}+o(1)$ -- as small as possible -- but $γ_{d,3}(F) \geq \frac{1}{2}+Ω(1)$. - Increasing $k$ doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16-38, 2000], we show that irrespective of the value of $k$, the Majority function $\mathrm{Maj}_n$ satisfies $γ_{d,k}(\mathrm{Maj}_n) \leq \frac{1}{2}+\frac{O(d)}{\sqrt{n}}$. In other words, polynomials over $\mathbb{Z}/2^k\mathbb{Z}$ for large $k$ do not approximate the majority function any better than polynomials over $\mathbb{Z}/2\mathbb{Z}$. We observe that the model we study subsumes the model of non-classical polynomials in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.

cs.CC↗

arXiv2016

Partition bound is quadratically tight for product distributions

Let $f : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a 2-party function. For every product distribution $μ$ on $\{0,1\}^n \times \{0,1\}^n$, we show that $$\mathsf{CC}^μ_{0.49}(f) = O\left(\left(\log \mathsf{prt}_{1/8}(f) \cdot \log \log \mathsf{prt}_{1/8}(f)\right)^2\right),$$ where $\mathsf{CC}^μ_\varepsilon(f)$ is the distributional communication complexity of $f$ with error at most $\varepsilon$ under the distribution $μ$ and $\mathsf{prt}_{1/8}(f)$ is the {\em partition bound} of $f$, as defined by Jain and Klauck [{\em Proc. 25th CCC}, 2010]. We also prove a similar bound in terms of $\mathsf{IC}_{1/8}(f)$, the {\em information complexity} of $f$, namely, $$\mathsf{CC}^μ_{0.49}(f) = O\left(\left(\mathsf{IC}_{1/8}(f) \cdot \log \mathsf{IC}_{1/8}(f)\right)^2\right).$$ The latter bound was recently and independently established by Kol [{\em Proc. 48th STOC}, 2016] using a different technique. We show a similar result for query complexity under product distributions. Let $g : \{0,1\}^n \rightarrow \{0,1\}$ be a function. For every bit-wise product distribution $μ$ on $\{0,1\}^n$, we show that $$\mathsf{QC}^μ_{0.49}(g) = O\left(\left( \log \mathsf{qprt}_{1/8}(g) \cdot \log \log\mathsf{qprt}_{1/8}(g) \right)^2 \right),$$ where $\mathsf{QC}^μ_{\varepsilon}(g)$ is the distributional query complexity of $f$ with error at most $\varepsilon$ under the distribution $μ$ and $\mathsf{qprt}_{1/8}(g))$ is the {\em query partition bound} of the function $g$. Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for {\em product} distributions.

cs.CC↗

arXiv2015

Embedding approximately low-dimensional $\ell_2^2$ metrics into $\ell_1$

Goemans showed that any $n$ points $x_1, \dotsc x_n$ in $d$-dimensions satisfying $\ell_2^2$ triangle inequalities can be embedded into $\ell_{1}$, with worst-case distortion at most $\sqrt{d}$. We extend this to the case when the points are approximately low-dimensional, albeit with average distortion guarantees. More precisely, we give an $\ell_{2}^{2}$-to-$\ell_{1}$ embedding with average distortion at most the stable rank, $\mathrm{sr}(M)$, of the matrix $M$ consisting of columns $\{x_i-x_j\}_{i<j}$. Average distortion embedding suffices for applications such as the Sparsest Cut problem. Our embedding gives an approximation algorithm for the \sparsestcut problem on low threshold-rank graphs, where earlier work was inspired by Lasserre SDP hierarchy, and improves on a previous result of the first and third author [Deshpande and Venkat, In Proc. 17th APPROX, 2014]. Our ideas give a new perspective on $\ell_{2}^{2}$ metric, an alternate proof of Goemans' theorem, and a simpler proof for average distortion $\sqrt{d}$. Furthermore, while the seminal result of Arora, Rao and Vazirani giving a $O(\sqrt{\log n})$ guarantee for Uniform Sparsest Cut can be seen to imply Goemans' theorem with average distortion, our work opens up the possibility of proving such a result directly via a Goemans'-like theorem.

cs.DS↗