arXiv subjects

Suprovat Ghoshal

Publications and source records attributed to Suprovat Ghoshal.

At least 19 recordsLinked to original sources

arXiv2026

Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP

We describe an $\widetildeΩ(1/d^4)$-improvement over threshold rounding schemes for a broad class of Boolean MAX 2-CSP instances in which every variable appears in at most $d$ constraints. In the case of MAX 2-SAT, we improve the ratio further and obtain an $(β_\star + \widetildeΩ(1/d^2))$-factor approximation algorithm for bounded-degree MAX 2-SAT instances, where $β_\star$ is the UGC-optimal approximation ratio for MAX 2-SAT achieved by the LLZ algorithm. Our result generalizes an $(α_{GW} + \widetildeΩ(1/d^2))$-factor approximation algorithm for MAX CUT on graphs with degrees bounded by $d$, due to Hsieh and Kothari. Together with the state-of-the-art approximability results for MAX DI-CUT and MAX 2-AND, our result suggests that similar improvements exist for bounded-degree instances of these problems as well.

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arXiv2026

Max Cut with Small-Dimensional SDP Solutions

We study the Max-Cut semidefinite programming (SDP) relaxation in the regime where a near-optimal solution admits a low-dimensional realization. While the Goemans--Williamson hyperplane rounding achieves the worst-case optimal approximation ratio $α_{GW}\approx 0.87856$, it is natural to ask whether one can beat $α_{GW}$ when the SDP solution lives in $\mathbb{R}^d$ for a small dimension $d$. We answer this in the affirmative for every fixed $d$: there is a polynomial-time rounding algorithm that, given a $d$-dimensional feasible solution to the standard Max-Cut SDP strengthened with triangle inequalities, produces a cut of expected value at least $(α_{GW}+2^{-O(d)})$ times the SDP value. Our improvement is driven by a new geometric anti-concentration lemma for signs of low-dimensional Gaussian projections.

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arXiv2026

On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets

Max-Cut is a classical graph-partitioning problem where given a graph $G = (V,E)$, the objective is to find a cut $(S,S^c)$ which maximizes the number of edges crossing the cut. In a seminal work, Goemans and Williamson gave an $α_{GW} \approx 0.87856$-factor approximation algorithm for the problem, which was later shown to be tight by the work of Khot, Kindler, Mossel, and O'Donnell. Since then, there has been a steady progress in understanding the approximability at even finer levels, and a fundamental goal in this context is to understand how the structure of the underlying graph affects the approximability of the Max-Cut problem. In this work, we investigate this question by exploring how the chromatic structure of a graph affects the Max-Cut problem. While it is well-known that Max-Cut can be solved perfectly and near-perfectly in $2$-colorable and almost $2$-colorable graphs in polynomial time, here we explore its approximability under much weaker structural conditions such as when the graph is $3$-colorable or contains a large independent set. Our main contributions in this context are as follows: 1. We show Max-Cut is $α_{GW}$-hard to approximate for $3$-colorable graphs. 2. We identify a natural threshold $α^*$ such that the following holds. Firstly, for graphs which contain an independent set of size up to $α^*$, Max-Cut continues to be $α_{GW}$-factor hard to approximate. Furthermore, for any graph that contains an independent set of size $> α^*$, there exists an efficient $>α_{GW}$-approximation algorithm for Max-Cut. Our hardness results are derived using various analytical tools and novel variants of the Majority-Is-Stablest theorem, which might be of independent interest. Our algorithmic results are based on a novel SDP relaxation, which is then rounded and analyzed using interval arithmetic.

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arXiv2024

Constraint Satisfaction Problems with Advice

We initiate the study of algorithms for constraint satisfaction problems with ML oracle advice. We introduce two models of advice and then design approximation algorithms for Max Cut, Max $2$-Lin, and Max $3$-Lin in these models. In particular, we show the following. 1. For Max-Cut and Max $2$-Lin, we design an algorithm that yields near-optimal solutions when the average degree is larger than a threshold degree, which only depends on the amount of advice and is independent of the instance size. We also give an algorithm for nearly satisfiable Max $3$-Lin instances with quantitatively similar guarantees. 2. Further, we provide impossibility results for algorithms in these models. In particular, under standard complexity assumptions, we show that Max $3$-Lin is still $1/2 + η$ hard to approximate given access to advice, when there are no assumptions on the instance degree distribution. Additionally, we also show that Max $4$-Lin is $1/2 + η$ hard to approximate even when the average degree of the instance is linear in the number of variables.

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arXiv2023

New Approximation Bounds for Small-Set Vertex Expansion

The vertex expansion of the graph is a fundamental graph parameter. Given a graph $G=(V,E)$ and a parameter $δ\in (0,1/2]$, its $δ$-Small-Set Vertex Expansion (SSVE) is defined as \[ \min_{S : |S| = δ|V|} \frac{|{\partial^V(S)}|}{ \min \{ |S|, |S^c| \} } \] where $\partial^V(S)$ is the vertex boundary of a set $S$. The SSVE~problem, in addition to being of independent interest as a natural graph partitioning problem, is also of interest due to its connections to the Strong Unique Games problem. We give a randomized algorithm running in time $n^{{\sf poly}(1/δ)}$, which outputs a set $S$ of size $Θ(δn)$, having vertex expansion at most \[ \max\left(O(\sqrt{ϕ^* \log d \log (1/δ)}) , \tilde{O}(d\log^2(1/δ)) \cdot ϕ^* \right), \] where $d$ is the largest vertex degree of the graph, and $ϕ^*$ is the optimal $δ$-SSVE. The previous best-known guarantees for this were the bi-criteria bounds of $\tilde{O}(1/δ)\sqrt{ϕ^* \log d}$ and $\tilde{O}(1/δ)ϕ^* \sqrt{\log n}$ due to Louis-Makarychev [TOC'16]. Our algorithm uses the basic SDP relaxation of the problem augmented with ${\rm poly}(1/δ)$ rounds of the Lasserre/SoS hierarchy. Our rounding algorithm is a combination of the rounding algorithms of Raghavendra-Tan [SODA'12] and Austrin-Benabbas-Georgiou [SODA'13]. A key component of our analysis is novel Gaussian rounding lemma for hyperedges which might be of independent interest.

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arXiv2023

A PTAS for $\ell_0$-Low Rank Approximation: Solving Dense CSPs over Reals

We consider the Low Rank Approximation problem, where the input consists of a matrix $A \in \mathbb{R}^{n_R \times n_C}$ and an integer $k$, and the goal is to find a matrix $B$ of rank at most $k$ that minimizes $\| A - B \|_0$, which is the number of entries where $A$ and $B$ differ. For any constant $k$ and $\varepsilon > 0$, we present a polynomial time $(1 + \varepsilon)$-approximation time for this problem, which significantly improves the previous best $poly(k)$-approximation. Our algorithm is obtained by viewing the problem as a Constraint Satisfaction Problem (CSP) where each row and column becomes a variable that can have a value from $\mathbb{R}^k$. In this view, we have a constraint between each row and column, which results in a {\em dense} CSP, a well-studied topic in approximation algorithms. While most of previous algorithms focus on finite-size (or constant-size) domains and involve an exhaustive enumeration over the entire domain, we present a new framework that bypasses such an enumeration in $\mathbb{R}^k$. We also use tools from the rich literature of Low Rank Approximation in different objectives (e.g., $\ell_p$ with $p \in (0, \infty)$) or domains (e.g., finite fields/generalized Boolean). We believe that our techniques might be useful to study other real-valued CSPs and matrix optimization problems. On the hardness side, when $k$ is part of the input, we prove that Low Rank Approximation is NP-hard to approximate within a factor of $Ω(\log n)$. This is the first superconstant NP-hardness of approximation for any $p \in [0, \infty]$ that does not rely on stronger conjectures (e.g., the Small Set Expansion Hypothesis).

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arXiv2023

On Lifting Integrality Gaps to SSEH Hardness for Globally Constrained CSPs

A $μ$-constrained Boolean Max-CSP$(ψ)$ instance is a Boolean Max-CSP instance on predicate $ψ:\{0,1\}^r \to \{0,1\}$ where the objective is to find a labeling of relative weight exactly $μ$ that maximizes the fraction of satisfied constraints. In this work, we study the approximability of constrained Boolean Max-CSPs via SDP hierarchies by relating the integrality gap of Max-CSP $(ψ)$ to its $μ$-dependent approximation curve. Formally, assuming the Small-Set Expansion Hypothesis, we show that it is NP-hard to approximate $μ$-constrained instances of Max-CSP($ψ$) up to factor ${\sf Gap}_{\ell,μ}(ψ)/\log(1/μ)^2$ (ignoring factors depending on $r$) for any $\ell \geq \ell(μ,r)$. Here, ${\sf Gap}_{\ell,μ}(ψ)$ is the optimal integrality gap of $\ell$-round Lasserre relaxation for $μ$-constrained Max-CSP($ψ$) instances. Our results are derived by combining the framework of Raghavendra [STOC 2008] along with more recent advances in rounding Lasserre relaxations and reductions from the Small-Set Expansion (SSE) problem. A crucial component of our reduction is a novel way of composing generic bias-dependent dictatorship tests with SSE, which could be of independent interest.

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arXiv2022

Approximating CSPs with Outliers

Constraint satisfaction problems (CSPs) are ubiquitous in theoretical computer science. We study the problem of StrongCSPs, i.e. instances where a large induced sub-instance has a satisfying assignment. More formally, given a CSP instance $Ψ(V, E, [k], \{Π_{ij}\}_{(i,j) \in E})$ consisting of a set of vertices $V$, a set of edges $E$, alphabet $[k]$, a constraint $Π_{ij} \subset [k] \times [k]$ for each $(i,j) \in E$, the goal of this problem is to compute the largest subset $S \subseteq V$ such that the instance induced on $S$ has an assignment that satisfies all the constraints. In this paper, we study approximation algorithms for Unique Games and related problems under the StrongCSP framework when the underlying constraint graph satisfies mild expansion properties. In particular, we show that given a Strong Unique Games instance whose optimal solution $S^*$ is supported on a regular low threshold rank graph, there exists an algorithm that runs in time exponential in the threshold rank, and recovers a large satisfiable sub-instance whose size is independent on the label set size and maximum degree of the graph. Our algorithm combines the techniques of Barak-Raghavendra-Steurer (FOCS'11), Guruswami-Sinop (FOCS'11) with several new ideas and runs in time exponential in the threshold rank of the optimal set. A key component of our algorithm is a new threshold rank based spectral decomposition, which is used to compute a "large" induced subgraph of "small" threshold rank; our techniques build on the work of Oveis Gharan and Rezaei (SODA'17) and could be of independent interest.

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arXiv2022

Tight Approximation Bounds for Maximum Multi-Coverage

In the classic maximum coverage problem, we are given subsets $T_1, \dots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := |\cup_{i \in S} T_i|$. It is well-known that the greedy algorithm for this problem achieves an approximation ratio of $(1-e^{-1})$ and there is a matching inapproximability result. We note that in the maximum coverage problem if an element $e \in [n]$ is covered by several sets, it is still counted only once. By contrast, if we change the problem and count each element $e$ as many times as it is covered, then we obtain a linear objective function, $C^{(\infty)}(S) = \sum_{i \in S} |T_i|$, which can be easily maximized under a cardinality constraint. We study the maximum $\ell$-multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted up to $\ell$ times but no more; hence, we consider maximizing the function $C^{(\ell)}(S) = \sum_{e \in [n]} \min\{\ell, |\{i \in S : e \in T_i\}| \}$, subject to the constraint $|S| \leq k$. Note that the case of $\ell = 1$ corresponds to the standard maximum coverage setting and $\ell = \infty$ gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of $1 - \frac{\ell^{\ell}e^{-\ell}}{\ell!}$ for the $\ell$-multi-coverage problem. In particular, when $\ell = 2$, this factor is $1-2e^{-2} \approx 0.73$ and as $\ell$ grows the approximation ratio behaves as $1 - \frac{1}{\sqrt{2π\ell}}$. We also prove that this approximation ratio is tight, i.e., establish a matching hardness-of-approximation result, under the Unique Games Conjecture.

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arXiv2022

Exploiting Correlation to Achieve Faster Learning Rates in Low-Rank Preference Bandits

We introduce the \emph{Correlated Preference Bandits} problem with random utility-based choice models (RUMs), where the goal is to identify the best item from a given pool of $n$ items through online subsetwise preference feedback. We investigate whether models with a simple correlation structure, e.g. low rank, can result in faster learning rates. While we show that the problem can be impossible to solve for the general `low rank' choice models, faster learning rates can be attained assuming more structured item correlations. In particular, we introduce a new class of \emph{Block-Rank} based RUM model, where the best item is shown to be $(ε,δ)$-PAC learnable with only $O(r ε^{-2} \log(n/δ))$ samples. This improves on the standard sample complexity bound of $\tilde{O}(nε^{-2} \log(1/δ))$ known for the usual learning algorithms which might not exploit the item-correlations ($r \ll n$). We complement the above sample complexity with a matching lower bound (up to logarithmic factors), justifying the tightness of our analysis. Surprisingly, we also show a lower bound of $Ω(nε^{-2}\log(1/δ))$ when the learner is forced to play just duels instead of larger subsetwise queries. Further, we extend the results to a more general `\emph{noisy Block-Rank}' model, which ensures robustness of our techniques. Overall, our results justify the advantage of playing subsetwise queries over pairwise preferences $(k=2)$, we show the latter provably fails to exploit correlation.

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arXiv2022

A Characterization of Approximability for Biased CSPs

A $μ$-biased Max-CSP instance with predicate $ψ:\{0,1\}^r \to \{0,1\}$ is an instance of Constraint Satisfaction Problem (CSP) where the objective is to find a labeling of relative weight at most $μ$ which satisfies the maximum fraction of constraints. Biased CSPs are versatile and express several well studied problems such as Densest-$k$-Sub(Hyper)graph and SmallSetExpansion. In this work, we explore the role played by the bias parameter $μ$ on the approximability of biased CSPs. We show that the approximability of such CSPs can be characterized (up to loss of factors of arity $r$) using the bias-approximation curve of Densest-$k$-SubHypergraph (DkSH). In particular, this gives a tight characterization of predicates which admit approximation guarantees that are independent of the bias parameter $μ$. Motivated by the above, we give new approximation and hardness results for DkSH. In particular, assuming the Small Set Expansion Hypothesis (SSEH), we show that DkSH with arity $r$ and $k = μn$ is NP-hard to approximate to a factor of $Ω(r^3μ^{r-1}\log(1/μ))$ for every $r \geq 2$ and $μ< 2^{-r}$. We also give a $O(μ^{r-1}\log(1/μ))$-approximation algorithm for the same setting. Our upper and lower bounds are tight up to constant factors, when the arity $r$ is a constant, and in particular, imply the first tight approximation bounds for the Densest-$k$-Subgraph problem in the linear bias regime. Furthermore, using the above characterization, our results also imply matching algorithms and hardness for every biased CSP of constant arity.

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arXiv2020

Approximation Algorithms and Hardness for Strong Unique Games

The UNIQUE GAMES problem is a central problem in algorithms and complexity theory. Given an instance of UNIQUE GAMES, the STRONG UNIQUE GAMES problem asks to find the largest subset of vertices, such that the UNIQUE GAMES instance induced on them is completely satisfiable. In this paper, we give new algorithmic and hardness results for STRONG UNIQUE GAMES. Given an instance with label set size $k$ where a set of $(1 - ε)$-fraction of the vertices induce an instance that is completely satisfiable, our first algorithm produces a set of $1 - \widetilde{O}({k^2}) ε\sqrt{\log n}$ fraction of the vertices such that the UNIQUE GAMES induced on them is completely satisfiable. In the same setting, our second algorithm produces a set of $1 - \widetilde{O}({k^2}) \sqrt{ε\log d}$ (here $d$ is the largest vertex degree of the graph) fraction of the vertices such that the UNIQUE GAMES induced on them is completely satisfiable. The technical core of our results is a new connection between STRONG UNIQUE GAMES and Small-Set-Vertex-Expansion in graphs. Complementing this, assuming the Unique Games Conjecture, we prove that it is NP-hard to compute a set of size larger than $1 - Ω( \sqrt{ε\log k \log d})$ for which all the constraints induced on this set are satisfied. Given an undirected graph $G(V,E)$ the ODD CYCLE TRANSVERSAL problem asks to delete the least fraction of vertices to make the induced graph on the remaining vertices bipartite. As a corollary to our main algorithmic results, we obtain an algorithm that outputs a set $S$ such the graph induced on $V \setminus S$ is bipartite, and $|S|/n \leq O(\sqrt{ε\log d})$ (here $d$ is the largest vertex degree and $ε$ is the optimal fraction of vertices that need to be deleted). Assuming the Unique Games Conjecture, we prove a matching (up to constant factors) hardness.

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arXiv2019

Combinatorial lower bounds for 3-query LDCs

A code is called a $q$-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index $i$ and a received word $w$ close to an encoding of a message $x$, outputs $x_i$ by querying only at most $q$ coordinates of $w$. Understanding the tradeoffs between the dimension, length and query complexity of LDCs is a fascinating and unresolved research challenge. In particular, for $3$-query binary LDCs of dimension $k$ and length $n$, the best known bounds are: $2^{k^{o(1)}} \geq n \geq \tildeΩ(k^2)$. In this work, we take a second look at binary $3$-query LDCs. We investigate a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query LDCs. We prove an upper bound on the number of edges in these hypergraphs, reproducing the known lower bound of $\tildeΩ(k^2)$ for the length of strong $3$-query LDCs. In contrast to previous work, our techniques are purely combinatorial and do not rely on a direct reduction to $2$-query LDCs, opening up a potentially different approach to analyzing 3-query LDCs.

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arXiv2019

Hardness of Learning DNFs using Halfspaces

The problem of learning $t$-term DNF formulas (for $t = O(1)$) has been studied extensively in the PAC model since its introduction by Valiant (STOC 1984). A $t$-term DNF can be efficiently learnt using a $t$-term DNF only if $t = 1$ i.e., when it is an AND, while even weakly learning a $2$-term DNF using a constant term DNF was shown to be NP-hard by Khot and Saket (FOCS 2008). On the other hand, Feldman et al. (FOCS 2009) showed the hardness of weakly learning a noisy AND using a halfspace -- the latter being a generalization of an AND, while Khot and Saket (STOC 2008) showed that an intersection of two halfspaces is hard to weakly learn using any function of constantly many halfspaces. The question of whether a $2$-term DNF is efficiently learnable using $2$ or constantly many halfspaces remained open. In this work we answer this question in the negative by showing the hardness of weakly learning a $2$-term DNF as well as a noisy AND using any function of a constant number of halfspaces. In particular we prove the following. For any constants $ν, ζ> 0$ and $\ell \in \mathbb{N}$, given a distribution over point-value pairs $\{0,1\}^n \times \{0,1\}$, it is NP-hard to decide whether, YES Case: There is a $2$-term DNF that classifies all the points of the distribution, and an AND that classifies at least $1-ζ$ fraction of the points correctly. NO Case: Any boolean function depending on at most $\ell$ halfspaces classifies at most $1/2 + ν$ fraction of the points of the distribution correctly. Our result generalizes and strengthens the previous best results mentioned above on the hardness of learning a $2$-term DNF, learning an intersection of two halfspaces, and learning a noisy AND.

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arXiv2019

Parameterized Intractability of Even Set and Shortest Vector Problem

The $k$-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over $\mathbb F_2$, which can be stated as follows: given a generator matrix $\mathbf A$ and an integer $k$, determine whether the code generated by $\mathbf A$ has distance at most $k$, or in other words, whether there is a nonzero vector $\mathbf{x}$ such that $\mathbf A \mathbf{x}$ has at most $k$ nonzero coordinates. The question of whether $k$-Even Set is fixed parameter tractable (FPT) parameterized by the distance $k$ has been repeatedly raised in literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows (1999). In this work, we show that $k$-Even Set is W[1]-hard under randomized reductions. We also consider the parameterized $k$-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer $k$, and the goal is to determine whether the norm of the shortest vector (in the $\ell_p$ norm for some fixed $p$) is at most $k$. Similar to $k$-Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any $p > 1$, $k$-SVP is W[1]-hard to approximate (under randomized reductions) to some constant factor.

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arXiv2019

Approximation Algorithms for Partially Colorable Graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $α\leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is $α$-partially $k$-colorable, if there exists a subset $S\subset V$ of cardinality $ |S | \geq α| V |$ such that the graph induced on $S$ is $k$-colorable. Partial $k$-colorability is a more robust structural property of a graph than $k$-colorability. For graphs that arise in practice, partial $k$-colorability might be a better notion to use than $k$-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a $(1 - ε)$-partially $3$-colorable graph $G$ and a constant $γ\in [ε, 1/10]$, and colors a $(1 - ε/γ)$ fraction of the vertices using $\tilde{O}\left(n^{0.25 + O(γ^{1/2})} \right)$ colors. We also study natural semi-random families of instances of partially $3$-colorable graphs and partially $2$-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances.

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arXiv2019

Average Bias and Polynomial Sources

We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1\}^n$, its average bias is: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim Z}(-1)^{\langle c, z\rangle}|$. A source with average bias at most $2^{-k}$ has min-entropy at least $k$, and so low average bias is a stronger condition than high min-entropy. We observe that the inner product function is an extractor for any source with average bias less than $2^{-n/2}$. The notion of average bias especially makes sense for polynomial sources, i.e., distributions sampled by low-degree $n$-variate polynomials over $\mathbb{F}_2$. For the well-studied case of affine sources, it is easy to see that min-entropy $k$ is exactly equivalent to average bias of $2^{-k}$. We show that for quadratic sources, min-entropy $k$ implies that the average bias is at most $2^{-Ω(\sqrt{k})}$. We use this relation to design dispersers for separable quadratic sources with a min-entropy guarantee.

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arXiv2018

Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH

The $k$-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over $\mathbb F_2$, which can be stated as follows: given a generator matrix $\mathbf A$ and an integer $k$, determine whether the code generated by $\mathbf A$ has distance at most $k$. Here, $k$ is the parameter of the problem. The question of whether $k$-Even Set is fixed parameter tractable (FPT) has been repeatedly raised in literature and has earned its place in Downey and Fellows' book (2013) as one of the "most infamous" open problems in the field of Parameterized Complexity. In this work, we show that $k$-Even Set does not admit FPT algorithms under the (randomized) Gap Exponential Time Hypothesis (Gap-ETH) [Dinur'16, Manurangsi-Raghavendra'16]. In fact, our result rules out not only exact FPT algorithms, but also any constant factor FPT approximation algorithms for the problem. Furthermore, our result holds even under the following weaker assumption, which is also known as the Parameterized Inapproximability Hypothesis (PIH) [Lokshtanov et al.'17]: no (randomized) FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $0.99$-satisfiable (where the parameter is the number of variables). We also consider the parameterized $k$-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer $k$, and the goal is to determine whether the norm of the shortest vector (in the $\ell_p$ norm for some fixed $p$) is at most $k$. Similar to $k$-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity. We show that, for any $p > 1$, $k$-SVP is hard to approximate (in FPT time) to some constant factor, assuming PIH. Furthermore, for the case of $p = 2$, the inapproximability factor can be amplified to any constant.

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