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Euiwoong Lee

Publications and source records attributed to Euiwoong Lee.

At least 55 records · Page 3Linked to original sources

arXiv2023

Handling Correlated Rounding Error via Preclustering: A 1.73-approximation for Correlation Clustering

We consider the classic Correlation Clustering problem: Given a complete graph where edges are labelled either $+$ or $-$, the goal is to find a partition of the vertices that minimizes the number of the \pedges across parts plus the number of the \medges within parts. Recently, Cohen-Addad, Lee and Newman [CLN22] presented a 1.994-approximation algorithm for the problem using the Sherali-Adams hierarchy, hence breaking through the integrality gap of 2 for the classic linear program and improving upon the 2.06-approximation of Chawla, Makarychev, Schramm and Yaroslavtsev [CMSY15]. We significantly improve the state-of-the-art by providing a 1.73-approximation for the problem. Our approach introduces a preclustering of Correlation Clustering instances that allows us to essentially ignore the error arising from the {\em correlated rounding} used by [CLN22]. This additional power simplifies the previous algorithm and analysis. More importantly, it enables a new {\em set-based rounding} that complements the previous roundings. A combination of these two rounding algorithms yields the improved bound.

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arXiv2023

Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove the following results for any sufficiently large $d$: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{d}{2} - o(d)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{d}{3} - o(d)\right)$. Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish the following hardness results for approximating Maximum Independent Set on $k$-claw-free graphs: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{k}{4} - o(k)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{k}{3 + 2\sqrt{2}} - o(k)\right) \geq \left(\frac{k}{5.829} - o(k)\right)$. In comparison, known approximation algorithms achieve $\left(\frac{k}{2} - o(k)\right)$-approximation in polynomial time [Neuwohner, STACS 2021; Thiery and Ward, SODA 2023] and $(\frac{k}{3} + o(k))$-approximation in quasi-polynomial time [Cygan et al., SODA 2013].

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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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arXiv2023

Correlation Clustering with Sherali-Adams

Given a complete graph $G = (V, E)$ where each edge is labeled $+$ or $-$, the Correlation Clustering problem asks to partition $V$ into clusters to minimize the number of $+$edges between different clusters plus the number of $-$edges within the same cluster. Correlation Clustering has been used to model a large number of clustering problems in practice, making it one of the most widely studied clustering formulations. The approximability of Correlation Clustering has been actively investigated [BBC04, CGW05, ACN08], culminating in a $2.06$-approximation algorithm [CMSY15], based on rounding the standard LP relaxation. Since the integrality gap for this formulation is 2, it has remained a major open question to determine if the approximation factor of 2 can be reached, or even breached. In this paper, we answer this question affirmatively by showing that there exists a $(1.994 + ε)$-approximation algorithm based on $O(1/ε^2$) rounds of the Sherali-Adams hierarchy. In order to round a solution to the Sherali-Adams relaxation, we adapt the {\em correlated rounding} originally developed for CSPs [BRS11, GS11, RT12]. With this tool, we reach an approximation ratio of $2+ε$ for Correlation Clustering. To breach this ratio, we go beyond the traditional triangle-based analysis by employing a global charging scheme that amortizes the total cost of the rounding across different triangles.

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arXiv2022

A Local Search-Based Approach for Set Covering

In the Set Cover problem, we are given a set system with each set having a weight, and we want to find a collection of sets that cover the universe, whilst having low total weight. There are several approaches known (based on greedy approaches, relax-and-round, and dual-fitting) that achieve a $H_k \approx \ln k + O(1)$ approximation for this problem, where the size of each set is bounded by $k$. Moreover, getting a $\ln k - O(\ln \ln k)$ approximation is hard. Where does the truth lie? Can we close the gap between the upper and lower bounds? An improvement would be particularly interesting for small values of $k$, which are often used in reductions between Set Cover and other combinatorial optimization problems. We consider a non-oblivious local-search approach: to the best of our knowledge this gives the first $H_k$-approximation for Set Cover using an approach based on local-search. Our proof fits in one page, and gives a integrality gap result as well. Refining our approach by considering larger moves and an optimized potential function gives an $(H_k - Ω(\log^2 k)/k)$-approximation, improving on the previous bound of $(H_k - Ω(1/k^8))$ (\emph{R.\ Hassin and A.\ Levin, SICOMP '05}) based on a modified greedy algorithm.

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arXiv2022

Fitting Metrics and Ultrametrics with Minimum Disagreements

Given $x \in (\mathbb{R}_{\geq 0})^{\binom{[n]}{2}}$ recording pairwise distances, the METRIC VIOLATION DISTANCE (MVD) problem asks to compute the $\ell_0$ distance between $x$ and the metric cone; i.e., modify the minimum number of entries of $x$ to make it a metric. Due to its large number of applications in various data analysis and optimization tasks, this problem has been actively studied recently. We present an $O(\log n)$-approximation algorithm for MVD, exponentially improving the previous best approximation ratio of $O(OPT^{1/3})$ of Fan et al. [ SODA, 2018]. Furthermore, a major strength of our algorithm is its simplicity and running time. We also study the related problem of ULTRAMETRIC VIOLATION DISTANCE (UMVD), where the goal is to compute the $\ell_0$ distance to the cone of ultrametrics, and achieve a constant factor approximation algorithm. The UMVD can be regarded as an extension of the problem of fitting ultrametrics studied by Ailon and Charikar [SIAM J. Computing, 2011] and by Cohen-Addad et al. [FOCS, 2021] from $\ell_1$ norm to $\ell_0$ norm. We show that this problem can be favorably interpreted as an instance of Correlation Clustering with an additional hierarchical structure, which we solve using a new $O(1)$-approximation algorithm for correlation clustering that has the structural property that it outputs a refinement of the optimum clusters. An algorithm satisfying such a property can be considered of independent interest. We also provide an $O(\log n \log \log n)$ approximation algorithm for weighted instances. Finally, we investigate the complementary version of these problems where one aims at choosing a maximum number of entries of $x$ forming an (ultra-)metric. In stark contrast with the minimization versions, we prove that these maximization versions are hard to approximate within any constant factor assuming the Unique Games Conjecture.

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arXiv2022

Improved Approximation Algorithms and Lower Bounds for Search-Diversification Problems

We study several questions related to diversifying search results. We give improved approximation algorithms in each of the following problems, together with some lower bounds. - We give a polynomial-time approximation scheme (PTAS) for a diversified search ranking problem [Bansal et al., ICALP 2010] whose objective is to minimizes the discounted cumulative gain. Our PTAS runs in time $n^{2^{O(\log(1/ε)/ε)}} \cdot m^{O(1)}$ where $n$ denotes the number of elements in the databases. Complementing this, we show that no PTAS can run in time $f(ε) \cdot (nm)^{2^{o(1/ε)}}$ assuming Gap-ETH; therefore our running time is nearly tight. Both of our bounds answer open questions of Bansal et al. - We next consider the Max-Sum Dispersion problem, whose objective is to select $k$ out of $n$ elements that maximizes the dispersion, which is defined as the sum of the pairwise distances under a given metric. We give a quasipolynomial-time approximation scheme for the problem which runs in time $n^{O_ε(\log n)}$. This improves upon previously known polynomial-time algorithms with approximate ratios 0.5 [Hassin et al., Oper. Res. Lett. 1997; Borodin et al., ACM Trans. Algorithms 2017]. Furthermore, we observe that known reductions rule out approximation schemes that run in $n^{\tilde{o}_ε(\log n)}$ time assuming ETH. - We consider a generalization of Max-Sum Dispersion called Max-Sum Diversification. In addition to the sum of pairwise distance, the objective includes another function $f$. For monotone submodular $f$, we give a quasipolynomial-time algorithm with approximation ratio arbitrarily close to $(1 - 1/e)$. This improves upon the best polynomial-time algorithm which has approximation ratio $0.5$ by Borodin et al. Furthermore, the $(1 - 1/e)$ factor is tight as achieving better-than-$(1 - 1/e)$ approximation is NP-hard [Feige, J. ACM 1998].

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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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arXiv2021

On Some Variants of Euclidean K-Supplier

The $k$-Supplier problem is an important location problem that has been actively studied in both general and Euclidean metrics. Many of its variants have also been studied, primarily on general metrics. We study two variants of $k$-Supplier, namely Priority $k$-Supplier and $k$-Supplier with Outliers, in Euclidean metrics. We obtain $(1+\sqrt{3})$-approximation algorithms for both variants, which are the first improvements over the previously-known factor-$3$ approximation (that is known to be best-possible for general metrics). We also study the Matroid Supplier problem on Euclidean metrics, and show that it cannot be approximated to a factor better than $3$ (assuming $P\ne NP$); so the Euclidean metric offers no improvement in this case.

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arXiv2021

Strong Hardness of Approximation for Tree Transversals

Let $H$ be a fixed graph. The $H$-Transversal problem, given a graph $G$, asks to remove the smallest number of vertices from $G$ so that $G$ does not contain $H$ as a subgraph. While a simple $|V(H)|$-approximation algorithm exists and is believed to be tight for every $2$-vertex-connected $H$, the best hardness of approximation for any tree was $Ω(\log |V(H)|)$-inapproximability when $H$ is a star. In this paper, we identify a natural parameter $Δ$ for every tree $T$ and show that $T$-Transversal is NP-hard to approximate within a factor $(Δ- 1 -\varepsilon)$ for an arbitrarily small constant $\varepsilon > 0$. As a corollary, we prove that there exists a tree $T$ such that $T$-Transversal is NP-hard to approximate within a factor $Ω(|V(T)|)$, exponentially improving the best known hardness of approximation for tree transversals.

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arXiv2021

Johnson Coverage Hypothesis: Inapproximability of k-means and k-median in L_p metrics

K-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives in $\ell_p$-metrics. We introduce a new hypothesis called the Johnson Coverage Hypothesis (JCH), which roughly asserts that the well-studied max k-coverage problem on set systems is hard to approximate to a factor greater than 1-1/e, even when the membership graph of the set system is a subgraph of the Johnson graph. We then show that together with generalizations of the embedding techniques introduced by Cohen-Addad and Karthik (FOCS '19), JCH implies hardness of approximation results for k-median and k-means in $\ell_p$-metrics for factors which are close to the ones obtained for general metrics. In particular, assuming JCH we show that it is hard to approximate the k-means objective: $\bullet$ Discrete case: To a factor of 3.94 in the $\ell_1$-metric and to a factor of 1.73 in the $\ell_2$-metric; this improves upon the previous factor of 1.56 and 1.17 respectively, obtained under UGC. $\bullet$ Continuous case: To a factor of 2.10 in the $\ell_1$-metric and to a factor of 1.36 in the $\ell_2$-metric; this improves upon the previous factor of 1.07 in the $\ell_2$-metric obtained under UGC. We also obtain similar improvements under JCH for the k-median objective. Additionally, we prove a weak version of JCH using the work of Dinur et al. (SICOMP '05) on Hypergraph Vertex Cover, and recover all the results stated above of Cohen-Addad and Karthik (FOCS '19) to (nearly) the same inapproximability factors but now under the standard NP$\neq$P assumption (instead of UGC).

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arXiv2021

Optimal Bounds for the $k$-cut Problem

In the $k$-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other hand, lower bounds from conjectures about the $k$-clique problem imply that $Ω(n^{(1-o(1))k})$ time is likely needed. Recent results of Gupta, Lee \& Li have given new algorithms for general $k$-cut in $n^{1.98k + O(1)}$ time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed $k$-cut of weight $αλ_k$ with probability $Ω_k(n^{-αk})$, where $λ_k$ denotes the minimum $k$-cut weight. This also gives an extremal bound of $O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute $λ_k$ with roughly $n^k \mathrm{polylog}(n)$ runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight $k$-clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than $2 λ_k/k$, using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process.

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arXiv2021

CoCoS: Fast and Accurate Distributed Triangle Counting in Graph Streams

Given a graph stream, how can we estimate the number of triangles in it using multiple machines with limited storage? Specifically, how should edges be processed and sampled across the machines for rapid and accurate estimation? The count of triangles (i.e., cliques of size three) has proven useful in numerous applications, including anomaly detection, community detection, and link recommendation. For triangle counting in large and dynamic graphs, recent work has focused largely on streaming algorithms and distributed algorithms but little on their combinations for "the best of both worlds". In this work, we propose CoCoS, a fast and accurate distributed streaming algorithm for estimating the counts of global triangles (i.e., all triangles) and local triangles incident to each node. Making one pass over the input stream, COCOS carefully processes and stores the edges across multiple machines so that the redundant use of computational and storage resources is minimized. Compared to baselines, CoCoS is (a) Accurate: giving up to 39X smaller estimation error, (b) Fast: up to 10.4X faster, scaling linearly with the size of the input stream, and (c) Theoretically sound: yielding unbiased estimates.

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arXiv2021

A PTAS for $\ell_p$-Low Rank Approximation

A number of recent works have studied algorithms for entrywise $\ell_p$-low rank approximation, namely, algorithms which given an $n \times d$ matrix $A$ (with $n \geq d$), output a rank-$k$ matrix $B$ minimizing $\|A-B\|_p^p=\sum_{i,j}|A_{i,j}-B_{i,j}|^p$ when $p > 0$; and $\|A-B\|_0=\sum_{i,j}[A_{i,j}\neq B_{i,j}]$ for $p=0$. On the algorithmic side, for $p \in (0,2)$, we give the first $(1+ε)$-approximation algorithm running in time $n^{\text{poly}(k/ε)}$. Further, for $p = 0$, we give the first almost-linear time approximation scheme for what we call the Generalized Binary $\ell_0$-Rank-$k$ problem. Our algorithm computes $(1+ε)$-approximation in time $(1/ε)^{2^{O(k)}/ε^{2}} \cdot nd^{1+o(1)}$. On the hardness of approximation side, for $p \in (1,2)$, assuming the Small Set Expansion Hypothesis and the Exponential Time Hypothesis (ETH), we show that there exists $δ:= δ(α) > 0$ such that the entrywise $\ell_p$-Rank-$k$ problem has no $α$-approximation algorithm running in time $2^{k^δ}$.

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arXiv2020

Inapproximability for Local Correlation Clustering and Dissimilarity Hierarchical Clustering

We present hardness of approximation results for Correlation Clustering with local objectives and for Hierarchical Clustering with dissimilarity information. For the former, we study the local objective of Puleo and Milenkovic (ICML '16) that prioritizes reducing the disagreements at data points that are worst off and for the latter we study the maximization version of Dasgupta's cost function (STOC '16). Our APX hardness results imply that the two problems are hard to approximate within a constant of 4/3 ~ 1.33 (assuming P vs NP) and 9159/9189 ~ 0.9967 (assuming the Unique Games Conjecture) respectively.

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arXiv2020

On Approximability of Clustering Problems Without Candidate Centers

The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this paper, we significantly improve upon the hardness of approximation factors known in the literature for these objectives. We show that if the input lies in a general metric space, it is NP-hard to approximate: $\bullet$ Continuous k-median to a factor of $2-o(1)$; this improves upon the previous inapproximability factor of 1.36 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ Continuous k-means to a factor of $4- o(1)$; this improves upon the previous inapproximability factor of 2.10 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ k-minsum to a factor of $1.415$; this improves upon the APX-hardness shown by Guruswami and Indyk (SODA '03). Our results shed new and perhaps counter-intuitive light on the differences between clustering problems in the continuous setting versus the discrete setting (where the candidate centers are given as part of the input).

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arXiv2020

Towards constant-factor approximation for chordal / distance-hereditary vertex deletion

For a family of graphs $\mathcal{F}$, Weighted $\mathcal{F}$-Deletion is the problem for which the input is a vertex weighted graph $G=(V,E)$ and the goal is to delete $S\subseteq V$ with minimum weight such that $G\setminus S\in\mathcal{F}$. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when $F$ is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.

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arXiv2020

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.

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