arXiv subjects

Kent Quanrud

Publications and source records attributed to Kent Quanrud.

31 records · Page 2Linked to original sources

arXiv2019

On Approximating Partial Set Cover and Generalizations

Partial Set Cover (PSC) is a generalization of the well-studied Set Cover problem (SC). In PSC the input consists of an integer $k$ and a set system $(U,S)$ where $U$ is a finite set, and $S \subseteq 2^U$ is a collection of subsets of $U$. The goal is to find a subcollection $S' \subseteq S$ of smallest cardinality such that sets in $S'$ cover at least $k$ elements of $U$; that is $|\cup_{A \in S'} A| \ge k$. SC is a special case of PSC when $k = |U|$. In the weighted version each set $X \in S$ has a non-negative weight $w(X)$ and the goal is to find a minimum weight subcollection to cover $k$ elements. Approximation algorithms for SC have been adapted to obtain comparable algorithms for PSC in various interesting cases. In recent work Inamdar and Varadarajan, motivated by geometric set systems, obtained a simple and elegant approach to reduce PSC to SC via the natural LP relaxation. They showed that if a deletion-closed family of SC admits a $β$-approximation via the natural LP relaxation, then one can obtain a $2(β+ 1)$-approximation for PSC on the same family. In a subsequent paper, they also considered a generalization of PSC that has multiple partial covering constraints which is partly inspired by and generalizes previous work of Bera et al on the Vertex Cover problem. Our main goal in this paper is to demonstrate some useful connections between the results in previous work and submodularity. This allows us to simplify, and in some cases improve their results. We improve the approximation for PSC to $(1-1/e)(β+ 1)$. We extend the previous work to the sparse setting.

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arXiv2019

$\ell_1$-sparsity Approximation Bounds for Packing Integer Programs

We consider approximation algorithms for packing integer programs (PIPs) of the form $\max\{\langle c, x\rangle : Ax \le b, x \in \{0,1\}^n\}$ where $c$, $A$, and $b$ are nonnegative. We let $W = \min_{i,j} b_i / A_{i,j}$ denote the width of $A$ which is at least $1$. Previous work by Bansal et al. \cite{bansal-sparse} obtained an $Ω(\frac{1}{Δ_0^{1/\lfloor W \rfloor}})$-approximation ratio where $Δ_0$ is the maximum number of nonzeroes in any column of $A$ (in other words the $\ell_0$-column sparsity of $A$). They raised the question of obtaining approximation ratios based on the $\ell_1$-column sparsity of $A$ (denoted by $Δ_1$) which can be much smaller than $Δ_0$. Motivated by recent work on covering integer programs (CIPs) \cite{cq,chs-16} we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. \cite{bansal-sparse} (but with a twist), yield approximation ratios for PIPs based on $Δ_1$. First, following an integrality gap example from \cite{bansal-sparse}, we observe that the case of $W=1$ is as hard as maximum independent set even when $Δ_1 \le 2$. In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width $W = 1 + ε$ where $ε\in (0,1]$, we obtain an $Ω(ε^2/Δ_1)$-approximation. In the large width regime, when $W \ge 2$, we obtain an $Ω((\frac{1}{1 + Δ_1/W})^{1/(W-1)})$-approximation. We also obtain a $(1-ε)$-approximation when $W = Ω(\frac{\log (Δ_1/ε)}{ε^2})$.

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arXiv2018

Parallelizing greedy for submodular set function maximization in matroids and beyond

We consider parallel, or low adaptivity, algorithms for submodular function maximization. This line of work was recently initiated by Balkanski and Singer and has already led to several interesting results on the cardinality constraint and explicit packing constraints. An important open problem is the classical setting of matroid constraint, which has been instrumental for developments in submodular function maximization. In this paper we develop a general strategy to parallelize the well-studied greedy algorithm and use it to obtain a randomized $\left(\frac{1}{2} - ε\right)$-approximation in $\operatorname{O}\left( \frac{\log^2 n}{ε^2} \right)$ rounds of adaptivity. We rely on this algorithm, and an elegant amplification approach due to Badanidiyuru and Vondrák to obtain a fractional solution that yields a near-optimal randomized $\left( 1 - 1/e - ε\right)$-approximation in $O\left( {\frac{\log^2 n}{ε^3}} \right) $ rounds of adaptivity. For non-negative functions we obtain a $\left( {3-2\sqrt{2}}\right)$-approximation and a fractional solution that yields a $\left( {\frac{1}{e} - ε}\right)$-approximation. Our approach for parallelizing greedy yields approximations for intersections of matroids and matchoids, and the approximation ratios are comparable to those known for sequential greedy.

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arXiv2018

Fast and Deterministic Approximations for $k$-Cut

In an undirected graph, a $k$-cut is a set of edges whose removal breaks the graph into at least $k$ connected components. The minimum weight $k$-cut can be computed in $O(n^{O(k)})$ time, but when $k$ is treated as part of the input, computing the minimum weight $k$-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For $\operatorname{poly}(m,n,k)$-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017]. Saran and Vazirani [1995] showed that a $(2 - 2/k)$-approximately minimum weight $k$-cut can be computed by $O(k)$ minimum cuts, which implies an $\tilde{O}(mk)$ randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that the minimum weight $k$-cut can be computed deterministically in $O(mn + n^2 \log n)$ time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate $k$-cuts matching the randomized running time of $\tilde{O}(mk)$? The second question qualitatively compares minimum cut to 2-approximate minimum $k$-cut. Can 2-approximate $k$-cuts be computed as fast as the (exact) minimum cut - in $\tilde{O}(m)$ randomized time? We make progress on these questions with a deterministic approximation algorithm that computes $(2 + ε)$-minimum $k$-cuts in $O(m \log^3(n) / ε^2)$ time, via a $(1 + ε)$-approximate for an LP relaxation of $k$-cut.

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arXiv2018

Submodular Function Maximization in Parallel via the Multilinear Relaxation

Balkanski and Singer [5] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Very recent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal $(1-1/e-ε)$-approximation in $O(\log n/ε^2)$ rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to packing constraints. Formally our problem is to maximize $F(x)$ over $x \in [0,1]^{n}$ subject to $Ax \le 1$ where $F$ is the multilinear relaxation of a monotone submodular function. Our algorithm achieves a near-optimal $(1-1/e-ε)$-approximation in $O(\log^2 m \log n/ε^4)$ rounds where $n$ is the cardinality of the ground set and $m$ is the number of packing constraints. For many constraints of interest, the resulting fractional solution can be rounded via known randomized rounding schemes that are oblivious to the specific submodular function. We thus derive randomized algorithms with poly-logarithmic adaptivity for a number of constraints including partition and laminar matroids, matchings, knapsack constraints, and their intersections.

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arXiv2018

On Approximating (Sparse) Covering Integer Programs

We consider approximation algorithms for covering integer programs of the form min $\langle c, x \rangle $ over $x \in \mathbb{N}^n $ subject to $A x \geq b $ and $x \leq d$; where $A \in \mathbb{R}_{\geq 0}^{m \times n}$, $b \in \mathbb{R}_{\geq 0}^m$, and $c, d \in \mathbb{R}_{\geq 0}^n$ all have nonnegative entries. We refer to this problem as $\operatorname{CIP}$, and the special case without the multiplicity constraints $x \le d$ as $\operatorname{CIP}_{\infty}$. These problems generalize the well-studied Set Cover problem. We make two algorithmic contributions. First, we show that a simple algorithm based on randomized rounding with alteration improves or matches the best known approximation algorithms for $\operatorname{CIP}$ and $\operatorname{CIP}_{\infty}$ in a wide range of parameter settings, and these bounds are essentially optimal. As a byproduct of the simplicity of the alteration algorithm and analysis, we can derandomize the algorithm without any loss in the approximation guarantee or efficiency. Previous work by Chen, Harris and Srinivasan [12] which obtained near-tight bounds is based on a resampling-based randomized algorithm whose analysis is complex. Non-trivial approximation algorithms for $\operatorname{CIP}$ are based on solving the natural LP relaxation strengthened with knapsack cover (KC) inequalities [5,24,12]. Our second contribution is a fast (essentially near-linear time) approximation scheme for solving the strengthened LP with a factor of $n$ speed up over the previous best running time [5]. Together, our contributions lead to near-optimal (deterministic) approximation bounds with near-linear running times for $\operatorname{CIP}$ and $\operatorname{CIP}_{\infty}$.

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arXiv2018

Approximating optimal transport with linear programs

In the regime of bounded transportation costs, additive approximations for the optimal transport problem are reduced (rather simply) to relative approximations for positive linear programs, resulting in faster additive approximation algorithms for optimal transport.

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arXiv2018

LP Relaxation and Tree Packing for Minimum $k$-cuts

Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm. Thorup developed a fast deterministic algorithm for the minimum $k$-cut problem via greedy recursive tree packings. In this paper we revisit properties of an LP relaxation for $k$-cut proposed by Naor and Rabani, and analyzed by Chekuri, Guha and Naor. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the $k$-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of $n$. We also improve the bound on the number of $α$-approximate $k$-cuts. Second, we give a simple proof that the integrality gap of the LP is $2(1-1/n)$. Third, we show that an optimum solution to the LP relaxation, for all values of $k$, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangian relaxation approach of Barahona and Ravi and Sinha; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup.

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arXiv2018

Fast Approximations for Metric-TSP via Linear Programming

We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017]. We show that the LP solution can be sparsified via cut-sparsification techniques such as those of Benczur and Karger [2015]. Given a weighted graph $G$ with $m$ edges and $n$ vertices, and $ε> 0$, our randomized algorithm outputs with high probability a $(1+ε)$-approximate solution to the LP relaxation whose support has $\operatorname{O}(n \log n /ε^2)$ edges. The running time of the algorithm is $\operatorname{Õ}(m/ε^2)$. This can be generically used to speed up algorithms that rely on the LP. For Metric-TSP, we obtain the following concrete result. For a weighted graph $G$ with $m$ edges and $n$ vertices, and $ε> 0$, we describe an algorithm that outputs with high probability a tour of $G$ with cost at most $(1 + ε) \frac{3}{2}$ times the minimum cost tour of $G$ in time $\operatorname{Õ}(m/ε^2 + n^{1.5}/ε^3)$. Previous implementations of Christofides' algorithm [Christofides, 1976] require, for a $\frac{3}{2}$-optimal tour, $\operatorname{Õ}(n^{2.5})$ time when the metric is explicitly given, or $\operatorname{Õ}(\min\{m^{1.5}, mn+n^{2.5}\})$ time when the metric is given implicitly as the shortest path metric of a weighted graph.

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arXiv2017

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $ε> 0$, the algorithm outputs in $O(m \log^4n /ε^2)$ time, with high probability, a $(1+ε)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/ε^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + ε\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm.

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arXiv2016

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density.

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arXiv2016

Notes on Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

This write-up contains some minor results and notes related to our work [HQ15] (some of them already known in the literature). In particular, it shows the following: - We show that a graph with polynomial expansion have sublinear separators. - We show that hereditary sublinear separators imply that a graph have small divisions. - We show a natural condition on a set of segments, such that they have low density. This might be of independent interest in trying to define a realistic input model for a set of segments. Unlike the previous two results, this is new. For context and more details, see the main paper.

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arXiv2015

Streaming Algorithms for Submodular Function Maximization

We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $Ω(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes value oracle access to $f$ and membership oracles for the matroids defining the $p$-matchoid constraint.

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