eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
1
1228
10.4230/LIPIcs.APPROX/RANDOM.2020
article
LIPIcs, Volume 176, APPROX/RANDOM 2020, Complete Volume
Byrka, Jarosław
1
https://orcid.org/0000-0002-3387-0913
Meka, Raghu
2
University of Wrocław, Poland
University of California, Los Angeles, USA
LIPIcs, Volume 176, APPROX/RANDOM 2020, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020/LIPIcs.APPROX-RANDOM.2020.pdf
LIPIcs, Volume 176, APPROX/RANDOM 2020, Complete Volume
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
0:i
0:xx
10.4230/LIPIcs.APPROX/RANDOM.2020.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Byrka, Jarosław
1
https://orcid.org/0000-0002-3387-0913
Meka, Raghu
2
University of Wrocław, Poland
University of California, Los Angeles, USA
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.0/LIPIcs.APPROX-RANDOM.2020.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
1:1
1:20
10.4230/LIPIcs.APPROX/RANDOM.2020.1
article
Extractor Lower Bounds, Revisited
Aggarwal, Divesh
1
Guo, Siyao
2
Obremski, Maciej
1
Ribeiro, João
3
https://orcid.org/0000-0002-9870-0501
Stephens-Davidowitz, Noah
4
National University of Singapore, Singapore
New York University Shanghai, China
Imperial College London, UK
Cornell University, Ithaca, NY, USA
We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a "change in quantifiers" over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input sources with sufficient min-entropy, a somewhere extractor only requires that there exists a seed whose output bias is small. More generally, we study what we call probable extractors, which on input a source with sufficient min-entropy guarantee that a large enough fraction of seeds have small enough associated output bias. Such extractors have played a key role in many constructions of pseudorandom objects, though they are often defined implicitly and have not been studied extensively.
Prior known techniques fail to yield good seed length lower bounds when applied to the variants above. Our novel approach yields significantly improved lower bounds for somewhere and probable extractors. To complement this, we construct a somewhere extractor that implies our lower bound for such functions is tight in the high min-entropy regime. Surprisingly, this means that a random function is far from an optimal somewhere extractor in this regime. The techniques that we develop also yield an alternative, simpler proof of the celebrated optimal lower bound for strong extractors originally due to Radhakrishnan and Ta-Shma (SIAM J. Discrete Math., 2000).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.1/LIPIcs.APPROX-RANDOM.2020.1.pdf
randomness extractors
lower bounds
explicit constructions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
2:1
2:15
10.4230/LIPIcs.APPROX/RANDOM.2020.2
article
A Simpler Strong Refutation of Random k-XOR
Ahn, Kwangjun
1
https://orcid.org/0000-0001-5516-5775
Department of EECS, Massachusetts Institute of Technology, Cambridge, MA, USA
Strong refutation of random CSPs is a fundamental question in theoretical computer science that has received particular attention due to the long-standing gap between the information-theoretic limit and the computational limit. This gap is recently bridged by Raghavendra, Rao and Schramm where they study sub-exponential algorithms for the regime between the two limits. In this work, we take a simpler approach to their algorithms and analyses.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.2/LIPIcs.APPROX-RANDOM.2020.2.pdf
Strong refutation
Random k-XOR
Spectral method
Trace power method
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
3:1
3:21
10.4230/LIPIcs.APPROX/RANDOM.2020.3
article
Iterated Decomposition of Biased Permutations via New Bounds on the Spectral Gap of Markov Chains
Miracle, Sarah
1
Streib, Amanda Pascoe
2
Streib, Noah
2
University of St. Thomas, St. Paul, MN, USA
Center for Computing Sciences, Bowie, MD, USA
In this paper, we address a conjecture of Fill [Fill03] about the spectral gap of a nearest-neighbor transposition Markov chain ℳ_nn over biased permutations of [n]. Suppose we are given a set of input probabilities 𝒫 = {p_{i,j}} for all 1 ≤ i, j ≤ n with p_{i, j} = 1-p_{j, i}. The Markov chain ℳ_nn operates by uniformly choosing a pair of adjacent elements, i and j, and putting i ahead of j with probability p_{i,j} and j ahead of i with probability p_{j,i}, independent of their current ordering.
We build on previous work [S. Miracle and A.P. Streib, 2018] that analyzed the spectral gap of ℳ_nn when the particles in [n] fall into k classes. There, the authors iteratively decomposed ℳ_nn into simpler chains, but incurred a multiplicative penalty of n^-2 for each application of the decomposition theorem of [Martin and Randall, 2000], leading to an exponentially small lower bound on the gap. We make progress by introducing a new complementary decomposition theorem. We introduce the notion of ε-orthogonality, and show that for ε-orthogonal chains, the complementary decomposition theorem may be iterated O(1/√ε) times while only giving away a constant multiplicative factor on the overall spectral gap. We show the decomposition given in [S. Miracle and A.P. Streib, 2018] of a related Markov chain ℳ_pp over k-class particle systems is 1/n²-orthogonal when the number of particles in each class is at least C log n, where C is a constant not depending on n. We then apply the complementary decomposition theorem iteratively n times to prove nearly optimal bounds on the spectral gap of ℳ_pp and to further prove the first inverse-polynomial bound on the spectral gap of ℳ_nn when k is as large as Θ(n/log n). The previous best known bound assumed k was at most a constant.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.3/LIPIcs.APPROX-RANDOM.2020.3.pdf
Markov chains
Permutations
Decomposition
Spectral Gap
Iterated Decomposition
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
4:1
4:16
10.4230/LIPIcs.APPROX/RANDOM.2020.4
article
Improved Explicit Hitting-Sets for ROABPs
Guo, Zeyu
1
Gurjar, Rohit
2
Department of Computer Science, University of Haifa, Israel
Department of Computer Science and Engineering, IIT Bombay, India
We give improved explicit constructions of hitting-sets for read-once oblivious algebraic branching programs (ROABPs) and related models. For ROABPs in an unknown variable order, our hitting-set has size polynomial in (nr)^{(log n)/(max{1, log log n-log log r})}d over a field whose characteristic is zero or large enough, where n is the number of variables, d is the individual degree, and r is the width of the ROABP. A similar improved construction works over fields of arbitrary characteristic with a weaker size bound.
Based on a result of Bisht and Saxena (2020), we also give an improved explicit construction of hitting-sets for sum of several ROABPs. In particular, when the characteristic of the field is zero or large enough, we give polynomial-size explicit hitting-sets for sum of constantly many log-variate ROABPs of width r = 2^{O(log d/log log d)}.
Finally, we give improved explicit hitting-sets for polynomials computable by width-r ROABPs in any variable order, also known as any-order ROABPs. Our hitting-set has polynomial size for width r up to 2^{O(log(nd)/log log(nd))} or 2^{O(log^{1-ε} (nd))}, depending on the characteristic of the field. Previously, explicit hitting-sets of polynomial size are unknown for r = ω(1).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.4/LIPIcs.APPROX-RANDOM.2020.4.pdf
polynomial identity testing
hitting-set
ROABP
arithmetic branching programs
derandomization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
5:1
5:20
10.4230/LIPIcs.APPROX/RANDOM.2020.5
article
Almost Optimal Testers for Concise Representations
Bshouty, Nader H.
1
Department of Computer Science, Technion, Haifa, Israel
We give improved and almost optimal testers for several classes of Boolean functions on n variables that have concise representation in the uniform and distribution-free model. Classes, such as k-Junta, k-Linear, s-Term DNF, s-Term Monotone DNF, r-DNF, Decision List, r-Decision List, size-s Decision Tree, size-s Boolean Formula, size-s Branching Program, s-Sparse Polynomial over the binary field and functions with Fourier Degree at most d.
The approach is new and combines ideas from Diakonikolas et al. [Ilias Diakonikolas et al., 2007], Bshouty [Nader H. Bshouty, 2018], Goldreich et al. [Oded Goldreich et al., 1998], and learning theory. The method can be extended to several other classes of functions over any domain that can be approximated by functions with a small number of relevant variables.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.5/LIPIcs.APPROX-RANDOM.2020.5.pdf
Property Testing
Boolean function
Junta
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
6:1
6:22
10.4230/LIPIcs.APPROX/RANDOM.2020.6
article
Palette Sparsification Beyond (Δ+1) Vertex Coloring
Alon, Noga
1
2
Assadi, Sepehr
3
Department of Mathematics, Princeton University, NJ, USA
Schools of Mathematics and Computer Science, Tel Aviv University, Israel
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every n-vertex graph G with maximum degree Δ, sampling O(log n) colors per each vertex independently from Δ+1 colors almost certainly allows for proper coloring of G from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown to have various applications to design of algorithms for (Δ+1) coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms.
In this paper, we focus on palette sparsification beyond (Δ+1) coloring, in both regimes when the number of available colors is much larger than (Δ+1), and when it is much smaller. In particular,
- We prove that for (1+ε) Δ coloring, sampling only O_ε(√{log n}) colors per vertex is sufficient and necessary to obtain a proper coloring from the sampled colors - this shows a separation between (1+ε) Δ and (Δ+1) coloring in the context of palette sparsification.
- A natural family of graphs with chromatic number much smaller than (Δ+1) are triangle-free graphs which are O(Δ/ln Δ) colorable. We prove a palette sparsification theorem tailored to these graphs: Sampling O(Δ^γ + √{log n}) colors per vertex is sufficient and necessary to obtain a proper O_γ(Δ/ln Δ) coloring of triangle-free graphs.
- We also consider the "local version" of graph coloring where every vertex v can only be colored from a list of colors with size proportional to the degree deg(v) of v. We show that sampling O_ε(log n) colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of (1+ε) ⋅ deg(v) arbitrary colors, or even only deg(v)+1 colors when the lists are the sets {1,…,deg(v)+1}.
Our new palette sparsification results naturally lead to a host of new and/or improved algorithms for vertex coloring in different models including streaming and sublinear-time algorithms.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.6/LIPIcs.APPROX-RANDOM.2020.6.pdf
Graph coloring
palette sparsification
sublinear algorithms
list-coloring
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
7:1
7:23
10.4230/LIPIcs.APPROX/RANDOM.2020.7
article
On Hitting-Set Generators for Polynomials That Vanish Rarely
Doron, Dean
1
Ta-Shma, Amnon
2
Tell, Roei
3
Department of Computer Science, Stanford University, CA, USA
The Blavatnik School of Computer Science, Tel-Aviv University, Israel
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel
The problem of constructing hitting-set generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct a hitting-set generator for polynomials p: 𝔽ⁿ → 𝔽 of degree d if we are guaranteed that the polynomial vanishes on at most an ε > 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/|𝔽|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017).
In this work our main interest is a systematic study of the relaxed problem, in its general form, and we prove results that significantly improve and extend the two previously-known results. Our contributions are of two types:
- Over fields of size 2 ≤ |𝔽| ≤ poly(n), we show that the seed length of any hitting-set generator for polynomials of degree d ≤ n^{.49} that vanish on at most ε = |𝔽|^{-t} of their inputs is at least Ω((d/t)⋅log(n)).
- Over 𝔽₂, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d ≤ n^{.99} that vanish on at most ε = |𝔽|^{-t} of their inputs with seed length O((d-t)⋅log(n)). We also show a polynomial-time computable hitting-set generator with seed length O((d-t)⋅(2^{d-t}+log(n))).
In addition, we prove that the problem we study is closely related to the following question: "Does there exist a small set S ⊆ 𝔽ⁿ whose degree-d closure is very large?", where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.7/LIPIcs.APPROX-RANDOM.2020.7.pdf
Hitting-set generators
Polynomials over finite fields
Quantified derandomization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
8:1
8:13
10.4230/LIPIcs.APPROX/RANDOM.2020.8
article
Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness
Bläser, Markus
1
Pandey, Anurag
2
Department of Computer Science, Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Max Planck Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany
We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting.
Our algorithm works for every field 𝔽 with | 𝔽 | ≥ d+1, and uses only d log n + log (1/ ε) + O(d log log n) random bits to achieve a success probability 1 - ε for some ε > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.8/LIPIcs.APPROX-RANDOM.2020.8.pdf
Algebraic Complexity theory
Polynomial Identity Testing
Hitting Set
Pseudorandomness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
9:1
9:21
10.4230/LIPIcs.APPROX/RANDOM.2020.9
article
Bounds for List-Decoding and List-Recovery of Random Linear Codes
Guruswami, Venkatesan
1
Li, Ray
2
Mosheiff, Jonathan
1
Resch, Nicolas
1
Silas, Shashwat
2
Wootters, Mary
2
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Department of Computer Science, Stanford University, CA, USA
A family of error-correcting codes is list-decodable from error fraction p if, for every code in the family, the number of codewords in any Hamming ball of fractional radius p is less than some integer L that is independent of the code length. It is said to be list-recoverable for input list size 𝓁 if for every sufficiently large subset of codewords (of size L or more), there is a coordinate where the codewords take more than 𝓁 values. The parameter L is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size L → ∞, is known to be 1-h_q(p) for list-decoding, and 1-log_q 𝓁 for list-recovery, where q is the alphabet size of the code family.
In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below q is the alphabet size, and ε > 0 is the gap to capacity).
- A random linear code of rate 1 - log_q(𝓁) - ε requires list size L ≥ 𝓁^{Ω(1/ε)} for list-recovery from input list size 𝓁. This is surprisingly in contrast to completely random codes, where L = O(𝓁/ε) suffices w.h.p.
- A random linear code of rate 1 - h_q(p) - ε requires list size L ≥ ⌊ {h_q(p)/ε+0.99}⌋ for list-decoding from error fraction p, when ε is sufficiently small.
- A random binary linear code of rate 1 - h₂(p) - ε is list-decodable from average error fraction p with list size with L ≤ ⌊ {h₂(p)/ε}⌋ + 2. (The average error version measures the average Hamming distance of the codewords from the center of the Hamming ball, instead of the maximum distance as in list-decoding.)
The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.
Our lower bounds follow by exhibiting an explicit subset of codewords so that this subset - or some symbol-wise permutation of it - lies in a random linear code with high probability. This uses a recent characterization of (Mosheiff, Resch, Ron-Zewi, Silas, Wootters, 2019) of configurations of codewords that are contained in random linear codes. Our upper bound follows from a refinement of the techniques of (Guruswami, Håstad, Sudan, Zuckerman, 2002) and strengthens a previous result of (Li, Wootters, 2018), which applied to list-decoding rather than average-radius list-decoding.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.9/LIPIcs.APPROX-RANDOM.2020.9.pdf
list-decoding
list-recovery
random linear codes
coding theory
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
10:1
10:20
10.4230/LIPIcs.APPROX/RANDOM.2020.10
article
Is It Possible to Improve Yao’s XOR Lemma Using Reductions That Exploit the Efficiency of Their Oracle?
Shaltiel, Ronen
1
University of Haifa, Israel
Yao’s XOR lemma states that for every function f:{0,1}^k → {0,1}, if f has hardness 2/3 for P/poly (meaning that for every circuit C in P/poly, Pr[C(X) = f(X)] ≤ 2/3 on a uniform input X), then the task of computing f(X₁) ⊕ … ⊕ f(X_t) for sufficiently large t has hardness 1/2 +ε for P/poly.
Known proofs of this lemma cannot achieve ε = 1/k^ω(1), and even for ε = 1/k, we do not know how to replace P/poly by AC⁰[parity] (the class of constant depth circuits with the gates {and,or,not,parity} of unbounded fan-in).
Recently, Grinberg, Shaltiel and Viola (FOCS 2018) (building on a sequence of earlier works) showed that these limitations cannot be circumvented by black-box reductions. Namely, by reductions Red^(⋅) that given oracle access to a function D that violates the conclusion of Yao’s XOR lemma, implement a circuit that violates the assumption of Yao’s XOR lemma.
There are a few known reductions in the related literature on worst-case to average case reductions that are non-black box. Specifically, the reductions of Gutfreund, Shaltiel and Ta Shma (Computational Complexity 2007) and Hirahara (FOCS 2018)) are "class reductions" that are only guaranteed to succeed when given oracle access to an oracle D from some efficient class of algorithms. These works seem to circumvent some black-box impossibility results.
In this paper we extend the previous limitations of Grinberg, Shaltiel and Viola to class reductions, giving evidence that class reductions cannot yield the desired improvements in Yao’s XOR lemma. To the best of our knowledge, this is the first limitation on reductions for hardness amplification that applies to class reductions.
Our technique imitates the previous lower bounds for black-box reductions, replacing the inefficient oracle used in that proof, with an efficient one that is based on limited independence, and developing tools to deal with the technical difficulties that arise following this replacement.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.10/LIPIcs.APPROX-RANDOM.2020.10.pdf
Yao’s XOR lemma
Hardness amplification
black-box reductions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
11:1
11:22
10.4230/LIPIcs.APPROX/RANDOM.2020.11
article
Balanced Allocation on Dynamic Hypergraphs
Greenhill, Catherine
1
Mans, Bernard
2
Pourmiri, Ali
2
UNSW Sydney, Australia
Macquarie University, Sydney, Australia
The {balls-into-bins model} randomly allocates n sequential balls into n bins, as follows: each ball selects a set D of d ⩾ 2 bins, independently and uniformly at random, then the ball is allocated to a least-loaded bin from D (ties broken randomly). The maximum load is the maximum number of balls in any bin. In 1999, Azar et al. showed that, provided ties are broken randomly, after n balls have been placed the maximum load, is log_d log n + 𝒪(1), with high probability. We consider this popular paradigm in a dynamic environment where the bins are structured as a dynamic hypergraph. A dynamic hypergraph is a sequence of hypergraphs, say ℋ^(t), arriving over discrete times t = 1,2,…, such that the vertex set of ℋ^(t)’s is the set of n bins, but (hyper)edges may change over time. In our model, the t-th ball chooses an edge from ℋ^(t) uniformly at random, and then chooses a set D of d ⩾ 2 random bins from the selected edge. The ball is allocated to a least-loaded bin from D, with ties broken randomly. We quantify the dynamicity of the model by introducing the notion of pair visibility, which measures the number of rounds in which a pair of bins appears within a (hyper)edge. We prove that if, for some ε > 0, a dynamic hypergraph has pair visibility at most n^{1-ε}, and some mild additional conditions hold, then with high probability the process has maximum load 𝒪(log_dlog n). Our proof is based on a variation of the witness tree technique, which is of independent interest. The model can also be seen as an adversarial model where an adversary decides the structure of the possible sets of d bins available to each ball.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.11/LIPIcs.APPROX-RANDOM.2020.11.pdf
balls-into-bins
balanced allocation
power of two choices
witness tree technique
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
12:1
12:20
10.4230/LIPIcs.APPROX/RANDOM.2020.12
article
The GaussianSketch for Almost Relative Error Kernel Distance
Phillips, Jeff M.
1
Tai, Wai Ming
1
School of Computing, University of Utah, Salt Lake City, UT, USA
We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the RecursiveTensorSketch [Ahle et al. SODA 2020]. After providing concentration and approximation properties of these sketches, we use them to approximate the kernel distance between points sets. These sketches yield almost (1+ε)-relative error, but with a small additive α term. In the first variants the dependence on 1/α is poly-logarithmic, but has higher degree of polynomial dependence on the original dimension d. In the second variant, the dependence on 1/α is still poly-logarithmic, but the dependence on d is linear.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.12/LIPIcs.APPROX-RANDOM.2020.12.pdf
Kernel Distance
Kernel Density Estimation
Sketching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
13:1
13:20
10.4230/LIPIcs.APPROX/RANDOM.2020.13
article
A Fast Binary Splitting Approach to Non-Adaptive Group Testing
Price, Eric
1
Scarlett, Jonathan
2
Department of Computer Science, University of Texas at Austin, TX, USA
Department of Computer Science & Department of Mathematics, National University of Singapore, Singapore
In this paper, we consider the problem of noiseless non-adaptive group testing under the for-each recovery guarantee, also known as probabilistic group testing. In the case of n items and k defectives, we provide an algorithm attaining high-probability recovery with O(k log n) scaling in both the number of tests and runtime, improving on the best known O(k² log k ⋅ log n) runtime previously available for any algorithm that only uses O(k log n) tests. Our algorithm bears resemblance to Hwang’s adaptive generalized binary splitting algorithm (Hwang, 1972); we recursively work with groups of items of geometrically vanishing sizes, while maintaining a list of "possibly defective" groups and circumventing the need for adaptivity. While the most basic form of our algorithm requires Ω(n) storage, we also provide a low-storage variant based on hashing, with similar recovery guarantees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.13/LIPIcs.APPROX-RANDOM.2020.13.pdf
Group testing
sparsity
sublinear-time decoding
binary splitting
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
14:1
14:13
10.4230/LIPIcs.APPROX/RANDOM.2020.14
article
Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size
Dreier, Jan
1
https://orcid.org/0000-0002-2662-5303
Kuinke, Philipp
1
https://orcid.org/0000-0001-9716-6346
Rossmanith, Peter
1
https://orcid.org/0000-0003-0177-8028
Department of Computer Science, RWTH Aachen University, Germany
Preferential attachment graphs are random graphs designed to mimic properties of real word networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We prove various structural asymptotic properties of this graph model. In particular, we show that the size of the largest r-shallow clique minor in Gⁿ_m is at most log(n)^{O(r²)}m^{O(r)}. Furthermore, there exists a one-subdivided clique of size log(n)^{1/4}. Therefore, preferential attachment graphs are asymptotically almost surely somewhere dense and algorithmic techniques developed for structurally sparse graph classes are not directly applicable. However, they are just barely somewhere dense. The removal of just slightly more than a polylogarithmic number of vertices asymptotically almost surely yields a graph with locally bounded treewidth.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.14/LIPIcs.APPROX-RANDOM.2020.14.pdf
Random Graphs
Preferential Attachment
Sparsity
Somewhere Dense
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
15:1
15:14
10.4230/LIPIcs.APPROX/RANDOM.2020.15
article
On Nonadaptive Security Reductions of Hitting Set Generators
Hirahara, Shuichi
1
Watanabe, Osamu
2
National Institute of Informatics, Tokyo, Japan
Tokyo Institute of Technology, Japan
One of the central open questions in the theory of average-case complexity is to establish the equivalence between the worst-case and average-case complexity of the Polynomial-time Hierarchy (PH). One general approach is to show that there exists a PH-computable hitting set generator whose security is based on some NP-hard problem. We present the limits of such an approach, by showing that there exists no exponential-time-computable hitting set generator whose security can be proved by using a nonadaptive randomized polynomial-time reduction from any problem outside AM ∩ coAM, which significantly improves the previous upper bound BPP^NP of Gutfreund and Vadhan (RANDOM/APPROX 2008 [Gutfreund and Vadhan, 2008]). In particular, any security proof of a hitting set generator based on some NP-hard problem must use either an adaptive or non-black-box reduction (unless the polynomial-time hierarchy collapses). To the best of our knowledge, this is the first result that shows limits of black-box reductions from an NP-hard problem to some form of a distributional problem in DistPH.
Based on our results, we argue that the recent worst-case to average-case reduction of Hirahara (FOCS 2018 [Hirahara, 2018]) is inherently non-black-box, without relying on any unproven assumptions. On the other hand, combining the non-black-box reduction with our simulation technique of black-box reductions, we exhibit the existence of a "non-black-box selector" for GapMCSP, i.e., an efficient algorithm that solves GapMCSP given as advice two circuits one of which is guaranteed to compute GapMCSP.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.15/LIPIcs.APPROX-RANDOM.2020.15.pdf
hitting set generator
black-box reduction
average-case complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
16:1
16:20
10.4230/LIPIcs.APPROX/RANDOM.2020.16
article
Testable Properties in General Graphs and Random Order Streaming
Czumaj, Artur
1
Fichtenberger, Hendrik
2
https://orcid.org/0000-0003-3246-5323
Peng, Pan
3
https://orcid.org/0000-0003-2700-5699
Sohler, Christian
4
Department of Computer Science and Centre for Discrete Mathematics and its Applications (DIMAP), University of Warwick, Coventry, UK
Department of Computer Science, TU Dortmund, Germany
Department of Computer Science, University of Sheffield, UK
Department of Mathematics and Computer Science, University of Cologne, Germany
We consider the fundamental question of understanding the relative power of two important computational models: property testing and data streaming. We present a novel framework closely linking these areas in the setting of general graphs in the context of constant-query complexity testing and constant-space streaming. Our main result is a generic transformation of a one-sided error property tester in the random-neighbor model with constant query complexity into a one-sided error property tester in the streaming model with constant space complexity. Previously such a generic transformation was only known for bounded-degree graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.16/LIPIcs.APPROX-RANDOM.2020.16.pdf
Graph property testing
sublinear algorithms
graph streaming algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
17:1
17:21
10.4230/LIPIcs.APPROX/RANDOM.2020.17
article
Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs
Beideman, Calvin
1
Chandrasekaran, Karthekeyan
1
Xu, Chao
2
University of Illinois, Urbana-Champaign, IL, USA
The Voleon Group, Berkeley, CA, USA
We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs.
1) For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2^{tr}n^{3t-1}). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne [Aissi et al., 2015]. In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time.
2) We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2^{r}n^{t+2}), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b ∈ ℝ^t_+ is O(n²).
3) We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne [Guinez and Queyranne, 2012]. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger [Karger, 1993]. Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.17/LIPIcs.APPROX-RANDOM.2020.17.pdf
Multiobjective Optimization
Hypergraph min-cut
Hypergraph-k-cut
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
18:1
18:15
10.4230/LIPIcs.APPROX/RANDOM.2020.18
article
On Testing and Robust Characterizations of Convexity
Blais, Eric
1
Bommireddi, Abhinav
1
University of Waterloo, Canada
A body K ⊂ ℝⁿ is convex if and only if the line segment between any two points in K is completely contained within K or, equivalently, if and only if the convex hull of a set of points in K is contained within K. We show that neither of those characterizations of convexity are robust: there are bodies in ℝⁿ that are far from convex - in the sense that the volume of the symmetric difference between the set K and any convex set C is a constant fraction of the volume of K - for which a line segment between two randomly chosen points x,y ∈ K or the convex hull of a random set X of points in K is completely contained within K except with exponentially small probability. These results show that any algorithms for testing convexity based on the natural line segment and convex hull tests have exponential query complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.18/LIPIcs.APPROX-RANDOM.2020.18.pdf
Convexity
Line segment test
Convex hull test
Intersecting cones
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
19:1
19:24
10.4230/LIPIcs.APPROX/RANDOM.2020.19
article
Distributed Testing of Graph Isomorphism in the CONGEST Model
Levi, Reut
1
https://orcid.org/0000-0003-3167-1766
Medina, Moti
2
https://orcid.org/0000-0002-5572-3754
Efi Arazi School of Computer Science, The Interdisciplinary Center, Herzliya, Israel
School of Electrical & Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel
In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, G_U, is isomorphic to G_K which is given as an input to all the nodes in the network, or alternatively, only to a single node.
We first consider the decision variant of the problem in which the algorithm should distinguish the case where G_U and G_K are isomorphic from the case where G_U and G_K are not isomorphic. Specifically, if G_U and G_K are not isomorphic then w.h.p. at least one node should output reject and otherwise all nodes should output accept . We provide a randomized algorithm with O(n) rounds for the setting in which G_K is given only to a single node. We prove that for this setting the number of rounds of any deterministic algorithm is Ω̃(n²) rounds, where n denotes the number of nodes, which implies a separation between the randomized and the deterministic complexities of deciding GI . Our algorithm can be adapted to the semi-streaming model, where a single pass is performed and Õ(n) bits of space are used.
We then consider the property testing variant of the problem, where the algorithm is only required to distinguish the case that G_U and G_K are isomorphic from the case that G_U and G_K are far from being isomorphic (according to some predetermined distance measure). We show that every (possibly randomized) algorithm, requires Ω(D) rounds, where D denotes the diameter of the network. This lower bound holds even if all the nodes are given G_K as an input, and even if the message size is unbounded. We provide a randomized algorithm with an almost matching round complexity of O(D+(ε^{-1}log n)²) rounds that is suitable for dense graphs (namely, graphs with Ω(n²) edges).
We also show that with the same number of rounds it is possible that each node outputs its mapping according to a bijection which is an approximate isomorphism.
We conclude with simple simulation arguments that allow us to adapt centralized property testing algorithms and obtain essentially tight algorithms with round complexity Õ(D) for special families of sparse graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.19/LIPIcs.APPROX-RANDOM.2020.19.pdf
the CONGEST model
graph isomorphism
distributed property testing
distributed decision
graph algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
20:1
20:15
10.4230/LIPIcs.APPROX/RANDOM.2020.20
article
Reaching a Consensus on Random Networks: The Power of Few
Tran, Linh
1
Vu, Van
1
Department of Mathematics, Yale University, New Haven, CT, USA
A community of n individuals splits into two camps, Red and Blue. The individuals are connected by a social network, which influences their colors. Everyday, each person changes his/her color according to the majority of his/her neighbors. Red (Blue) wins if everyone in the community becomes Red (Blue) at some point.
We study this process when the underlying network is the random Erdos-Renyi graph G(n, p). With a balanced initial state (n/2 persons in each camp), it is clear that each color wins with the same probability.
Our study reveals that for any constants p and ε, there is a constant c such that if one camp has n/2 + c individuals at the initial state, then it wins with probability at least 1 - ε. The surprising fact here is that c does not depend on n, the population of the community. When p = 1/2 and ε = .1, one can set c = 6, meaning one camp has n/2 + 6 members initially. In other words, it takes only 6 extra people to win an election with overwhelming odds. We also generalize the result to p = p_n = o(1) in a separate paper.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.20/LIPIcs.APPROX-RANDOM.2020.20.pdf
Random Graphs Majority Dynamics Consensus
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
21:1
21:18
10.4230/LIPIcs.APPROX/RANDOM.2020.21
article
Time-Space Tradeoffs for Distinguishing Distributions and Applications to Security of Goldreich’s PRG
Garg, Sumegha
1
Kothari, Pravesh K.
2
Raz, Ran
1
Department of Computer Science, Princeton University, NJ, USA
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
In this work, we establish lower-bounds against memory bounded algorithms for distinguishing between natural pairs of related distributions from samples that arrive in a streaming setting.
Our first result applies to the problem of distinguishing the uniform distribution on {0,1}ⁿ from uniform distribution on some unknown linear subspace of {0,1}ⁿ. As a specific corollary, we show that any algorithm that distinguishes between uniform distribution on {0,1}ⁿ and uniform distribution on an n/2-dimensional linear subspace of {0,1}ⁿ with non-negligible advantage needs 2^Ω(n) samples or Ω(n²) memory (tight up to constants in the exponent).
Our second result applies to distinguishing outputs of Goldreich’s local pseudorandom generator from the uniform distribution on the output domain. Specifically, Goldreich’s pseudorandom generator G fixes a predicate P:{0,1}^k → {0,1} and a collection of subsets S₁, S₂, …, S_m ⊆ [n] of size k. For any seed x ∈ {0,1}ⁿ, it outputs P(x_S₁), P(x_S₂), …, P(x_{S_m}) where x_{S_i} is the projection of x to the coordinates in S_i. We prove that whenever P is t-resilient (all non-zero Fourier coefficients of (-1)^P are of degree t or higher), then no algorithm, with < n^ε memory, can distinguish the output of G from the uniform distribution on {0,1}^m with a large inverse polynomial advantage, for stretch m ≤ (n/t) ^{(1-ε)/36 ⋅ t} (barring some restrictions on k). The lower bound holds in the streaming model where at each time step i, S_i ⊆ [n] is a randomly chosen (ordered) subset of size k and the distinguisher sees either P(x_{S_i}) or a uniformly random bit along with S_i.
An important implication of our second result is the security of Goldreich’s generator with super linear stretch (in the streaming model), against memory-bounded adversaries, whenever the predicate P satisfies the necessary condition of t-resiliency identified in various prior works.
Our proof builds on the recently developed machinery for proving time-space trade-offs (Raz 2016 and follow-ups). Our key technical contribution is to adapt this machinery to work for distinguishing problems in contrast to prior works on similar results for search/learning problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.21/LIPIcs.APPROX-RANDOM.2020.21.pdf
memory-sample tradeoffs
bounded storage cryptography
Goldreich’s local PRG
distinguishing problems
refuting CSPs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
22:1
22:23
10.4230/LIPIcs.APPROX/RANDOM.2020.22
article
Streaming Verification for Graph Problems: Optimal Tradeoffs and Nonlinear Sketches
Chakrabarti, Amit
1
https://orcid.org/0000-0003-3633-9180
Ghosh, Prantar
1
Thaler, Justin
2
Dartmouth College, Hanover, NH, USA
Georgetown University, Washington, DC, USA
We study graph computations in an enhanced data streaming setting, where a space-bounded client reading the edge stream of a massive graph may delegate some of its work to a cloud service. We seek algorithms that allow the client to verify a purported proof sent by the cloud service that the work done in the cloud is correct. A line of work starting with Chakrabarti et al. (ICALP 2009) has provided such algorithms, which we call schemes, for several statistical and graph-theoretic problems, many of which exhibit a tradeoff between the length of the proof and the space used by the streaming verifier.
This work designs new schemes for a number of basic graph problems - including triangle counting, maximum matching, topological sorting, and single-source shortest paths - where past work had either failed to obtain smooth tradeoffs between these two key complexity measures or only obtained suboptimal tradeoffs. Our key innovation is having the verifier compute certain nonlinear sketches of the input stream, leading to either new or improved tradeoffs. In many cases, our schemes in fact provide optimal tradeoffs up to logarithmic factors.
Specifically, for most graph problems that we study, it is known that the product of the verifier’s space cost v and the proof length h must be at least Ω(n²) for n-vertex graphs. However, matching upper bounds are only known for a handful of settings of h and v on the curve h ⋅ v = Θ̃(n²). For example, for counting triangles and maximum matching, schemes with costs lying on this curve are only known for (h = Õ(n²), v = Õ(1)), (h = Õ(n), v = Õ(n)), and the trivial (h = Õ(1), v = Õ(n²)). A major message of this work is that by exploiting nonlinear sketches, a significant "portion" of costs on the tradeoff curve h ⋅ v = n² can be achieved.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.22/LIPIcs.APPROX-RANDOM.2020.22.pdf
data streams
interactive proofs
Arthur-Merlin
graph algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
23:1
23:15
10.4230/LIPIcs.APPROX/RANDOM.2020.23
article
Disjointness Through the Lens of Vapnik–Chervonenkis Dimension: Sparsity and Beyond
Bhattacharya, Anup
1
Chakraborty, Sourav
1
Ghosh, Arijit
1
Mishra, Gopinath
1
Paraashar, Manaswi
1
Indian Statistical Institute, Kolkata, India
The disjointness problem - where Alice and Bob are given two subsets of {1, … , n} and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be Θ(n), it is also known that if the sets are assumed to be drawn from some restricted set systems then the communication complexity can be much lower. In this work, we explore how communication complexity measures change with respect to the complexity of the underlying set system. The complexity measure for the set system that we use in this work is the Vapnik–Chervonenkis (VC) dimension. More precisely, on any set system with VC dimension bounded by d, we analyze how large can the deterministic and randomized communication complexities be, as a function of d and n. The d-sparse set disjointness problem, where the sets have size at most d, is one such set system with VC dimension d. The deterministic and the randomized communication complexities of the d-sparse set disjointness problem have been well studied and is known to be Θ (d log ({n}/{d})) and Θ(d), respectively, in the multi-round communication setting. In this paper, we address the question of whether the randomized communication complexity is always upper bounded by a function of the VC dimension of the set system, and does there always exist a gap between the deterministic and randomized communication complexity for set systems with small VC dimension.
In this paper, we construct two natural set systems of VC dimension d, motivated from geometry. Using these set systems we show that the deterministic and randomized communication complexity can be Θ̃(dlog (n/d)) for set systems of VC dimension d and this matches the deterministic upper bound for all set systems of VC dimension d. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exists set systems of VC dimension d such that both deterministic and randomized (one-way and multi-round) complexities for the set intersection problem can be as high as Θ(dlog (n/d)), and this is tight among all set systems of VC dimension d.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.23/LIPIcs.APPROX-RANDOM.2020.23.pdf
Communication complexity
VC dimension
Sparsity
and Geometric Set System
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
24:1
24:13
10.4230/LIPIcs.APPROX/RANDOM.2020.24
article
Testing Data Binnings
Canonne, Clément L.
1
https://orcid.org/0000-0001-7153-5211
Wimmer, Karl
2
IBM Research, Almaden, CA, USA
Duquesne University, Pittsburgh, PA, USA
Motivated by the question of data quantization and "binning," we revisit the problem of identity testing of discrete probability distributions. Identity testing (a.k.a. one-sample testing), a fundamental and by now well-understood problem in distribution testing, asks, given a reference distribution (model) 𝐪 and samples from an unknown distribution 𝐩, both over [n] = {1,2,… ,n}, whether 𝐩 equals 𝐪, or is significantly different from it.
In this paper, we introduce the related question of identity up to binning, where the reference distribution 𝐪 is over k ≪ n elements: the question is then whether there exists a suitable binning of the domain [n] into k intervals such that, once "binned," 𝐩 is equal to 𝐪. We provide nearly tight upper and lower bounds on the sample complexity of this new question, showing both a quantitative and qualitative difference with the vanilla identity testing one, and answering an open question of Canonne [Clément L. Canonne, 2019]. Finally, we discuss several extensions and related research directions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.24/LIPIcs.APPROX-RANDOM.2020.24.pdf
property testing
distribution testing
identity testing
hypothesis testing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
25:1
25:22
10.4230/LIPIcs.APPROX/RANDOM.2020.25
article
Chernoff Bound for High-Dimensional Expanders
Kaufman, Tali
1
Sharakanski, Ella
1
Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
We generalize the expander Chernoff bound to high-dimensional expanders. The expander Chernoff bound is an essential property of expanders, first proved by Gillman [Gillman, 1993]. Given a graph G and a function f on the vertices, it states that the probability of f’s mean sampled via a random walk on G to deviate from its actual mean, has a bound that depends on the spectral gap of the walk and decreases exponentially as the walk’s length increases.
We are interested in obtaining an analog Chernoff bound for high order walks on high-dimensional expanders. A naive generalization of the expander Chernoff bound from expander graphs to high-dimensional expanders gives a very poor bound due to obstructions that occur in high-dimensional expanders and are not present in (one-dimensional) expander graphs. Because of these obstructions, the spectral gap of high-order random walks is inherently small.
A natural question that arises is how to get a meaningful Chernoff bound for high-dimensional expanders. In this paper, we manage to get a strong Chernoff bound for high-dimensional expanders by looking beyond the spectral gap.
First, we prove an expander Chernoff bound that depends on a notion that we call the "shrinkage of a function" instead of the spectral gap. In one-dimensional expanders, the shrinkage of any function with zero-mean is bounded by λ(M). Therefore, the spectral gap is just the one-dimensional manifestation of the shrinkage.
Next, we show that in good high-dimensional expanders, the shrinkage of functions that "do not come from below" is good. A function does not come from below if from any local point of view (called "link") its mean is zero.
Finally, we prove a high-dimensional Chernoff bound that captures the expansion of the complex. When the function on the faces has a small variance and does not "come from below", our bound is better than the naive high-dimensional expander Chernoff bound.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.25/LIPIcs.APPROX-RANDOM.2020.25.pdf
High Dimensional Expanders
Random Walks
Tail Bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
26:1
26:20
10.4230/LIPIcs.APPROX/RANDOM.2020.26
article
Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems
Rashtchian, Cyrus
1
Woodruff, David P.
2
Zhu, Hanlin
3
Department of Computer Science & Engineering, UC San Diego, CA, USA
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of u^{T}Mv over a fixed field 𝔽 for a specified pair of vectors u,v ∈ 𝔽ⁿ. To motivate these queries, we observe that they generalize many previously studied models, such as independent set queries, cut queries, and standard graph queries. They also specialize the recently studied matrix-vector query model. Our work is exploratory and broad, and we provide new upper and lower bounds for a wide variety of problems, spanning linear algebra, statistics, and graphs. Many of our results are nearly tight, and we use diverse techniques from linear algebra, randomized algorithms, and communication complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.26/LIPIcs.APPROX-RANDOM.2020.26.pdf
Query complexity
property testing
vector-matrix-vector
linear algebra
statistics
graph parameter estimation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
27:1
27:19
10.4230/LIPIcs.APPROX/RANDOM.2020.27
article
Almost Optimal Distribution-Free Sample-Based Testing of k-Modality
Ron, Dana
1
https://orcid.org/0000-0001-6576-7200
Rosin, Asaf
1
Tel Aviv University, Israel
For an integer k ≥ 0, a sequence σ = σ₁,… ,σ_n over a fully ordered set is k-modal, if there exist indices 1 = a₀ < a₁ < … < a_{k+1} = n such that for each i, the subsequence σ_{a_i},… ,σ_{a_{i+1}} is either monotonically non-decreasing or monotonically non-increasing. The property of k-modality is a natural extension of monotonicity, which has been studied extensively in the area of property testing. We study one-sided error property testing of k-modality in the distribution-free sample-based model. We prove an upper bound of O({√{kn}log k}/ε) on the sample complexity, and an almost matching lower bound of Ω(√{kn}/ε). When the underlying distribution is uniform, we obtain a completely tight bound of Θ(√{kn/ε}), which generalizes what is known for sample-based testing of monotonicity under the uniform distribution.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.27/LIPIcs.APPROX-RANDOM.2020.27.pdf
Sample-based property testing
Distribution-free property testing
k-modality
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
28:1
28:16
10.4230/LIPIcs.APPROX/RANDOM.2020.28
article
When Is Amplification Necessary for Composition in Randomized Query Complexity?
Ben-David, Shalev
1
Göös, Mika
2
Kothari, Robin
3
Watson, Thomas
4
University of Waterloo, Canada
Stanford University, CA, USA
Microsoft Quantum and Microsoft Research, Redmond, WA, USA
University of Memphis, TN, USA
Suppose we have randomized decision trees for an outer function f and an inner function g. The natural approach for obtaining a randomized decision tree for the composed function (f∘ gⁿ)(x¹,…,xⁿ) = f(g(x¹),…,g(xⁿ)) involves amplifying the success probability of the decision tree for g, so that a union bound can be used to bound the error probability over all the coordinates. The amplification introduces a logarithmic factor cost overhead. We study the question: When is this log factor necessary? We show that when the outer function is parity or majority, the log factor can be necessary, even for models that are more powerful than plain randomized decision trees. Our results are related to, but qualitatively strengthen in various ways, known results about decision trees with noisy inputs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.28/LIPIcs.APPROX-RANDOM.2020.28.pdf
Amplification
composition
query complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
29:1
29:23
10.4230/LIPIcs.APPROX/RANDOM.2020.29
article
On Multilinear Forms: Bias, Correlation, and Tensor Rank
Bhrushundi, Abhishek
1
Harsha, Prahladh
2
Hatami, Pooya
3
Kopparty, Swastik
4
Kumar, Mrinal
5
Rutgers University, Piscataway, NJ, USA
Tata Institute of Fundamental Research, Mumbai, India
Dept. of Computer Science & Engineering, The Ohio State University, Columbus, OH, USA
Dept. of Computer Science & Dept. of Mathematics, Rutgers University, Piscataway, NJ, USA
Dept. of Computer Science & Engineering, IIT Bombay, India
In this work, 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 F₂. We show the following results for multilinear forms and tensors.
Correlation bounds. We show that a random d-linear form has exponentially low correlation with low-degree polynomials. More precisely, for d = 2^{o(k)}, we show that a random d-linear form f(X₁,X₂, … , X_d) : (F₂^{k}) ^d → F₂ has correlation 2^{-k(1-o(1))} with any polynomial of degree at most d/2 with high probability. This result is proved by giving near-optimal bounds on the bias of a random d-linear form, which is in turn proved by giving near-optimal bounds on the probability that a sum of t random d-dimensional rank-1 tensors is identically zero.
Tensor rank vs Bias. We show that if a 3-dimensional tensor has small rank then its bias, when viewed as a 3-linear form, is large. More precisely, given any 3-dimensional tensor T: [k]³ → F₂ of rank at most t, the bias of the 3-linear form f_T(X₁, X₂, X₃) : = ∑_{(i₁, i₂, i₃) ∈ [k]³} T(i₁, i₂, i₃)⋅ X_{1,i₁}⋅ X_{2,i₂}⋅ X_{3,i₃} is at least (3/4)^t. This bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds. In particular, we use this approach to give a new proof that the finite field multiplication tensor has tensor rank at least 3.52 k, which is the best known rank lower bound for any explicit tensor in three dimensions over F₂. Moreover, this relation between bias and tensor rank holds for d-dimensional tensors for any fixed d.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.29/LIPIcs.APPROX-RANDOM.2020.29.pdf
polynomials
Boolean functions
tensor rank
bias
correlation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
30:1
30:11
10.4230/LIPIcs.APPROX/RANDOM.2020.30
article
On the List Recoverability of Randomly Punctured Codes
Lund, Ben
1
https://orcid.org/0000-0002-0141-0621
Potukuchi, Aditya
2
https://orcid.org/0000-0001-7233-7532
Department of Mathematics, Princeton University, NJ, USA
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.30/LIPIcs.APPROX-RANDOM.2020.30.pdf
List recovery
randomly punctured codes
Reed-Solomon codes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
31:1
31:22
10.4230/LIPIcs.APPROX/RANDOM.2020.31
article
On Perturbation Resilience of Non-Uniform k-Center
Bandyapadhyay, Sayan
1
https://orcid.org/0000-0001-8875-0102
Department of Informatics, University of Bergen, Norway
The Non-Uniform k-center (NUkC) problem has recently been formulated by Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the classical k-center clustering problem. In NUkC, given a set of n points P in a metric space and non-negative numbers r₁, r₂, … , r_k, the goal is to find the minimum dilation α and to choose k balls centered at the points of P with radius α⋅ r_i for 1 ≤ i ≤ k, such that all points of P are contained in the union of the chosen balls. They showed that the problem is NP-hard to approximate within any factor even in tree metrics. On the other hand, they designed a "bi-criteria" constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the r_i’s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial [Combinatorics, Probability and Computing, 2012]. We show that the problem under 2-perturbation resilience is polynomial time solvable when the r_i’s belong to a constant sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any "good" approximation for the problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.31/LIPIcs.APPROX-RANDOM.2020.31.pdf
Non-Uniform k-center
stability
clustering
perturbation resilience
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
32:1
32:18
10.4230/LIPIcs.APPROX/RANDOM.2020.32
article
Low-Rank Binary Matrix Approximation in Column-Sum Norm
Fomin, Fedor V.
1
https://orcid.org/0000-0003-1955-4612
Golovach, Petr A.
1
https://orcid.org/0000-0002-2619-2990
Panolan, Fahad
2
https://orcid.org/0000-0001-6213-8687
Simonov, Kirill
1
https://orcid.org/0000-0001-9436-7310
Department of Informatics, University of Bergen, Norway
Department of Computer Science and Engineering, IIT Hyderabad, India
We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show that for every ε ∈ (0, 1), there is a {randomized} (1+ε)-approximation algorithm for 𝓁₁-Rank-r Approximation over {GF}(2) of running time m^{O(1)}n^{O(2^{4r}⋅ ε^{-4})}. This is the first polynomial time approximation scheme (PTAS) for this problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.32/LIPIcs.APPROX-RANDOM.2020.32.pdf
Binary Matrix Factorization
PTAS
Column-sum norm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
33:1
33:21
10.4230/LIPIcs.APPROX/RANDOM.2020.33
article
Pinning down the Strong Wilber 1 Bound for Binary Search Trees
Chalermsook, Parinya
1
Chuzhoy, Julia
2
Saranurak, Thatchaphol
2
Aalto University, Finland
Toyota Technological Institute at Chicago, IL, USA
The dynamic optimality conjecture, postulating the existence of an O(1)-competitive online algorithm for binary search trees (BSTs), is among the most fundamental open problems in dynamic data structures. Despite extensive work and some notable progress, including, for example, the Tango Trees (Demaine et al., FOCS 2004), that give the best currently known O(log log n)-competitive algorithm, the conjecture remains widely open. One of the main hurdles towards settling the conjecture is that we currently do not have approximation algorithms achieving better than an O(log log n)-approximation, even in the offline setting. All known non-trivial algorithms for BST’s so far rely on comparing the algorithm’s cost with the so-called Wilber’s first bound (WB-1). Therefore, establishing the worst-case relationship between this bound and the optimal solution cost appears crucial for further progress, and it is an interesting open question in its own right.
Our contribution is two-fold. First, we show that the gap between the WB-1 bound and the optimal solution value can be as large as Ω(log log n/ log log log n); in fact, we show that the gap holds even for several stronger variants of the bound. Second, we provide a simple algorithm, that, given an integer D > 0, obtains an O(D)-approximation in time exp (O (n^{1/2^{Ω(D)}}log n)). In particular, this yields a constant-factor approximation algorithm with sub-exponential running time. Moreover, we obtain a simpler and cleaner efficient O(log log n)-approximation algorithm that can be used in an online setting. Finally, we suggest a new bound, that we call the Guillotine Bound, that is stronger than WB-1, while maintaining its algorithm-friendly nature, that we hope will lead to better algorithms. All our results use the geometric interpretation of the problem, leading to cleaner and simpler analysis.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.33/LIPIcs.APPROX-RANDOM.2020.33.pdf
Binary search trees
Dynamic optimality
Wilber bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
34:1
34:14
10.4230/LIPIcs.APPROX/RANDOM.2020.34
article
Revisiting Alphabet Reduction in Dinur’s PCP
Guruswami, Venkatesan
1
https://orcid.org/0000-0001-7926-3396
Opršal, Jakub
2
https://orcid.org/0000-0003-1245-3456
Sandeep, Sai
1
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Computer Science Department, Durham University, UK
Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.34/LIPIcs.APPROX-RANDOM.2020.34.pdf
PCP theorem
CSP
discrete Fourier analysis
label cover
long code
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
35:1
35:23
10.4230/LIPIcs.APPROX/RANDOM.2020.35
article
L_p Pattern Matching in a Stream
Starikovskaya, Tatiana
1
Svagerka, Michal
2
Uznański, Przemysław
3
DIENS, École normale supérieure, PSL Research University, Paris, France
ETH Zürich, Switzerland
Institute of Computer Science, University of Wrocław, Poland
We consider the problem of computing distance between a pattern of length n and all n-length subwords of a text in the streaming model.
In the streaming setting, only the Hamming distance (L₀) has been studied. It is known that computing the exact Hamming distance between a pattern and a streaming text requires Ω(n) space (folklore). Therefore, to develop sublinear-space solutions, one must relax their requirements. One possibility to do so is to compute only the distances bounded by a threshold k, see [SODA'19, Clifford, Kociumaka, Porat] and references therein. The motivation for this variant of this problem is that we are interested in subwords of the text that are similar to the pattern, i.e. in subwords such that the distance between them and the pattern is relatively small.
On the other hand, the main application of the streaming setting is processing large-scale data, such as biological data. Recent advances in hardware technology allow generating such data at a very high speed, but unfortunately, the produced data may contain about 10% of noise [Biol. Direct.'07, Klebanov and Yakovlev]. To analyse such data, it is not sufficient to consider small distances only. A possible workaround for this issue is the (1±ε)-approximation. This line of research was initiated in [ICALP'16, Clifford and Starikovskaya] who gave a (1±ε)-approximation algorithm with space 𝒪~(ε^{-5}√n).
In this work, we show a suite of new streaming algorithms for computing the Hamming, L₁, L₂ and general L_p (0 < p < 2) distances between the pattern and the text. Our results significantly extend over the previous result in this setting. In particular, for the Hamming distance and for the L_p distance when 0 < p ≤ 1 we show a streaming algorithm that uses 𝒪~(ε^{-2}√n) space for polynomial-size alphabets.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.35/LIPIcs.APPROX-RANDOM.2020.35.pdf
streaming algorithms
approximate pattern matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
36:1
36:21
10.4230/LIPIcs.APPROX/RANDOM.2020.36
article
Computing Bi-Lipschitz Outlier Embeddings into the Line
Chubarian, Karine
1
Sidiropoulos, Anastasios
2
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, IL, USA
Department of Computer Science, University of Illinois at Chicago, IL, USA
The problem of computing a bi-Lipschitz embedding of a graphical metric into the line with minimum distortion has received a lot of attention. The best-known approximation algorithm computes an embedding with distortion O(c²), where c denotes the optimal distortion [Bădoiu et al. 2005]. We present a bi-criteria approximation algorithm that extends the above results to the setting of outliers.
Specifically, we say that a metric space (X,ρ) admits a (k,c)-embedding if there exists K ⊂ X, with |K| = k, such that (X⧵ K, ρ) admits an embedding into the line with distortion at most c. Given k ≥ 0, and a metric space that admits a (k,c)-embedding, for some c ≥ 1, our algorithm computes a (poly(k, c, log n), poly(c))-embedding in polynomial time. This is the first algorithmic result for outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier embeddings where known only for the case of additive distortion.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.36/LIPIcs.APPROX-RANDOM.2020.36.pdf
metric embeddings
outliers
distortion
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
37:1
37:16
10.4230/LIPIcs.APPROX/RANDOM.2020.37
article
Online Minimum Cost Matching with Recourse on the Line
Megow, Nicole
1
https://orcid.org/0000-0002-3531-7644
Nölke, Lukas
1
https://orcid.org/0000-0003-0523-0668
Department for Mathematics and Computer Science, University of Bremen, Germany
In online minimum cost matching on the line, n requests appear one by one and have to be matched immediately and irrevocably to a given set of servers, all on the real line. The goal is to minimize the sum of distances from the requests to their respective servers. Despite all research efforts, it remains an intriguing open question whether there exists an O(1)-competitive algorithm. The best known online algorithm by Raghvendra [S. Raghvendra, 2018] achieves a competitive factor of Θ(log n). This result matches a lower bound of Ω(log n) [A. Antoniadis et al., 2018] that holds for a quite large class of online algorithms, including all deterministic algorithms in the literature.
In this work, we approach the problem in a recourse model where we allow to revoke online decisions to some extent, i.e., we allow to reassign previously matched edges. We show an O(1)-competitive algorithm for online matching on the line with amortized recourse of O(log n). This is the first non-trivial result for min-cost bipartite matching with recourse. For so-called alternating instances, with no more than one request between two servers, we obtain a near-optimal result. We give a (1+ε)-competitive algorithm that reassigns any request at most O(ε^{-1.001}) times. This special case is interesting as the aforementioned quite general lower bound Ω(log n) holds for such instances.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.37/LIPIcs.APPROX-RANDOM.2020.37.pdf
min-cost matching in bipartite graphs
recourse
competitive analysis
online
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
38:1
38:16
10.4230/LIPIcs.APPROX/RANDOM.2020.38
article
Hardness of Approximation of (Multi-)LCS over Small Alphabet
Bhangale, Amey
1
Chakraborty, Diptarka
2
Kumar, Rajendra
3
2
University of California Riverside, CA, USA
National University of Singapore, Singapore
IIT Kanpur, India
The problem of finding longest common subsequence (LCS) is one of the fundamental problems in computer science, which finds application in fields such as computational biology, text processing, information retrieval, data compression etc. It is well known that (decision version of) the problem of finding the length of a LCS of an arbitrary number of input sequences (which we refer to as Multi-LCS problem) is NP-complete. Jiang and Li [SICOMP'95] showed that if Max-Clique is hard to approximate within a factor of s then Multi-LCS is also hard to approximate within a factor of Θ(s). By the NP-hardness of the problem of approximating Max-Clique by Zuckerman [ToC'07], for any constant δ > 0, the length of a LCS of arbitrary number of input sequences of length n each, cannot be approximated within an n^{1-δ}-factor in polynomial time unless {P}={NP}. However, the reduction of Jiang and Li assumes the alphabet size to be Ω(n). So far no hardness result is known for the problem of approximating Multi-LCS over sub-linear sized alphabet. On the other hand, it is easy to get 1/|Σ|-factor approximation for strings of alphabet Σ.
In this paper, we make a significant progress towards proving hardness of approximation over small alphabet by showing a polynomial-time reduction from the well-studied densest k-subgraph problem with perfect completeness to approximating Multi-LCS over alphabet of size poly(n/k). As a consequence, from the known hardness result of densest k-subgraph problem (e.g. [Manurangsi, STOC'17]) we get that no polynomial-time algorithm can give an n^{-o(1)}-factor approximation of Multi-LCS over an alphabet of size n^{o(1)}, unless the Exponential Time Hypothesis is false.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.38/LIPIcs.APPROX-RANDOM.2020.38.pdf
Longest common subsequence
Hardness of approximation
ETH-hardness
Densest k-subgraph problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
39:1
39:21
10.4230/LIPIcs.APPROX/RANDOM.2020.39
article
On Approximating Degree-Bounded Network Design Problems
Guo, Xiangyu
1
Kortsarz, Guy
2
Laekhanukit, Bundit
3
https://orcid.org/0000-0002-4476-8914
Li, Shi
1
Vaz, Daniel
4
Xian, Jiayi
1
Department of Computer Science and Engineering, University at Buffalo, NY, USA
Department of Computer Science, Rutgers University Camden, NJ, USA
ITCS, Shanghai University of Finance and Economics, China
Operations Research Group, TU Munich, Germany
Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/log log k)-approximation algorithms for the problem, which are tight under popular complexity assumptions.
In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasi-polynomial time (O(log n log k), O(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem.
While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)-factor away from the approximation lower bound of Ω(log n) from the Set Cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.39/LIPIcs.APPROX-RANDOM.2020.39.pdf
Directed Steiner Tree
Group Steiner Tree
degree-bounded
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
40:1
40:20
10.4230/LIPIcs.APPROX/RANDOM.2020.40
article
Permutation Strikes Back: The Power of Recourse in Online Metric Matching
Gupta, Varun
1
Krishnaswamy, Ravishankar
2
Sandeep, Sai
3
University of Chicago, IL, USA
Microsoft Research India, Bangalore, India
Carnegie Mellon University, Pittsburgh, PA, USA
In this paper, we study the online metric matching with recourse (OMM-Recourse) problem. Given a metric space with k servers, a sequence of clients is revealed online. A client must be matched to an available server on arrival. Unlike the classical online matching model where the match is irrevocable, the recourse model permits the algorithm to rematch existing clients upon the arrival of a new client. The goal is to maintain an online matching with a near-optimal total cost, while at the same time not rematching too many clients.
For the classical online metric matching problem without recourse, the optimal competitive ratio for deterministic algorithms is 2k-1, and the best-known randomized algorithms have competitive ratio O(log² k). For the much-studied special case of line metric, the best-known algorithms have competitive ratios of O(log k). Improving these competitive ratios (or showing lower bounds) are important open problems in this line of work.
In this paper, we show that logarithmic recourse significantly improves the quality of matchings we can maintain online. For general metrics, we show a deterministic O(log k)-competitive algorithm, with O(log k) recourse per client, an exponential improvement over the 2k-1 lower bound without recourse. For line metrics we show a deterministic 3-competitive algorithm with O(log k) amortized recourse, again improving the best-known O(log k)-competitive algorithms without recourse. The first result (general metrics) simulates a batched version of the classical algorithm for OMM called Permutation. The second result (line metric) also uses Permutation as the foundation but makes non-trivial changes to the matching to balance the competitive ratio and recourse.
Finally, we also consider the model when both clients and servers may arrive or depart dynamically, and exhibit a simple randomized O(log n)-competitive algorithm with O(log Δ) recourse, where n and Δ are the number of points and the aspect ratio of the underlying metric. We remark that no non-trivial bounds are possible in this fully-dynamic model when no recourse is allowed.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.40/LIPIcs.APPROX-RANDOM.2020.40.pdf
online algorithms
bipartite matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
41:1
41:17
10.4230/LIPIcs.APPROX/RANDOM.2020.41
article
How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut
Chlamtáč, Eden
1
https://orcid.org/0000-0002-0296-0107
Kolman, Petr
2
https://orcid.org/0000-0003-2235-0506
Ben Gurion University of the Negev, Beer Sheva, Israel
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic
The Minimum Length Bounded Cut problem is a natural variant of Minimum Cut: given a graph, terminal nodes s,t and a parameter L, find a minimum cardinality set of nodes (other than s,t) whose removal ensures that the distance from s to t is greater than L. We focus on the approximability of the problem for bounded values of the parameter L.
The problem is solvable in polynomial time for L ≤ 4 and NP-hard for L ≥ 5. The best known algorithms have approximation factor ⌈ (L-1)/2⌉. It is NP-hard to approximate the problem within a factor of 1.17175 and Unique Games hard to approximate it within Ω(L), for any L ≥ 5. Moreover, for L = 5 the problem is 4/3-ε Unique Games hard for any ε > 0.
Our first result matches the hardness for L = 5 with a 4/3-approximation algorithm for this case, improving over the previous 2-approximation. For 6-bounded cuts we give a 7/4-approximation, improving over the previous best 3-approximation. More generally, we achieve approximation ratios that always outperform the previous ⌈ (L-1)/2⌉ guarantee for any (fixed) value of L, while for large values of L, we achieve a significantly better ((11/25)L+O(1))-approximation.
All our algorithms apply in the weighted setting, in both directed and undirected graphs, as well as for edge-cuts, which easily reduce to the node-cut variant. Moreover, by rounding the natural linear programming relaxation, our algorithms also bound the corresponding bounded-length flow-cut gaps.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.41/LIPIcs.APPROX-RANDOM.2020.41.pdf
Approximation Algorithms
Length Bounded Cuts
Cut-Flow Duality
Rounding of Linear Programms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
42:1
42:23
10.4230/LIPIcs.APPROX/RANDOM.2020.42
article
On the Facility Location Problem in Online and Dynamic Models
Guo, Xiangyu
1
Kulkarni, Janardhan
2
Li, Shi
1
Xian, Jiayi
1
Department of Computer Science and Engineering, University at Buffalo, NY, USA
The Algorithms Group, Microsoft Research, Redmond, WA, USA
In this paper we study the facility location problem in the online with recourse and dynamic algorithm models. In the online with recourse model, clients arrive one by one and our algorithm needs to maintain good solutions at all time steps with only a few changes to the previously made decisions (called recourse). We show that the classic local search technique can lead to a (1+√2+ε)-competitive online algorithm for facility location with only O(log n/ε log 1/ε) amortized facility and client recourse, where n is the total number of clients arrived during the process.
We then turn to the dynamic algorithm model for the problem, where the main goal is to design fast algorithms that maintain good solutions at all time steps. We show that the result for online facility location, combined with the randomized local search technique of Charikar and Guha [Charikar and Guha, 2005], leads to a (1+√2+ε)-approximation dynamic algorithm with total update time of Õ(n²) in the incremental setting against adaptive adversaries. The approximation factor of our algorithm matches the best offline analysis of the classic local search algorithm.
Finally, we study the fully dynamic model for facility location, where clients can both arrive and depart. Our main result is an O(1)-approximation algorithm in this model with O(|F|) preprocessing time and O(nlog³ D) total update time for the HST metric spaces, where |F| is the number of potential facility locations. Using the seminal results of Bartal [Bartal, 1996] and Fakcharoenphol, Rao and Talwar [Fakcharoenphol et al., 2003], which show that any arbitrary N-point metric space can be embedded into a distribution over HSTs such that the expected distortion is at most O(log N), we obtain an O(log |F|) approximation with preprocessing time of O(|F|²log |F|) and O(nlog³ D) total update time. The approximation guarantee holds in expectation for every time step of the algorithm, and the result holds in the oblivious adversary model.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.42/LIPIcs.APPROX-RANDOM.2020.42.pdf
Facility location
online algorithm
recourse
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
43:1
43:14
10.4230/LIPIcs.APPROX/RANDOM.2020.43
article
Nearly Optimal Embeddings of Flat Tori
Agarwal, Ishan
1
Regev, Oded
1
Tang, Yi
1
Courant Institute of Mathematical Sciences, New York University, NY, USA
We show that for any n-dimensional lattice ℒ ⊆ ℝⁿ, the torus ℝⁿ/ℒ can be embedded into Hilbert space with O(√{nlog n}) distortion. This improves the previously best known upper bound of O(n√{log n}) shown by Haviv and Regev (APPROX 2010, J. Topol. Anal. 2013) and approaches the lower bound of Ω(√n) due to Khot and Naor (FOCS 2005, Math. Ann. 2006).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.43/LIPIcs.APPROX-RANDOM.2020.43.pdf
Lattices
metric embeddings
flat torus
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
44:1
44:18
10.4230/LIPIcs.APPROX/RANDOM.2020.44
article
A Tight (3/2+ε) Approximation for Skewed Strip Packing
Gálvez, Waldo
1
https://orcid.org/0000-0002-6395-3322
Grandoni, Fabrizio
2
https://orcid.org/0000-0002-9676-4931
Ameli, Afrouz Jabal
2
https://orcid.org/0000-0001-5620-9039
Jansen, Klaus
3
https://orcid.org/0000-0001-8358-6796
Khan, Arindam
4
https://orcid.org/0000-0001-7505-1687
Rau, Malin
5
https://orcid.org/0000-0002-5710-560X
Technical University of Munich, Germany
IDSIA, USI-SUPSI, Manno, Switzerland
University of Kiel, Germany
Indian Institute of Science, Bangalore, India
Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP*, LIG, Grenoble, France
In the Strip Packing problem, we are given a vertical half-strip [0,W]× [0,+∞) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. Strip Packing generalizes classical well-studied problems such as Makespan Minimization on identical machines (when rectangle widths are identical) and Bin Packing (when rectangle heights are identical). It has applications in manufacturing, scheduling and energy consumption in smart grids among others. It is NP-hard to approximate this problem within a factor (3/2-ε) for any constant ε > 0 by a simple reduction from the Partition problem. The current best approximation factor for Strip Packing is (5/3+ε) by Harren et al. [Computational Geometry '14], and it is achieved with a fairly complex algorithm and analysis.
It seems plausible that Strip Packing admits a (3/2+ε)-approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ > 0, we call large the rectangles with width at least δ W and height at least δ OPT, and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, it is NP-hard to approximate within a factor (3/2-ε) and we provide an (almost) tight (3/2+ε)-approximation algorithm.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.44/LIPIcs.APPROX-RANDOM.2020.44.pdf
strip packing
approximation algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
45:1
45:15
10.4230/LIPIcs.APPROX/RANDOM.2020.45
article
Learning Lines with Ordinal Constraints
Fan, Bohan
1
Ihara, Diego
1
https://orcid.org/0000-0002-8468-0845
Mohammadi, Neshat
1
Sgherzi, Francesco
1
Sidiropoulos, Anastasios
1
Valizadeh, Mina
1
Department of Computer Science, University of Illinois at Chicago, IL, USA
We study the problem of finding a mapping f from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points (u,v,w) asserts that |f(u)-f(v)| < |f(u)-f(w)|. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies (1-ε)-fraction of all constraints, our algorithm computes a solution that satisfies (1-O(ε^{1/8}))-fraction of all constraints, in time O(n⁷) + (1/ε)^{O(1/ε^{1/8})} n.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.45/LIPIcs.APPROX-RANDOM.2020.45.pdf
metric learning
embedding into the line
ordinal constraints
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
46:1
46:24
10.4230/LIPIcs.APPROX/RANDOM.2020.46
article
Improved Circular k-Mismatch Sketches
Golan, Shay
1
https://orcid.org/0000-0001-8357-2802
Kociumaka, Tomasz
1
https://orcid.org/0000-0002-2477-1702
Kopelowitz, Tsvi
1
https://orcid.org/0000-0002-3525-8314
Porat, Ely
1
https://orcid.org/0000-0001-6912-5766
Uznański, Przemysław
2
https://orcid.org/0000-0002-8652-0490
Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Institute of Computer Science, University of Wrocław, Poland
The shift distance sh(S₁,S₂) between two strings S₁ and S₂ of the same length is defined as the minimum Hamming distance between S₁ and any rotation (cyclic shift) of S₂. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings S₁ and S₂ of length n are given to two identical players (encoders), who independently compute sketches (summaries) sk(S₁) and sk(S₂), respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) sh(S₁,S₂) with high probability.
This paper primarily focuses on the more general k-mismatch version of the problem, where the decoder is allowed to declare a failure if sh(S₁,S₂) > k, where k is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular k-mismatch sketches of size Õ(k+D(n)), where D(n) is the number of divisors of n. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches.
We circumvent this lower bound by designing a (non-linear) exact circular k-mismatch sketch of size Õ(k); this size matches communication-complexity lower bounds. We also design (1± ε)-approximate circular k-mismatch sketch of size Õ(min(ε^{-2}√k, ε^{-1.5}√n)), which improves upon an Õ(ε^{-2}√n)-size sketch of Crouch and McGregor (APPROX'11).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.46/LIPIcs.APPROX-RANDOM.2020.46.pdf
Hamming distance
k-mismatch
sketches
rotation
cyclic shift
communication complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
47:1
47:22
10.4230/LIPIcs.APPROX/RANDOM.2020.47
article
On Guillotine Separability of Squares and Rectangles
Khan, Arindam
1
Pittu, Madhusudhan Reddy
2
Indian Institute of Science, Bangalore, India
Indian Institute of Technology, Kharagpur, India
Guillotine separability of rectangles has recently gained prominence in combinatorial optimization, computational geometry, and combinatorics. Consider a given large stock unit (say glass or wood) and we need to cut out a set of required rectangles from it. Many cutting technologies allow only end-to-end cuts called guillotine cuts. Guillotine cuts occur in stages. Each stage consists of either only vertical cuts or only horizontal cuts. In k-stage packing, the number of cuts to obtain each rectangle from the initial packing is at most k (plus an additional trimming step to separate the rectangle itself from a waste area). Pach and Tardos [Pach and Tardos, 2000] studied the following question: Given a set of n axis-parallel rectangles (in the weighted case, each rectangle has an associated weight), cut out as many rectangles (resp. weight) as possible using a sequence of guillotine cuts. They provide a guillotine cutting sequence that recovers 1/(2 log n)-fraction of rectangles (resp. weights). Abed et al. [Fidaa Abed et al., 2015] claimed that a guillotine cutting sequence can recover a constant fraction for axis-parallel squares. They also conjectured that for any set of rectangles, there exists a sequence of axis-parallel guillotine cuts that recovers a constant fraction of rectangles. This conjecture, if true, would yield a combinatorial O(1)-approximation for Maximum Independent Set of Rectangles (MISR), a long-standing open problem. We show the conjecture is not true, if we only allow o(log log n) stages (resp. o(log n/log log n)-stages for the weighted case). On the positive side, we show a simple O(n log n)-time 2-stage cut sequence that recovers 1/(1+log n)-fraction of rectangles. We improve the extraction of squares by showing that 1/40-fraction (resp. 1/160 in the weighted case) of squares can be recovered using guillotine cuts. We also show O(1)-fraction of rectangles, even in the weighted case, can be recovered for many special cases of rectangles, e.g. fat (bounded width/height), δ-large (large in one of the dimensions), etc. We show that this implies O(1)-factor approximation for Maximum Weighted Independent Set of Rectangles, the weighted version of MISR, for these classes of rectangles.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.47/LIPIcs.APPROX-RANDOM.2020.47.pdf
Guillotine cuts
Rectangles
Squares
Packing
k-stage packing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
48:1
48:18
10.4230/LIPIcs.APPROX/RANDOM.2020.48
article
Maximizing Throughput in Flow Shop Real-Time Scheduling
Ben Yamin, Lior
1
Li, Jing
2
Sarpatwar, Kanthi
3
https://orcid.org/0000-0002-7737-1200
Schieber, Baruch
2
Shachnai, Hadas
1
Computer Science Department, Technion, Haifa, Israel
Department of Computer Science, New Jersey Institute of Technology, Newark, NJ, USA
IBM T. J. Watson Research Center, Yorktown Heights, NY, USA
We consider scheduling real-time jobs in the classic flow shop model. The input is a set of n jobs, each consisting of m segments to be processed on m machines in the specified order, such that segment I_i of a job can start processing on machine M_i only after segment I_{i-1} of the same job completed processing on machine M_{i-1}, for 2 ≤ i ≤ m. Each job also has a release time, a due date, and a weight. The objective is to maximize the throughput (or, profit) of the n jobs, i.e., to find a subset of the jobs that have the maximum total weight and can complete processing on the m machines within their time windows. This problem has numerous real-life applications ranging from manufacturing to cloud and embedded computing platforms, already in the special case where m = 2. Previous work in the flow shop model has focused on makespan, flow time, or tardiness objectives. However, little is known for the flow shop model in the real-time setting. In this work, we give the first nontrivial results for this problem and present a pseudo-polynomial time (2m+1)-approximation algorithm for the problem on m ≥ 2 machines, where m is a constant. This ratio is essentially tight due to a hardness result of Ω(m/(log m)) for the approximation ratio. We further give a polynomial-time algorithm for the two-machine case, with an approximation ratio of (9+ε) where ε = O(1/n). We obtain better bounds for some restricted subclasses of inputs with two machines. To the best of our knowledge, this fundamental problem of throughput maximization in the flow shop scheduling model is studied here for the first time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.48/LIPIcs.APPROX-RANDOM.2020.48.pdf
Flow shop
real-time scheduling
throughput maximization
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
49:1
49:18
10.4230/LIPIcs.APPROX/RANDOM.2020.49
article
Maximizing the Correlation: Extending Grothendieck’s Inequality to Large Domains
Katzelnick, Dor
1
Schwartz, Roy
1
Department of Computer Science, Technion, Haifa, Israel
Correlation Clustering is an elegant model where given a graph with edges labeled + or -, the goal is to produce a clustering that agrees the most with the labels: + edges should reside within clusters and - edges should cross between clusters. In this work we study the MaxCorr objective, aiming to find a clustering that maximizes the difference between edges classified correctly and incorrectly. We focus on the case of bipartite graphs and present an improved approximation of 0.254, improving upon the known approximation of 0.219 given by Charikar and Wirth [FOCS`2004] and going beyond the 0.2296 barrier imposed by their technique. Our algorithm is inspired by Krivine’s method for bounding Grothendieck’s constant, and we extend this method to allow for more than two clusters in the output. Moreover, our algorithm leads to two additional results: (1) the first known approximation guarantees for MaxCorr where the output is constrained to have a bounded number of clusters; and (2) a natural extension of Grothendieck’s inequality to large domains.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.49/LIPIcs.APPROX-RANDOM.2020.49.pdf
Correlation Clustering
Grothendieck’s Inequality
Approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
50:1
50:22
10.4230/LIPIcs.APPROX/RANDOM.2020.50
article
Streaming Complexity of SVMs
Andoni, Alexandr
1
Burns, Collin
1
Li, Yi
2
Mahabadi, Sepideh
3
Woodruff, David P.
4
Columbia University, New York, NY, USA
Nanyang Technological University, Singapore, Singapore
Toyota Technological Institute at Chicago, IL, USA
Carnegie Mellon University, Pittsburgh, PA, USA
We study the space complexity of solving the bias-regularized SVM problem in the streaming model. In particular, given a data set (x_i,y_i) ∈ ℝ^d× {-1,+1}, the objective function is F_λ(θ,b) = λ/2‖(θ,b)‖₂² + 1/n∑_{i=1}ⁿ max{0,1-y_i(θ^Tx_i+b)} and the goal is to find the parameters that (approximately) minimize this objective. This is a classic supervised learning problem that has drawn lots of attention, including for developing fast algorithms for solving the problem approximately: i.e., for finding (θ,b) such that F_λ(θ,b) ≤ min_{(θ',b')} F_λ(θ',b')+ε.
One of the most widely used algorithms for approximately optimizing the SVM objective is Stochastic Gradient Descent (SGD), which requires only O(1/λε) random samples, and which immediately yields a streaming algorithm that uses O(d/λε) space. For related problems, better streaming algorithms are only known for smooth functions, unlike the SVM objective that we focus on in this work.
We initiate an investigation of the space complexity for both finding an approximate optimum of this objective, and for the related "point estimation" problem of sketching the data set to evaluate the function value F_λ on any query (θ, b). We show that, for both problems, for dimensions d = 1,2, one can obtain streaming algorithms with space polynomially smaller than 1/λε, which is the complexity of SGD for strongly convex functions like the bias-regularized SVM [Shalev-Shwartz et al., 2007], and which is known to be tight in general, even for d = 1 [Agarwal et al., 2009]. We also prove polynomial lower bounds for both point estimation and optimization. In particular, for point estimation we obtain a tight bound of Θ(1/√{ε}) for d = 1 and a nearly tight lower bound of Ω̃(d/{ε}²) for d = Ω(log(1/ε)). Finally, for optimization, we prove a Ω(1/√{ε}) lower bound for d = Ω(log(1/ε)), and show similar bounds when d is constant.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.50/LIPIcs.APPROX-RANDOM.2020.50.pdf
support vector machine
streaming algorithm
space lower bound
sketching algorithm
point estimation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
51:1
51:19
10.4230/LIPIcs.APPROX/RANDOM.2020.51
article
On the Parameterized Approximability of Contraction to Classes of Chordal Graphs
Gunda, Spoorthy
1
Jain, Pallavi
2
Lokshtanov, Daniel
3
Saurabh, Saket
4
5
Tale, Prafullkumar
6
Simon Fraser University, Burnaby, Canada
Indian Institute of Technology Jodhpur, India
University of California, Santa Barbara, CA, USA
The Institute of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Norway
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this paper, we study the F-Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k, F-Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. We obtain the following results for the F-Contraction problem.
- Clique Contraction is known to be FPT. However, unless NP ⊆ coNP/poly, it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (PSAKS). That is, it admits a (1 + ε)-approximate kernel with {O}(k^{f(ε)}) vertices for every ε > 0.
- Split Contraction is known to be W[1]-Hard. We deconstruct this intractability result in two ways. Firstly, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)-FPT-approximation algorithm for Split Contraction). Furthermore, we show that, assuming Gap-ETH, there is no (5/4-δ)-FPT-approximation algorithm for Split Contraction. Here, ε, δ > 0 are fixed constants.
- Chordal Contraction is known to be W[2]-Hard. We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1], there is no F(k)-FPT-approximation algorithm for Chordal Contraction. Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k)-FPT-approximation algorithm for the F-Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and |X| ≤ k, it outputs an edge set Y of size at most h(k) ⋅ k for which G/Y is in F. We find it extremely interesting that three closely related problems have different behavior with respect to FPT-approximation.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.51/LIPIcs.APPROX-RANDOM.2020.51.pdf
Graph Contraction
FPT-Approximation
Inapproximability
Lossy Kernels
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
52:1
52:18
10.4230/LIPIcs.APPROX/RANDOM.2020.52
article
Online Coloring of Short Intervals
Chybowska-Sokół, Joanna
1
https://orcid.org/0000-0002-4180-4342
Gutowski, Grzegorz
2
https://orcid.org/0000-0003-3313-1237
Junosza-Szaniawski, Konstanty
1
https://orcid.org/0000-0003-0352-8583
Mikos, Patryk
2
https://orcid.org/0000-0002-0519-0830
Polak, Adam
2
https://orcid.org/0000-0003-4925-774X
Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in [1,σ]. For σ = 1 it is the class of unit interval graphs, and for σ = ∞ the class of all interval graphs. Our focus is on intermediary classes. We present a (1+σ)-competitive algorithm, which beats the state of the art for 1 < σ < 2, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than 5/3-competitive for any σ > 1, nor better than 7/4-competitive for any σ > 2, and that no algorithm beats the 5/2 asymptotic competitive ratio for all, arbitrarily large, values of σ. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than 2.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.52/LIPIcs.APPROX-RANDOM.2020.52.pdf
Online algorithms
graph coloring
interval graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
53:1
53:16
10.4230/LIPIcs.APPROX/RANDOM.2020.53
article
Approximating Requirement Cut via a Configuration LP
Schwartz, Roy
1
Sharoni, Yotam
1
Department of Computer Science, Technion, Haifa, Israel
We consider the {Requirement Cut} problem, where given an undirected graph G = (V,E) equipped with non-negative edge weights c:E → R_{+}, and g groups of vertices X₁,…,X_{g} ⊆ V each equipped with a requirement r_i, the goal is to find a collection of edges F ⊆ E, with total minimum weight, such that once F is removed from G in the resulting graph every X_{i} is broken into at least r_{i} connected components. {Requirement Cut} captures multiple classic cut problems in graphs, e.g., {Multicut}, {Multiway Cut}, {Min k-Cut}, {Steiner k-Cut}, {Steiner Multicut}, and {Multi-Multiway Cut}. Nagarajan and Ravi [Algoritmica`10] presented an approximation of O(log{n}log{R}) for the problem, which was subsequently improved to O(log{g} log{k}) by Gupta, Nagarajan and Ravi [Operations Research Letters`10] (here R = ∑ _{i = 1}^g r_i and k = |∪ _{i = 1}^g X_i |). We present an approximation of O(Xlog{R} √{log{k}}log{log{k}}) for {Requirement Cut} (here X = max _{i = 1,…,g} {|X_i|}). Our approximation in general is incomparable to the above mentioned previous results, however when all groups are not too large, i.e., X = o((√{log{k}}log{g})/(log{R}log{log{k}})), it is better. Our algorithm is based on a new configuration linear programming relaxation for the problem, which is accompanied by a remarkably simple randomized rounding procedure.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.53/LIPIcs.APPROX-RANDOM.2020.53.pdf
Approximation
Requirement Cut
Sparsest Cut
Metric Embedding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
54:1
54:14
10.4230/LIPIcs.APPROX/RANDOM.2020.54
article
Parametrized Metrical Task Systems
Bubeck, Sébastien
1
Rabani, Yuval
2
Microsoft Research, Redmond, WA, USA
Hebrew University of Jerusalem, Israel
We consider parametrized versions of metrical task systems and metrical service systems, two fundamental models of online computing, where the constrained parameter is the number of possible distinct requests m. Such parametrization occurs naturally in a wide range of applications. Striking examples are certain power management problems, which are modeled as metrical task systems with m = 2. We characterize the competitive ratio in terms of the parameter m for both deterministic and randomized algorithms on hierarchically separated trees. Our findings uncover a rich and unexpected picture that differs substantially from what is known or conjectured about the unparametrized versions of these problems. For metrical task systems, we show that deterministic algorithms do not exhibit any asymptotic gain beyond one-level trees (namely, uniform metric spaces), whereas randomized algorithms do not exhibit any asymptotic gain even for one-level trees. In contrast, the special case of metrical service systems (subset chasing) behaves very differently. Both deterministic and randomized algorithms exhibit gain, for m sufficiently small compared to n, for any number of levels. Most significantly, they exhibit a large gain for uniform metric spaces and a smaller gain for two-level trees. Moreover, it turns out that in these cases (as well as in the case of metrical task systems for uniform metric spaces with m being an absolute constant), deterministic algorithms are essentially as powerful as randomized algorithms. This is surprising and runs counter to the ubiquitous intuition/conjecture that, for most problems that can be modeled as metrical task systems, the randomized competitive ratio is polylogarithmic in the deterministic competitive ratio.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.54/LIPIcs.APPROX-RANDOM.2020.54.pdf
online computing
competitive analysis
metrical task systems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
55:1
55:13
10.4230/LIPIcs.APPROX/RANDOM.2020.55
article
A Constant Factor Approximation for Capacitated Min-Max Tree Cover
Das, Syamantak
1
Jain, Lavina
1
Kumar, Nikhil
2
IIIT Delhi, India
IIT Delhi, India
Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (λ) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is kλ [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover λ vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.55/LIPIcs.APPROX-RANDOM.2020.55.pdf
Approximation Algorithms
Graph Algorithms
Min-Max Tree Cover
Vehicle Routing
Steiner Tree
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
56:1
56:16
10.4230/LIPIcs.APPROX/RANDOM.2020.56
article
An Extension of Plücker Relations with Applications to Subdeterminant Maximization
Anari, Nima
1
Vuong, Thuy-Duong
1
Department of Computer Science, Stanford University, CA, USA
Given a matrix A and k ≥ 0, we study the problem of finding the k × k submatrix of A with the maximum determinant in absolute value. This problem is motivated by the question of computing the determinant-based lower bound of cite{LSV86} on hereditary discrepancy, which was later shown to be an approximate upper bound as well [Matoušek, 2013]. The special case where k coincides with one of the dimensions of A has been extensively studied. Nikolov gave a 2^{O(k)}-approximation algorithm for this special case, matching known lower bounds; he also raised as an open problem the question of designing approximation algorithms for the general case.
We make progress towards answering this question by giving the first efficient approximation algorithm for general k× k subdeterminant maximization with an approximation ratio that depends only on k. Our algorithm finds a k^{O(k)}-approximate solution by performing a simple local search. Our main technical contribution, enabling the analysis of the approximation ratio, is an extension of Plücker relations for the Grassmannian, which may be of independent interest; Plücker relations are quadratic polynomial equations involving the set of k× k subdeterminants of a k× n matrix. We find an extension of these relations to k× k subdeterminants of general m× n matrices.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.56/LIPIcs.APPROX-RANDOM.2020.56.pdf
Plücker relations
determinant maximization
local search
exchange property
discrete concavity
discrepancy
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
57:1
57:19
10.4230/LIPIcs.APPROX/RANDOM.2020.57
article
Approximating Star Cover Problems
Gamlath, Buddhima
1
Grinberg, Vadim
2
École Polytechnique Fédérale de Lausanne, Switzerland
Toyota Technological Institute at Chicago, Chicago, IL, USA
Given a metric space (F ∪ C, d), we consider star covers of C with balanced loads. A star is a pair (i, C_i) where i ∈ F and C_i ⊆ C, and the load of a star is ∑_{j ∈ C_i} d(i, j). In minimum load k-star cover problem (MLkSC), one tries to cover the set of clients C using k stars that minimize the maximum load of a star, and in minimum size star cover (MSSC) one aims to find the minimum number of stars of load at most T needed to cover C, where T is a given parameter.
We obtain new bicriteria approximations for the two problems using novel rounding algorithms for their standard LP relaxations. For MLkSC, we find a star cover with (1+O(ε))k stars and O(1/ε²)OPT_MLk load where OPT_MLk is the optimum load. For MSSC, we find a star cover with O(1/ε²) OPT_MS stars of load at most (2 + O(ε)) T where OPT_MS is the optimal number of stars for the problem. Previously, non-trivial bicriteria approximations were known only when F = C.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.57/LIPIcs.APPROX-RANDOM.2020.57.pdf
star cover
approximation algorithms
lp rounding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
58:1
58:20
10.4230/LIPIcs.APPROX/RANDOM.2020.58
article
On the Approximability of Presidential Type Predicates
Huang, Neng
1
Potechin, Aaron
1
University of Chicago, IL, USA
Given a predicate P: {-1, 1}^k → {-1, 1}, let CSP(P) be the set of constraint satisfaction problems whose constraints are of the form P. We say that P is approximable if given a nearly satisfiable instance of CSP(P), there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that P is approximation resistant.
In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential type predicates P are approximable. More precisely, we prove the following result: for any δ₀ > 0, there exists a k₀ such that if k ≥ k₀, δ ∈ (δ₀,1 - 2/k], and {δ}k + k - 1 is an odd integer then the presidential type predicate P(x) = sign({δ}k{x₁} + ∑_{i = 2}^{k} {x_i}) is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.58/LIPIcs.APPROX-RANDOM.2020.58.pdf
constraint satisfaction problems
approximation algorithms
presidential type predicates
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
59:1
59:18
10.4230/LIPIcs.APPROX/RANDOM.2020.59
article
An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem
Hallgren, Sean
1
Lee, Eunou
1
Parekh, Ojas
2
Pennsylvania State University, State College, University Park, PA, USA
Sandia National Laboratories, Albuquerque, NM, USA
We present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.59/LIPIcs.APPROX-RANDOM.2020.59.pdf
approximation algorithm
quantum computing
local Hamiltonian
mean-field theory
randomized rounding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
60:1
60:17
10.4230/LIPIcs.APPROX/RANDOM.2020.60
article
Better and Simpler Learning-Augmented Online Caching
Wei, Alexander
1
Harvard University, Cambridge, MA, USA
Lykouris and Vassilvitskii (ICML 2018) introduce a model of online caching with machine-learned advice that marries the predictive power of machine learning with the robustness guarantees of competitive analysis. In this model, each page request is augmented with a prediction for when that page will next be requested. The goal is to design algorithms that (1) perform well when the predictions are accurate and (2) are robust in the sense of worst-case competitive analysis.
We continue the study of algorithms for online caching with machine-learned advice, following the work of Lykouris and Vassilvitskii as well as Rohatgi (SODA 2020). Our main contribution is a substantially simpler algorithm that outperforms all existing approaches. This algorithm is a black-box combination of an algorithm that just naïvely follows the predictions with an optimal competitive algorithm for online caching. We further show that combining the naïve algorithm with LRU in a black-box manner is optimal among deterministic algorithms for this problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.60/LIPIcs.APPROX-RANDOM.2020.60.pdf
Online caching
learning-augmented algorithms
beyond worst-case analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
61:1
61:12
10.4230/LIPIcs.APPROX/RANDOM.2020.61
article
A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case
Boyd, Sylvia
1
Cheriyan, Joseph
2
Cummings, Robert
2
Grout, Logan
2
Ibrahimpur, Sharat
2
https://orcid.org/0000-0002-1575-9648
Szigeti, Zoltán
3
Wang, Lu
2
School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Department of Combinatorics and Optimization, University of Waterloo, Canada
University Grenoble Alpes, CNRS, G-SCOP, France
Given a connected undirected graph G ̅ on n vertices, and non-negative edge costs c, the 2ECM problem is that of finding a 2-edge connected spanning multisubgraph of G ̅ of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the Traveling Salesman Problem on the metric closure of G ̅, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution x, Carr and Ravi (1998) showed that the integrality gap is at most 4/3: they show that the vector 4/3 x dominates a convex combination of incidence vectors of 2-edge connected spanning multisubgraphs of G ̅.
We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lovász’s splitting-off theorem. Our proof naturally leads to a 4/3-approximation algorithm for half-integral instances. Given a half-integral solution x to the LP for 2ECM, we give an O(n²)-time algorithm to obtain a 2-edge connected spanning multisubgraph of G ̅ whose cost is at most 4/3 c^T x.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.61/LIPIcs.APPROX-RANDOM.2020.61.pdf
2-Edge Connectivity
Approximation Algorithms
Subtour LP for TSP
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
62:1
62:19
10.4230/LIPIcs.APPROX/RANDOM.2020.62
article
Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints
Huang, Chien-Chung
1
Thiery, Theophile
2
Ward, Justin
2
CNRS, DI ENS, Université PSL, Paris, France
School of Mathematical Sciences, Queen Mary University of London, UK
We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general p-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of p arbitrary matroid constraints and p-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of p+1+ε using O(p/ε)-passes and requires storing only O(k) elements, where k is the maximum size of feasible solution. This immediately gives an O(1/ε)-pass (2+ε)-approximation for monotone submodular maximization in a matroid and (3+ε)-approximation for monotone submodular matching. Our algorithm is oblivious to the choice ε and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithms for general, non-negative submodular functions subject to a p-matchoid constraint. We show that a randomized O(p/ε)-pass algorithm storing O(p³klog(k)/ε³) elements gives a (p+1+γ+O(ε))-approximation, where γ is the guarantee of the best-known offline algorithm for the same problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.62/LIPIcs.APPROX-RANDOM.2020.62.pdf
submodular maximization
streaming algorithms
matroid
matchoid
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
63:1
63:20
10.4230/LIPIcs.APPROX/RANDOM.2020.63
article
Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs
Chan, Chun-Hsiang
1
Laekhanukit, Bundit
2
https://orcid.org/0000-0002-4476-8914
Wei, Hao-Ting
3
Zhang, Yuhao
4
Department of Computer Science, University of Michigan, Ann Arbor, MI, USA
Institute for Theoretical Computer Science, Shanghai University of Finance & Economics, China
Department of IEOR, Columbia University, New York, NY, USA
Department of Computer Science, The University of Hong Kong, China
In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ≥ 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ≥ 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.63/LIPIcs.APPROX-RANDOM.2020.63.pdf
Approximation Algorithms
Network Design
Directed Graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
176
64:1
64:22
10.4230/LIPIcs.APPROX/RANDOM.2020.64
article
Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams
Bakshi, Ainesh
1
Chepurko, Nadiia
2
Woodruff, David P.
1
Carnegie Mellon University, Pittsburgh, PA, USA
MIT, Cambridge, MA, USA
We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let α be the cardinality of the largest independent set. Our goal is to estimate α in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating β, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating α and β.
For unit-length intervals, we obtain a (2+ε)-approximation to α and β in poly(log(n)/ε) space. We also show a matching lower bound. Combined with the 3/2-approximation for insertion-only streams by Cabello and Perez-Lanterno [Cabello and Pérez-Lantero, 2017], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a (8√3/π)-approximation to α and β in poly(log(n)/ε) space, which is closely related to the hexagonal circle packing constant.
Finally, we provide algorithms for estimating α for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating α and β, where the parameter is the ratio of maximum to minimum interval length.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.64/LIPIcs.APPROX-RANDOM.2020.64.pdf
Weighted Maximum Independent Set
Geometric Graphs
Turnstile Streams