eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
1
1304
10.4230/LIPIcs.APPROX/RANDOM.2023
article
LIPIcs, Volume 275, APPROX/RANDOM 2023, Complete Volume
Megow, Nicole
1
https://orcid.org/0000-0002-3531-7644
Smith, Adam
2
https://orcid.org/0000-0001-9393-1127
University of Bremen, Germany
Boston University, MA, USA
LIPIcs, Volume 275, APPROX/RANDOM 2023, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023/LIPIcs.APPROX-RANDOM.2023.pdf
LIPIcs, Volume 275, APPROX/RANDOM 2023, Complete Volume
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
0:i
0:xxiv
10.4230/LIPIcs.APPROX/RANDOM.2023.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Megow, Nicole
1
https://orcid.org/0000-0002-3531-7644
Smith, Adam
2
https://orcid.org/0000-0001-9393-1127
University of Bremen, Germany
Boston University, MA, USA
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.0/LIPIcs.APPROX-RANDOM.2023.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
1:1
1:19
10.4230/LIPIcs.APPROX/RANDOM.2023.1
article
On Complexity of 1-Center in Various Metrics
Abboud, Amir
1
https://orcid.org/0000-0002-0502-4517
Bateni, MohammadHossein
2
https://orcid.org/0000-0003-1814-1293
Cohen-Addad, Vincent
3
https://orcid.org/0000-0002-0779-8962
Karthik C. S.
4
https://orcid.org/0000-0001-9105-364X
Seddighin, Saeed
5
https://orcid.org/0000-0002-3951-7096
Weizmann Institute of Science, Rehovot, Israel
Google Research, Mountain View, CA, USA
Google Research, Zürich, Switzerland
Rutgers University, New Brunswick, NJ, USA
Toyota Technological Institute at Chicago, IL, USA
We consider the classic 1-center problem: Given a set P of n points in a metric space find the point in P that minimizes the maximum distance to the other points of P. We study the complexity of this problem in d-dimensional 𝓁_p-metrics and in edit and Ulam metrics over strings of length d. Our results for the 1-center problem may be classified based on d as follows.
- Small d. Assuming the hitting set conjecture (HSC), we show that when d = ω(log n), no subquadratic algorithm can solve the 1-center problem in any of the 𝓁_p-metrics, or in the edit or Ulam metrics.
- Large d. When d = Ω(n), we extend our conditional lower bound to rule out subquartic algorithms for the 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a (1+ε)-approximation for 1-center in the Ulam metric with running time O_{ε}̃(nd+n²√d).
We also strengthen some of the above lower bounds by allowing approximation algorithms or by reducing the dimension d, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of n strings each of length n, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.1/LIPIcs.APPROX-RANDOM.2023.1.pdf
Center
Clustering
Edit metric
Ulam metric
Hamming metric
Fine-grained Complexity
Approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
2:1
2:10
10.4230/LIPIcs.APPROX/RANDOM.2023.2
article
Probabilistic Metric Embedding via Metric Labeling
Munagala, Kamesh
1
Sankar, Govind S.
1
Taylor, Erin
1
Department of Computer Science, Duke University, Durham, NC, USA
We consider probabilistic embedding of metric spaces into ultra-metrics (or equivalently to a constant factor, into hierarchically separated trees) to minimize the expected distortion of any pairwise distance. Such embeddings have been widely used in network design and online algorithms. Our main result is a polynomial time algorithm that approximates the optimal distortion on any instance to within a constant factor. We achieve this via a novel LP formulation that reduces this problem to a probabilistic version of uniform metric labeling.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.2/LIPIcs.APPROX-RANDOM.2023.2.pdf
Metric Embedding
Approximation Algorithms
Ultrametrics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
3:1
3:16
10.4230/LIPIcs.APPROX/RANDOM.2023.3
article
Approximating Submodular k-Partition via Principal Partition Sequence
Chandrasekaran, Karthekeyan
1
https://orcid.org/0000-0002-3421-7238
Wang, Weihang
1
https://orcid.org/0000-0002-0628-5532
University of Illinois, Urbana-Champaign, IL, USA
In submodular k-partition, the input is a submodular function f:2^V → ℝ_{≥ 0} (given by an evaluation oracle) along with a positive integer k and the goal is to find a partition of the ground set V into k non-empty parts V_1, V_2, …, V_k in order to minimize ∑_{i=1}^k f(V_i). Narayanan, Roy, and Patkar [Narayanan et al., 1996] designed an algorithm for submodular k-partition based on the principal partition sequence and showed that the approximation factor of their algorithm is 2 for the special case of graph cut functions (which was subsequently rediscovered by Ravi and Sinha [R. Ravi and A. Sinha, 2008]). In this work, we study the approximation factor of their algorithm for three subfamilies of submodular functions - namely monotone, symmetric, and posimodular and show the following results:
1) The approximation factor of their algorithm for monotone submodular k-partition is 4/3. This result improves on the 2-factor that was known to be achievable for monotone submodular k-partition via other algorithms. Moreover, our upper bound of 4/3 matches the recently shown lower bound under polynomial number of function evaluation queries [Santiago, 2021]. Our upper bound of 4/3 is also the first improvement beyond 2 for a certain graph partitioning problem that is a special case of monotone submodular k-partition.
2) The approximation factor of their algorithm for symmetric submodular k-partition is 2. This result generalizes their approximation factor analysis beyond graph cut functions.
3) The approximation factor of their algorithm for posimodular submodular k-partition is 2. We also construct an example to show that the approximation factor of their algorithm for arbitrary submodular functions is Ω(n/k).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.3/LIPIcs.APPROX-RANDOM.2023.3.pdf
Approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
4:1
4:21
10.4230/LIPIcs.APPROX/RANDOM.2023.4
article
Experimental Design for Any p-Norm
Lau, Lap Chi
1
Wang, Robert
1
Zhou, Hong
2
https://orcid.org/0000-0003-0784-4073
David R. Cheriton School of Computer Science, University of Waterloo, Canada
School of Mathematics and Statistics, Fuzhou University, China
We consider a general p-norm objective for experimental design problems that captures some well-studied objectives (D/A/E-design) as special cases. We prove that a randomized local search approach provides a unified algorithm to solve this problem for all nonnegative integer p. This provides the first approximation algorithm for the general p-norm objective, and a nice interpolation of the best known bounds of the special cases.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.4/LIPIcs.APPROX-RANDOM.2023.4.pdf
Approximation Algorithm
Optimal Experimental Design
Randomized Local Search
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
5:1
5:17
10.4230/LIPIcs.APPROX/RANDOM.2023.5
article
Approximation Algorithms for Maximum Weighted Throughput on Unrelated Machines
Karakostas, George
1
Kolliopoulos, Stavros G.
2
Department of Computing & Software, McMaster University, Hamilton, Canada
Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece
We study the classic weighted maximum throughput problem on unrelated machines. We give a (1-1/e-ε)-approximation algorithm for the preemptive case. To our knowledge this is the first ever approximation result for this problem. It is an immediate consequence of a polynomial-time reduction we design, that uses any ρ-approximation algorithm for the single-machine problem to obtain an approximation factor of (1-1/e)ρ -ε for the corresponding unrelated-machines problem, for any ε > 0. On a single machine we present a PTAS for the non-preemptive version of the problem for the special case of a constant number of distinct due dates or distinct release dates. By our reduction this yields an approximation factor of (1-1/e) -ε for the non-preemptive problem on unrelated machines when there is a constant number of distinct due dates or release dates on each machine.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.5/LIPIcs.APPROX-RANDOM.2023.5.pdf
scheduling
maximum weighted throughput
unrelated machines
approximation algorithm
PTAS
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
6:1
6:24
10.4230/LIPIcs.APPROX/RANDOM.2023.6
article
Facility Location in the Sublinear Geometric Model
Monemizadeh, Morteza
1
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
In the sublinear geometric model, we are provided with an oracle access to a point set P of n points in a bounded discrete space [Δ]², where Δ = n^O(1) is a polynomially bounded number in n. That is, we do not have direct access to the points, but we can make certain types of queries and there is an oracle that responds to our queries. The type of queries that we assume we can make in this paper, are range counting queries where ranges are axis-aligned rectangles (that are basic primitives in database [Srikanta Tirthapura and David P. Woodruff, 2012; Bentley, 1975; Mark de Berg et al., 2008], computational geometry [Pankaj K. Agarwal, 2004; Pankaj K. Agarwal et al., 1996; Boris Aronov et al., 2010; Boris Aronov et al., 2009], and machine learning [Menachem Sadigurschi and Uri Stemmer, 2021; Long and Tan, 1998; Michael J. Kearns and Umesh V. Vazirani, 1995; Michael J. Kearns and Umesh V. Vazirani, 1994]). The oracle then answers these queries by returning the number of points that are in queried ranges. Let {Alg} be an algorithm that (exactly or approximately) solves a problem 𝒫 in the sublinear geometric model. The query complexity of Alg is measured in terms of the number of queries that Alg makes to solve 𝒫. In this paper, we study the complexity of the (uniform) Euclidean facility location problem in the sublinear geometric model. We develop a randomized sublinear algorithm that with high probability, (1+ε)-approximates the cost of the Euclidean facility location problem of the point set P in the sublinear geometric model using Õ(√n) range counting queries. We complement this result by showing that approximating the cost of the Euclidean facility location problem within o(log(n))-factor in the sublinear geometric model using the sampling strategy that we propose for our sublinear algorithm needs Ω̃(n^{1/4}) RangeCount queries. We leave it as an open problem whether such a polynomial lower bound on the number of RangeCount queries exists for any randomized sublinear algorithm that approximates the cost of the facility location problem within a constant factor.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.6/LIPIcs.APPROX-RANDOM.2023.6.pdf
Facility Location
Sublinear Geometric Model
Range Counting Queries
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
7:1
7:22
10.4230/LIPIcs.APPROX/RANDOM.2023.7
article
Bicriteria Approximation Algorithms for Priority Matroid Median
Bajpai, Tanvi
1
Chekuri, Chandra
1
Department of Computer Science, University of Illinois, Urbana-Champaign, Urbana, IL, USA
Fairness considerations have motivated new clustering problems and algorithms in recent years. In this paper we consider the Priority Matroid Median problem which generalizes the Priority k-Median problem that has recently been studied. The input consists of a set of facilities ℱ and a set of clients 𝒞 that lie in a metric space (ℱ ∪ 𝒞,d), and a matroid ℳ = (ℱ,ℐ) over the facilities. In addition, each client j has a specified radius r_j ≥ 0 and each facility i ∈ ℱ has an opening cost f_i > 0. The goal is to choose a subset S ⊆ ℱ of facilities to minimize ∑_{i ∈ ℱ} f_i + ∑_{j ∈ 𝒞} d(j,S) subject to two constraints: (i) S is an independent set in ℳ (that is S ∈ ℐ) and (ii) for each client j, its distance to an open facility is at most r_j (that is, d(j,S) ≤ r_j). For this problem we describe the first bicriteria (c₁,c₂) approximations for fixed constants c₁,c₂: the radius constraints of the clients are violated by at most a factor of c₁ and the objective cost is at most c₂ times the optimum cost. We also improve the previously known bicriteria approximation for the uniform radius setting (r_j : = L ∀ j ∈ 𝒞).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.7/LIPIcs.APPROX-RANDOM.2023.7.pdf
k-median
fair clustering
matroid
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
8:1
8:23
10.4230/LIPIcs.APPROX/RANDOM.2023.8
article
Approximation Algorithms for Directed Weighted Spanners
Grigorescu, Elena
1
https://orcid.org/0000-0001-9673-4313
Kumar, Nithish
1
Lin, Young-San
2
https://orcid.org/0000-0002-5719-6708
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Melbourne Business School, Australia
In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied.
We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting:
- An Õ(n^{4/5 + ε})-approximation algorithm for the weighted pairwise spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives an Õ(n^{3/5 + ε})-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on Algorithms, 2020).
- An Õ(n^{1/2+ε})-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives an Õ(n^{1/2})-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. Our results include the following:
- An Õ(k^{1/2 + ε})-competitive algorithm for the online weighted pairwise spanner problem. The state-of-the-art results are an Õ(n^{4/5})-competitive algorithm when edges have unit costs and arbitrary positive lengths, and a min{Õ(k^{1/2 + ε}), Õ(n^{2/3 + ε})}-competitive algorithm when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021).
- An Õ(k^ε)-competitive algorithm for the online weighted single-source (or single-sink) spanner problem. Without distance constraints, this problem is equivalent to the online directed Steiner tree problem. The best previous algorithm for online directed Steiner trees is an Õ(k^ε)-competitive algorithm, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018). Our online results also imply efficient approximation algorithms for the corresponding offline problems. To the best of our knowledge, these are the first approximation (online) polynomial-time algorithms with sublinear approximation (competitive) ratios for the weighted spanner problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.8/LIPIcs.APPROX-RANDOM.2023.8.pdf
directed weighted spanners
linear programming
junction tree
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
9:1
9:17
10.4230/LIPIcs.APPROX/RANDOM.2023.9
article
Approximation Algorithms and Lower Bounds for Graph Burning
Lieskovský, Matej
1
https://orcid.org/0000-0002-0058-3133
Sgall, Jiří
1
https://orcid.org/0000-0003-3658-4848
Feldmann, Andreas Emil
2
https://orcid.org/0000-0001-6229-5332
Faculty of Mathematics and Physics, Computer Science Institute of Charles University, Prague, Czech Republic
Department of Computer Science, University of Sheffield, UK
Graph Burning models information spreading in a given graph as a process such that in each step one node is infected (informed) and also the infection spreads to all neighbors of previously infected nodes. Formally, given a graph G = (V,E), possibly with edge lengths, the burning number b(G) is the minimum number g such that there exist nodes v_0,…,v_{g-1} ∈ V satisfying the property that for each u ∈ V there exists i ∈ {0,…,g-1} so that the distance between u and v_i is at most i.
We present a randomized 2.314-approximation algorithm for computing the burning number of a general graph, even with arbitrary edge lengths. We complement this by an approximation lower bound of 2 for the case of equal length edges, and a lower bound of 4/3 for the case when edges are restricted to have length 1.
This improves on the previous 3-approximation algorithm and an APX-hardness result.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.9/LIPIcs.APPROX-RANDOM.2023.9.pdf
Graph Algorithms
approximation Algorithms
randomized Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
10:1
10:20
10.4230/LIPIcs.APPROX/RANDOM.2023.10
article
The (Im)possibility of Simple Search-To-Decision Reductions for Approximation Problems
Golovnev, Alexander
1
Guo, Siyao
2
Peters, Spencer
3
Stephens-Davidowitz, Noah
3
Georgetown University, Washington, D. C., USA
Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning, NYU Shanghai, China
Cornell University, Ithaca, NY, USA
We study the question of when an approximate search optimization problem is harder than the associated decision problem. Specifically, we study a natural and quite general model of black-box search-to-decision reductions, which we call branch-and-bound reductions (in analogy with branch-and-bound algorithms). In this model, an algorithm attempts to minimize (or maximize) a function f: D → ℝ_{≥ 0} by making oracle queries to h_f : 𝒮 → ℝ_{≥ 0} satisfying
min_{x ∈ S} f(x) ≤ h_f(S) ≤ γ ⋅ min_{x ∈ S} f(x) (*)
for some γ ≥ 1 and any subset S in some allowed class of subsets 𝒮 of the domain D. (When the goal is to maximize f, h_f instead yields an approximation to the maximal value of f over S.) We show tight upper and lower bounds on the number of queries q needed to find even a γ'-approximate minimizer (or maximizer) for quite large γ' in a number of interesting settings, as follows.
- For arbitrary functions f : {0,1}ⁿ → ℝ_{≥ 0}, where 𝒮 contains all subsets of the domain, we show that no branch-and-bound reduction can achieve γ' ≲ γ^{n/log q}, while a simple greedy approach achieves essentially γ^{n/log q}.
- For a large class of MAX-CSPs, where 𝒮 := {S_w} contains each set of assignments to the variables induced by a partial assignment w, we show that no branch-and-bound reduction can do significantly better than essentially a random guess, even when the oracle h_f guarantees an approximation factor of γ ≈ 1+√{log(q)/n}.
- For the Traveling Salesperson Problem (TSP), where 𝒮 := {S_p} contains each set of tours extending a path p, we show that no branch-and-bound reduction can achieve γ' ≲ (γ-1) n/log q. We also prove a nearly matching upper bound in our model.
These results show an oracle model in which approximate search and decision are strongly separated. (In particular, our result for TSP can be viewed as a negative answer to a question posed by Bellare and Goldwasser (SIAM J. Comput. 1994), though only in an oracle model.) We also note two alternative interpretations of our results. First, if we view h_f as a data structure, then our results unconditionally rule out black-box search-to-decision reductions for certain data structure problems. Second, if we view h_f as an efficiently computable heuristic, then our results show that any reasonably efficient branch-and-bound algorithm requires more guarantees from its heuristic than simply Eq. (*).
Behind our results is a "useless oracle lemma," which allows us to argue that under certain conditions the oracle h_f is "useless," and which might be of independent interest. See also the full version [Alexander Golovnev et al., 2022].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.10/LIPIcs.APPROX-RANDOM.2023.10.pdf
search-to-decision reductions
oracles
constraint satisfaction
traveling salesman
discrete optimization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
11:1
11:19
10.4230/LIPIcs.APPROX/RANDOM.2023.11
article
Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment
Chlamtáč, Eden
1
https://orcid.org/0000-0002-0296-0107
Makarychev, Yury
2
https://orcid.org/0000-0003-3114-3947
Vakilian, Ali
2
https://orcid.org/0000-0001-5049-7594
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Toyota Technological Institute at Chicago (TTIC), IL, USA
We provide new approximation algorithms for the Red-Blue Set Cover and Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm for Red-Blue Set Cover achieves Õ(m^{1/3})-approximation improving on the Õ(m^{1/2})-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSA_t (for circuits of depth t) gives an Õ(N^{1-δ}) approximation for δ = 1/32^{3-⌈t/2⌉}, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSA_t with t ≥ 4 were previously known.
We complement these results with lower bounds for these problems: For Red-Blue Set Cover, we provide a nearly approximation preserving reduction from Min k-Union that gives an ̃Ω(m^{1/4 - ε}) hardness under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an SDP relaxation strengthened by Sherali-Adams has an integrality gap of N^{1-ε} where ε → 0 as the circuit depth t → ∞.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.11/LIPIcs.APPROX-RANDOM.2023.11.pdf
Red-Blue Set Cover Problem
Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem
LP Rounding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
12:1
12:19
10.4230/LIPIcs.APPROX/RANDOM.2023.12
article
Efficient Algorithms and Hardness Results for the Weighted k-Server Problem
Gupta, Anupam
1
Kumar, Amit
2
Panigrahi, Debmalya
3
Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Computer Science and Engineering Department, Indian Institute of Technology, Delhi, India
Computer Science, Duke University, Durham, NC, USA
In this paper, we study the weighted k-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) k-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted k-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use c-resource augmentation for c < 2. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least 𝓁 resource augmentation, where 𝓁 is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of (2+ε)𝓁 for any constant ε > 0.
In the online setting, an exp(k) lower bound is known for the competitive ratio of any randomized algorithm for the weighted k-server problem on the uniform metric. In contrast, we show that 2𝓁-resource augmentation can bring the competitive ratio down by an exponential factor to only O(𝓁² log 𝓁). Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.12/LIPIcs.APPROX-RANDOM.2023.12.pdf
Online Algorithms
Weighted k-server
Integrality Gap
Hardness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
13:1
13:18
10.4230/LIPIcs.APPROX/RANDOM.2023.13
article
A Constant-Factor Approximation for Quasi-Bipartite Directed Steiner Tree on Minor-Free Graphs
Friggstad, Zachary
1
Mousavi, Ramin
1
Department of Computing Science, University of Alberta, Canada
We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for K_r-minor-free graphs our approximation guarantee is O(r⋅√(log r)) and, further, for planar graphs our approximation guarantee is 20.
Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the Directed Steiner Tree instances we consider.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.13/LIPIcs.APPROX-RANDOM.2023.13.pdf
Directed Steiner tree
Combinatorial optimization
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
14:1
14:14
10.4230/LIPIcs.APPROX/RANDOM.2023.14
article
Algorithms for 2-Connected Network Design and Flexible Steiner Trees with a Constant Number of Terminals
Bansal, Ishan
1
Cheriyan, Joe
2
Grout, Logan
1
Ibrahimpur, Sharat
3
https://orcid.org/0000-0002-1575-9648
Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA
Department of Combinatorics and Optimization, University of Waterloo, Canada
Department of Mathematics, London School of Economics and Political Science, UK
The k-Steiner-2NCS problem is as follows: Given a constant (positive integer) k, and an undirected connected graph G = (V,E), non-negative costs c on the edges, and a partition (T, V⧵T) of V into a set of terminals, T, and a set of non-terminals (or, Steiner nodes), where |T| = k, find a min-cost two-node connected subgraph that contains the terminals. The k-Steiner-2ECS problem has the same inputs; the algorithmic goal is to find a min-cost two-edge connected subgraph that contains the terminals.
We present a randomized polynomial-time algorithm for the unweighted k-Steiner-2NCS problem, and a randomized FPTAS for the weighted k-Steiner-2NCS problem. We obtain similar results for a capacitated generalization of the k-Steiner-2ECS problem.
Our methods build on results by Björklund, Husfeldt, and Taslaman (SODA 2012) that give a randomized polynomial-time algorithm for the unweighted k-Steiner-cycle problem; this problem has the same inputs as the unweighted k-Steiner-2NCS problem, and the algorithmic goal is to find a min-cost simple cycle C that contains the terminals (C may contain any number of Steiner nodes).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.14/LIPIcs.APPROX-RANDOM.2023.14.pdf
Approximation algorithms
Capacitated network design
Network design
Parametrized algorithms
Steiner cycle problem
Steiner 2-edge connected subgraphs
Steiner 2-node connected subgraphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
15:1
15:19
10.4230/LIPIcs.APPROX/RANDOM.2023.15
article
Oblivious Algorithms for the Max-kAND Problem
Singer, Noah G.
1
https://orcid.org/0000-0002-0076-521X
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-kAND problem. This is a class of simple, combinatorial algorithms which round each variable with probability depending only on a quantity called the variable’s bias. Our definition generalizes a class of algorithms defined by Feige and Jozeph (Algorithmica '15) for Max-DICUT, a special case of Max-2AND.
For each oblivious algorithm, we design a so-called factor-revealing linear program (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all k, oblivious algorithms for Max-kAND provably outperform a special subclass of algorithms we call "superoblivious" algorithms.
Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every k, O(log n)-space sketching algorithms for Max-kAND known to be optimal in o(√n)-space can be beaten in (a) O(log n)-space under a random-ordering assumption, and (b) O(n^{1-1/k} D^{1/k}) space under a maximum-degree-D assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.15/LIPIcs.APPROX-RANDOM.2023.15.pdf
streaming algorithm
approximation algorithm
constraint satisfaction problem (CSP)
factor-revealing linear program
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
16:1
16:19
10.4230/LIPIcs.APPROX/RANDOM.2023.16
article
A 10/7-Approximation for Discrete Bamboo Garden Trimming and Continuous Trimming on Star Graphs
Höhne, Felix
1
van Stee, Rob
2
Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern, Germany
University of Siegen, Germany
In the discrete bamboo garden trimming problem we are given n bamboo that grow at rates v_1,… ,v_n per day. Each day a robotic gardener cuts down one bamboo to height 0. The goal is to find a schedule that minimizes the height of the tallest bamboo that ever exists.
We present a 10/7-approximation algorithm that is based on a reduction to the pinwheel problem. This is consistent with the approach of earlier algorithms, but some new techniques are used that lead to a better approximation ratio.
We also consider the continuous version of the problem where the gardener travels in a metric space between plants and cuts down a plant each time he reaches one. We show that on the star graph the previously proposed algorithm Reduce-Fastest is a 6-approximation and the known Deadline-Driven Strategy is a (3+2√2)-approximation. The Deadline-Driven Strategy is also a (9+2√5)-approximation on star graphs with multiple plants on each branch.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.16/LIPIcs.APPROX-RANDOM.2023.16.pdf
bamboo garden trimming
approximation algorithms
scheduling
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
17:1
17:17
10.4230/LIPIcs.APPROX/RANDOM.2023.17
article
Online Matching with Set and Concave Delays
Deryckere, Lindsey
1
Umboh, Seeun William
2
https://orcid.org/0000-0001-6984-4007
School of Computer Science, The University of Sydney, Australia
School of Computing and Information Systems, The University of Melbourne, Australia
We initiate the study of online problems with set delay, where the delay cost at any given time is an arbitrary function of the set of pending requests. In particular, we study the online min-cost perfect matching with set delay (MPMD-Set) problem, which generalises the online min-cost perfect matching with delay (MPMD) problem introduced by Emek et al. (STOC 2016). In MPMD, m requests arrive over time in a metric space of n points. When a request arrives the algorithm must choose to either match or delay the request. The goal is to create a perfect matching of all requests while minimising the sum of distances between matched requests, and the total delay costs incurred by each of the requests. In contrast to previous work we study MPMD-Set in the non-clairvoyant setting, where the algorithm does not know the future delay costs. We first show no algorithm is competitive in n or m. We then study the natural special case of size-based delay where the delay is a non-decreasing function of the number of unmatched requests. Our main result is the first non-clairvoyant algorithms for online min-cost perfect matching with size-based delay that are competitive in terms of m. In fact, these are the first non-clairvoyant algorithms for any variant of MPMD. A key technical ingredient is an analog of the symmetric difference of matchings that may be useful for other special classes of set delay. Furthermore, we prove a lower bound of Ω(n) for any deterministic algorithm and Ω(log n) for any randomised algorithm. These lower bounds also hold for clairvoyant algorithms. Finally, we also give an m-competitive deterministic algorithm for uniform concave delays in the clairvoyant setting.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.17/LIPIcs.APPROX-RANDOM.2023.17.pdf
online algorithms
matching
delay
non-clairvoyant
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
18:1
18:21
10.4230/LIPIcs.APPROX/RANDOM.2023.18
article
An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs
Dürr, Anita
1
https://orcid.org/0000-0003-0440-5008
El Maalouly, Nicolas
1
https://orcid.org/0000-0002-1037-0203
Wulf, Lasse
2
https://orcid.org/0000-0001-7139-4092
Department of Computer Science, ETH Zürich, Switzerland
Institute of Discrete Mathematics, TU Graz, Austria
In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis.
In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS'23). It outputs a perfect matching with k' red edges with the guarantee that 0.5k ≤ k' ≤ 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k' red edges such that k/3 ≤ k' ≤ k.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.18/LIPIcs.APPROX-RANDOM.2023.18.pdf
Perfect Matching
Exact Matching
Red-Blue Matching
Approximation Algorithms
Bounded Color Matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
19:1
19:16
10.4230/LIPIcs.APPROX/RANDOM.2023.19
article
Tighter Approximation for the Uniform Cost-Distance Steiner Tree Problem
Foos, Josefine
1
Held, Stephan
1
https://orcid.org/0000-0003-2188-1559
Spitzley, Yannik Kyle Dustin
1
https://orcid.org/0009-0001-2389-0048
Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany
Uniform cost-distance Steiner trees minimize the sum of the total length and weighted path lengths from a dedicated root to the other terminals. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks. They are a special case of general cost-distance Steiner trees, where different distance functions are used for total length and path lengths.
We improve the best published approximation factor for the uniform cost-distance Steiner tree problem from 2.39 [Khazraei and Held, 2021] to 2.05. If we can approximate the minimum-length Steiner tree problem arbitrarily well, our algorithm achieves an approximation factor arbitrarily close to 1+1/√2. This bound is tight in the following sense. We also prove the gap 1+1/√2 between optimum solutions and the lower bound which we and all previous approximation algorithms for this problem use.
Similarly to previous approaches, we start with an approximate minimum-length Steiner tree and split it into subtrees that are later re-connected. To improve the approximation factor, we split it into components more carefully, taking the cost structure into account, and we significantly enhance the analysis.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.19/LIPIcs.APPROX-RANDOM.2023.19.pdf
cost-distance Steiner tree
approximation algorithm
uniform
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
20:1
20:20
10.4230/LIPIcs.APPROX/RANDOM.2023.20
article
Round and Bipartize for Vertex Cover Approximation
Kashaev, Danish
1
Schäfer, Guido
1
2
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
ILLC, University of Amsterdam, The Netherlands
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (𝒢, S), consisting of a graph with an odd cycle transversal.
If S is a stable set, we prove a tight approximation ratio of 1 + 1/ρ, where 2ρ -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph 𝒢̃ : = 𝒢/S and satisfies ρ ∈ [2,∞], with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/ρ) (1 - α) + 2 α, where α ∈ [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph 𝒢̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.20/LIPIcs.APPROX-RANDOM.2023.20.pdf
Combinatorial optimization
approximation algorithms
rounding algorithms
beyond the worst-case analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
21:1
21:13
10.4230/LIPIcs.APPROX/RANDOM.2023.21
article
On Minimizing Generalized Makespan on Unrelated Machines
Ayyadevara, Nikhil
1
Bansal, Nikhil
1
Prabhu, Milind
1
University of Michigan, Ann Arbor, MI, USA
We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given n jobs and m machines and each job j has arbitrary processing time p_{ij} on machine i. Additionally, there is a general symmetric monotone norm ψ_i for each machine i, that determines the load on machine i as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load.
Recently, Deng, Li, and Rabani [Deng et al., 2023] gave a 3 approximation for GMP when the ψ_i are top-k norms, and they ask the question whether an O(1) approximation exists for general norms ψ? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant δ > 0, such that GMP is Ω(log^δ n) hard to approximate. We also give an Ω(log^{1/2} n) integrality gap for the natural configuration LP.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.21/LIPIcs.APPROX-RANDOM.2023.21.pdf
Hardness of Approximation
Generalized Makespan
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
22:1
22:16
10.4230/LIPIcs.APPROX/RANDOM.2023.22
article
An AFPTAS for Bin Packing with Partition Matroid via a New Method for LP Rounding
Doron-Arad, Ilan
1
Kulik, Ariel
2
Shachnai, Hadas
1
Computer Science Department, Technion, Haifa, Israel
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
We consider the Bin Packing problem with a partition matroid constraint. The input is a set of items of sizes in [0,1], and a partition matroid over the items. The goal is to pack the items in a minimum number of unit-size bins, such that each bin forms an independent set in the matroid. This variant of classic Bin Packing has natural applications in secure storage on the Cloud, as well as in equitable scheduling and clustering with fairness constraints.
Our main result is an asymptotic fully polynomial-time approximation scheme (AFPTAS) for Bin Packing with a partition matroid constraint. This scheme generalizes the known AFPTAS for Bin Packing with Cardinality Constraints and improves the existing asymptotic polynomial-time approximation scheme (APTAS) for Group Bin Packing, which are both special cases of Bin Packing with partition matroid. We derive the scheme via a new method for rounding a (fractional) solution for a configuration-LP. Our method uses this solution to obtain prototypes, in which items are interpreted as placeholders for other items, and applies fractional grouping to modify a fractional solution (prototype) into one having desired integrality properties.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.22/LIPIcs.APPROX-RANDOM.2023.22.pdf
bin packing
partition-matroid
AFPTAS
LP-rounding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
23:1
23:22
10.4230/LIPIcs.APPROX/RANDOM.2023.23
article
Submodular Norms with Applications To Online Facility Location and Stochastic Probing
Patton, Kalen
1
Russo, Matteo
2
Singla, Sahil
3
School of Mathematics, Georgia Tech, Atlanta, GA, USA
DIAG, Sapienza Università di Roma, Italy
School of Computer Science, Georgia Tech, Atlanta, GA, USA
Optimization problems often involve vector norms, which has led to extensive research on developing algorithms that can handle objectives beyond 𝓁_p norms. Our work introduces the concept of submodular norms, which are a versatile type of norms that possess marginal properties similar to submodular set functions. We show that submodular norms can either accurately represent or approximate well-known classes of norms, such as 𝓁_p norms, ordered norms, and symmetric norms. Furthermore, we establish that submodular norms can be applied to optimization problems such as online facility location and stochastic probing. This allows us to develop a logarithmic-competitive algorithm for online facility location with symmetric norms, and to prove logarithmic adaptivity gap for stochastic probing with symmetric norms.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.23/LIPIcs.APPROX-RANDOM.2023.23.pdf
Submodularity
Monotone Norms
Online Facility Location
Stochastic Probing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
24:1
24:22
10.4230/LIPIcs.APPROX/RANDOM.2023.24
article
Independent Sets in Elimination Graphs with a Submodular Objective
Chekuri, Chandra
1
Quanrud, Kent
2
Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Maximum weight independent set (MWIS) admits a 1/k-approximation in inductively k-independent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)-approximation in k-perfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize k-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others [Ye and Borodin, 2012; Kammer and Tholey, 2014]. We consider a generalization of MWIS to a submodular objective. Given a graph G = (V,E) and a non-negative submodular function f: 2^V → ℝ_+, the goal is to approximately solve max_{S ∈ ℐ_G} f(S) where ℐ_G is the set of independent sets of G. We obtain an Ω(1/k)-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least 1/e(k+1). This approach also yields parallel (or low-adaptivity) approximations.
Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively k-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.24/LIPIcs.APPROX-RANDOM.2023.24.pdf
elimination graphs
independent set
submodular maximization
primal-dual
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
25:1
25:22
10.4230/LIPIcs.APPROX/RANDOM.2023.25
article
Improved Diversity Maximization Algorithms for Matching and Pseudoforest
Mahabadi, Sepideh
1
Narayanan, Shyam
2
Microsoft Research, Redmond, WA, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
In this work we consider the diversity maximization problem, where given a data set X of n elements, and a parameter k, the goal is to pick a subset of X of size k maximizing a certain diversity measure. Chandra and Halldórsson [Barun Chandra and Magnús M. Halldórsson, 2001] defined a variety of diversity measures based on pairwise distances between the points. A constant factor approximation algorithm was known for all those diversity measures except "remote-matching", where only an O(log k) approximation was known. In this work we present an O(1) approximation for this remaining notion. Further, we consider these notions from the perpective of composable coresets. Indyk et al. [Piotr Indyk et al., 2014] provided composable coresets with a constant factor approximation for all but "remote-pseudoforest" and "remote-matching", which again they only obtained a O(log k) approximation. Here we also close the gap up to constants and present a constant factor composable coreset algorithm for these two notions. For remote-matching, our coreset has size only O(k), and for remote-pseudoforest, our coreset has size O(k^{1+ε}) for any ε > 0, for an O(1/ε)-approximate coreset.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.25/LIPIcs.APPROX-RANDOM.2023.25.pdf
diversity maximization
approximation algorithms
composable coresets
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
26:1
26:24
10.4230/LIPIcs.APPROX/RANDOM.2023.26
article
Approximating Pandora’s Box with Correlations
Chawla, Shuchi
1
https://orcid.org/0000-0001-5583-2320
Gergatsouli, Evangelia
2
https://orcid.org/0000-0001-9905-2457
McMahan, Jeremy
2
https://orcid.org/0000-0002-1933-4719
Tzamos, Christos
2
3
https://orcid.org/0000-0002-7560-5069
University of Texas - Austin, TX, USA
University of Wisconsin - Madison, WI, USA
University of Athens, Greece
We revisit the classic Pandora’s Box (PB) problem under correlated distributions on the box values. Recent work of [Shuchi Chawla et al., 2020] obtained constant approximate algorithms for a restricted class of policies for the problem that visit boxes in a fixed order. In this work, we study the complexity of approximating the optimal policy which may adaptively choose which box to visit next based on the values seen so far.
Our main result establishes an approximation-preserving equivalence of PB to the well studied Uniform Decision Tree (UDT) problem from stochastic optimization and a variant of the Min-Sum Set Cover (MSSC_f) problem. For distributions of support m, UDT admits a log m approximation, and while a constant factor approximation in polynomial time is a long-standing open problem, constant factor approximations are achievable in subexponential time [Ray Li et al., 2020]. Our main result implies that the same properties hold for PB and MSSC_f.
We also study the case where the distribution over values is given more succinctly as a mixture of m product distributions. This problem is again related to a noisy variant of the Optimal Decision Tree which is significantly more challenging. We give a constant-factor approximation that runs in time n^Õ(m²/ε²) when the mixture components on every box are either identical or separated in TV distance by ε.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.26/LIPIcs.APPROX-RANDOM.2023.26.pdf
Pandora’s Box
Min Sum Set Cover
stochastic optimization
approximation preserving reduction
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
27:1
27:19
10.4230/LIPIcs.APPROX/RANDOM.2023.27
article
Stable Approximation Algorithms for Dominating Set and Independent Set
de Berg, Mark
1
Sadhukhan, Arpan
1
https://orcid.org/0000-0003-4048-7143
Spieksma, Frits
1
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
We study Dominating Set and Independent Set for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is k-stable when it makes at most k changes to its output independent set or dominating set upon the arrival of each vertex. We study trade-offs between the stability parameter k of the algorithm and the approximation ratio it achieves. We obtain the following results.
- We show that there is a constant ε^* > 0 such that any dynamic (1+ε^*)-approximation algorithm for Dominating Set has stability parameter Ω(n), even for bipartite graphs of maximum degree 4.
- We present algorithms with very small stability parameters for Dominating Set in the setting where the arrival degree of each vertex is upper bounded by d. In particular, we give a 1-stable (d+1)²-approximation, and a 3-stable (9d/2)-approximation algorithm.
- We show that there is a constant ε^* > 0 such that any dynamic (1+ε^*)-approximation algorithm for Independent Set has stability parameter Ω(n), even for bipartite graphs of maximum degree 3.
- Finally, we present a 2-stable O(d)-approximation algorithm for Independent Set, in the setting where the average degree of the graph is upper bounded by some constant d at all times.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.27/LIPIcs.APPROX-RANDOM.2023.27.pdf
Dynamic algorithms
approximation algorithms
stability
dominating set
independent set
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
28:1
28:24
10.4230/LIPIcs.APPROX/RANDOM.2023.28
article
Scalable Auction Algorithms for Bipartite Maximum Matching Problems
Liu, Quanquan C.
1
https://orcid.org/0000-0003-1230-2754
Ke, Yiduo
1
https://orcid.org/0009-0000-8118-948X
Khuller, Samir
1
https://orcid.org/0000-0002-5408-8023
Northwestern University, Evanston, IL, USA
Bipartite maximum matching and its variants are well-studied problems under various models of computation with the vast majority of approaches centering around various methods to find and eliminate augmenting paths. Beginning with the seminal papers of Demange, Gale and Sotomayor [DGS86] and Bertsekas [Ber81], bipartite maximum matching problems have also been studied in the context of auction algorithms. These algorithms model the maximum matching problem as an auction where one side of the bipartite graph consists of bidders and the other side consists of items; as such, these algorithms offer a very different approach to solving this problem that do not use classical methods. Dobzinski, Nisan and Oren [DNO14] demonstrated the utility of such algorithms in distributed, interactive settings by providing a simple and elegant O(log n/ε²) round maximum cardinality bipartite matching (MCM) algorithm that has small round and communication complexity and gives a (1-ε)-approximation for any (not necessarily constant) ε > 0. They leave as an open problem whether an auction algorithm, with similar guarantees, can be found for the maximum weighted bipartite matching (MWM) problem. Very recently, Assadi, Liu, and Tarjan [ALT21] extended the utility of auction algorithms for MCM into the semi-streaming and massively parallel computation (MPC) models, by cleverly using maximal matching as a subroutine, to give a new auction algorithm that uses O(1/ε²) rounds and achieves the state-of-the-art bipartite MCM results in the streaming and MPC settings.
In this paper, we give new auction algorithms for maximum weighted bipartite matching (MWM) and maximum cardinality bipartite b-matching (MCbM). Our algorithms run in O(log n/ε⁸) and O(log n/ε²) rounds, respectively, in the distributed setting. We show that our MWM algorithm can be implemented in the distributed, interactive setting using O(log² n) and O(log n) bit messages, respectively, directly answering the open question posed by Demange, Gale and Sotomayor [DNO14]. Furthermore, we implement our algorithms in a variety of other models including the the semi-streaming model, the shared-memory work-depth model, and the massively parallel computation model. Our semi-streaming MWM algorithm uses O(1/ε⁸) passes in O(n log n ⋅ log(1/ε)) space and our MCbM algorithm runs in O(1/ε²) passes using O((∑_{i ∈ L} b_i + |R|) log(1/ε)) space (where parameters b_i represent the degree constraints on the b-matching and L and R represent the left and right side of the bipartite graph, respectively). Both of these algorithms improves exponentially the dependence on ε in the space complexity in the semi-streaming model against the best-known algorithms for these problems, in addition to improvements in round complexity for MCbM. Finally, our algorithms eliminate the large polylogarithmic dependence on n in depth and number of rounds in the work-depth and massively parallel computation models, respectively, improving on previous results which have large polylogarithmic dependence on n (and exponential dependence on ε in the MPC model).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.28/LIPIcs.APPROX-RANDOM.2023.28.pdf
auction algorithms
maximum weight bipartite matching
maximum b-matching
distributed blackboard model
parallel work-depth model
streaming model
massively parallel computation model
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
29:1
29:17
10.4230/LIPIcs.APPROX/RANDOM.2023.29
article
Giant Components in Random Temporal Graphs
Becker, Ruben
1
https://orcid.org/0000-0002-3495-3753
Casteigts, Arnaud
2
https://orcid.org/0000-0002-7819-7013
Crescenzi, Pierluigi
3
https://orcid.org/0000-0001-8789-3195
Kodric, Bojana
1
https://orcid.org/0000-0001-7242-0096
Renken, Malte
4
https://orcid.org/0000-0002-1450-1901
Raskin, Michael
5
https://orcid.org/0000-0002-6660-5673
Zamaraev, Viktor
6
https://orcid.org/0000-0001-5755-4141
Ca’ Foscari University of Venice, Italy
University of Geneva, Switzerland
Gran Sasso Science Institute, L’Aquila, Italy
Technical University of Berlin, Germany
LaBRI, CNRS, University of Bordeaux, France
University of Liverpool, UK
A temporal graph is a graph whose edges appear only at certain points in time. In these graphs, reachability among nodes relies on paths that traverse edges in chronological order (temporal paths). Unlike standard paths, temporal paths may not be composable, thus the reachability relation is not transitive and connected components (i.e., sets of pairwise temporally connected nodes) do not form equivalence classes, a fact with far-reaching consequences.
Recently, Casteigts et al. [FOCS 2021] proposed a natural temporal analog of the seminal Erdős-Rényi random graph model, based on the same parameters n and p. The proposed model is obtained by randomly permuting the edges of an Erdős-Rényi random graph and interpreting this permutation as an ordering of presence times. Casteigts et al. showed that the well-known single threshold for connectivity in the Erdős-Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting.
The second most basic phenomenon studied by Erdős and Rényi in static (i.e., non-temporal) random graphs is the emergence of a giant connected component. However, the existence of a similar phase transition in the temporal model was left open until now. In this paper, we settle this question. We identify a sharp threshold at p = log n/n, where the size of the largest temporally connected component increases from o(n) to n-o(n) nodes. This transition occurs significantly later than in the static setting, where a giant component of size n-o(n) already exists for any p ∈ ω(1/n) (i.e., as soon as p is larger than a constant fraction of n). Interestingly, the threshold that we obtain holds for both open and closed connected components, i.e., components that allow, respectively forbid, their connecting paths to use external nodes - a distinction arising from the absence of transitivity.
We achieve these results by strengthening the tools from Casteigts et al. and developing new techniques that provide means to decouple dependencies between past and future events in temporal Erdős-Rényi graphs, which could be of general interest in future investigations of these objects.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.29/LIPIcs.APPROX-RANDOM.2023.29.pdf
random temporal graph
Erdős–Rényi random graph
sharp threshold
temporal connectivity
temporal connected component
edge-ordered graph
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
30:1
30:22
10.4230/LIPIcs.APPROX/RANDOM.2023.30
article
On Connectivity in Random Graph Models with Limited Dependencies
Lengler, Johannes
1
https://orcid.org/0000-0003-0004-7629
Martinsson, Anders
1
https://orcid.org/0000-0002-7943-9280
Petrova, Kalina
1
https://orcid.org/0009-0006-1753-6962
Schnider, Patrick
1
https://orcid.org/0000-0002-2172-9285
Steiner, Raphael
1
https://orcid.org/0000-0002-4234-6136
Weber, Simon
1
https://orcid.org/0000-0003-1901-3621
Welzl, Emo
1
Department of Computer Science, ETH Zürich, Switzerland
For any positive edge density p, a random graph in the Erdős-Rényi G_{n,p} model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability ρ(n), such that for any distribution 𝒢 (in this model) on graphs with n vertices in which each potential edge has a marginal probability of being present at least ρ(n), a graph drawn from 𝒢 is connected with non-zero probability?
As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold ρ(n). For each condition, we provide upper and lower bounds for ρ(n). In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that ρ(n) → 2-ϕ ≈ 0.38 for n → ∞, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that ρ(n) > 0.5-o(n). In stark contrast to the coloring model, for our weakest independence condition - pairwise independence of non-adjacent edges - we show that ρ(n) lies within O(1/n²) of the threshold 1-2/n for completely arbitrary distributions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.30/LIPIcs.APPROX-RANDOM.2023.30.pdf
Random Graphs
Independence
Dependency
Connectivity
Threshold
Probabilistic Method
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
31:1
31:21
10.4230/LIPIcs.APPROX/RANDOM.2023.31
article
Synergy Between Circuit Obfuscation and Circuit Minimization
Impagliazzo, Russell
1
https://orcid.org/0000-0003-3236-9796
Kabanets, Valentine
2
Volkovich, Ilya
3
https://orcid.org/0000-0002-7616-0751
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
School of Computing Science, Simon Fraser University, Burnaby, Canada
Computer Science Department, Boston College, Chestnut Hill, MA, USA
We study close connections between Indistinguishability Obfuscation (IO) and the Minimum Circuit Size Problem (MCSP), and argue that efficient algorithms/construction for MCSP and IO create a synergy. Some of our main results are:
- If there exists a perfect (imperfect) IO that is computationally secure against nonuniform polynomial-size circuits, then for all k ∈ ℕ: NP ∩ ZPP^{MCSP} ⊈ SIZE[n^k] (MA ∩ ZPP^{MCSP} ⊈ SIZE[n^k]).
- In addition, if there exists a perfect IO that is computationally secure against nonuniform polynomial-size circuits, then NEXP ∩ ZPEXP^{MCSP} ⊈ P/poly.
- If MCSP ∈ BPP, then statistical security and computational security for IO are equivalent.
- If computationally-secure perfect IO exists, then MCSP ∈ BPP iff NP = ZPP.
- If computationally-secure perfect IO exists, then ZPEXP ≠ BPP.
To the best of our knowledge, this is the first consequence of strong circuit lower bounds from the existence of an IO. The results are obtained via a construction of an optimal universal distinguisher, computable in randomized polynomial time with access to the MCSP oracle, that will distinguish any two circuit-samplable distributions with the advantage that is the statistical distance between these two distributions minus some negligible error term. This is our main technical contribution. As another immediate application, we get a simple proof of the result by Allender and Das (Inf. Comput., 2017) that SZK ⊆ BPP^{MCSP}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.31/LIPIcs.APPROX-RANDOM.2023.31.pdf
Minimal Circuit Size Problem (MCSP)
Circuit Lower Bounds
Complexity Classes
Indistinguishability Obfuscation
Separation of Classes
Statistical Distance
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
32:1
32:14
10.4230/LIPIcs.APPROX/RANDOM.2023.32
article
Interactive Error Correcting Codes: New Constructions and Impossibility Bounds
Gupta, Meghal
1
https://orcid.org/0000-0001-7657-2847
Zhang, Rachel Yun
2
https://orcid.org/0000-0001-6341-3505
University of California Berkeley, CA, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
An interactive error correcting code (iECC) is an interactive protocol with the guarantee that the receiver can correctly determine the sender’s message, even in the presence of noise. It was shown in works by Gupta, Kalai, and Zhang (STOC 2022) and by Efremenko, Kol, Saxena, and Zhang (FOCS 2022) that there exist iECC’s that are resilient to a larger fraction of errors than is possible in standard error-correcting codes without interaction. In this work, we improve upon these existing works in two ways:
- First, we improve upon the erasure iECC of Kalai, Gupta, and Zhang, which has communication complexity quadratic in the message size. In our work, we construct the first iECC resilient to > 1/2 adversarial erasures that is also positive rate. For any ε > 0, our iECC is resilient to 6/11 - ε adversarial erasures and has size O_ε(k).
- Second, we prove a better upper bound on the maximal possible error resilience of any iECC in the case of bit flip errors. It is known that an iECC can achieve 1/4 + 10^{-5} error resilience (Efremenko, Kol, Saxena, and Zhang), while the best known upper bound was 2/7 ≈ 0.2857 (Gupta, Kalai, and Zhang). We improve upon the upper bound, showing that no iECC can be resilient to more than 13/47 ≈ 0.2766 fraction of errors.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.32/LIPIcs.APPROX-RANDOM.2023.32.pdf
Code
Interactive Protocol
Error Resilience
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
33:1
33:16
10.4230/LIPIcs.APPROX/RANDOM.2023.33
article
Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees
Efthymiou, Charilaos
1
Hayes, Thomas P.
2
Štefankovič, Daniel
3
Vigoda, Eric
4
Computer Science, University of Warwick, Coventry, UK
Dept. of Computer Science and Engineering, University at Buffalo, NY, USA
Department of Computer Science, University of Rochester, NY, USA
Department of Computer Science, University of California, Santa Barbara, CA, USA
We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter λ > 0; the special case λ = 1 corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete Δ-regular tree for all λ. However, Restrepo et al. (2014) showed that for sufficiently large λ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width.
We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of O(n) for the Glauber dynamics for unweighted independent sets on arbitrary trees. Moreover, for λ ≤ .44 we prove an optimal mixing time bound of O(n log n). We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree Δ. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order λ = O(1/Δ). Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance/entropy via a non-trivial inductive proof.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.33/LIPIcs.APPROX-RANDOM.2023.33.pdf
MCMC
Mixing Time
Independent Sets
Hard-Core Model
Approximate Counting Algorithms
Sampling Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
34:1
34:20
10.4230/LIPIcs.APPROX/RANDOM.2023.34
article
Superpolynomial Lower Bounds for Learning Monotone Classes
Bshouty, Nader H.
1
Department of Computer Science, Technion, Haifa, Israel
Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time n^Õ(log log s) for the classes of n-variable size-s DNF, size-s Decision Tree, and log s-Junta by DNF (that returns a DNF hypothesis). Assuming a natural conjecture on the hardness of set cover, they give the lower bound n^Ω(log s). This matches the best known upper bound for n-variable size-s Decision Tree, and log s-Junta.
In this paper, we give the same lower bounds for PAC-learning of n-variable size-s Monotone DNF, size-s Monotone Decision Tree, and Monotone log s-Junta by DNF. This solves the open problem proposed by Koch, Strassle, and Tan and subsumes the above results.
The lower bound holds, even if the learner knows the distribution, can draw a sample according to the distribution in polynomial time, and can compute the target function on all the points of the support of the distribution in polynomial time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.34/LIPIcs.APPROX-RANDOM.2023.34.pdf
PAC Learning
Monotone DNF
Monotone Decision Tree
Monotone Junta
Lower Bound
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
35:1
35:22
10.4230/LIPIcs.APPROX/RANDOM.2023.35
article
An Embarrassingly Parallel Optimal-Space Cardinality Estimation Algorithm
Karayel, Emin
1
https://orcid.org/0000-0003-3290-5034
Department of Computer Science, Technische Universität München, Germany
In 2020 Błasiok (ACM Trans. Algorithms 16(2) 3:1-3:28) constructed an optimal space streaming algorithm for the cardinality estimation problem with the space complexity of O(ε^{-2} ln(δ^{-1}) + ln n) where ε, δ and n denote the relative accuracy, failure probability and universe size, respectively. However, his solution requires the stream to be processed sequentially. On the other hand, there are algorithms that admit a merge operation; they can be used in a distributed setting, allowing parallel processing of sections of the stream, and are highly relevant for large-scale distributed applications. The best-known such algorithm, unfortunately, has a space complexity exceeding Ω(ln(δ^{-1}) (ε^{-2} ln ln n + ln n)). This work presents a new algorithm that improves on the solution by Błasiok, preserving its space complexity, but with the benefit that it admits such a merge operation, thus providing an optimal solution for the problem for both sequential and parallel applications. Orthogonally, the new algorithm also improves algorithmically on Błasiok’s solution (even in the sequential setting) by reducing its implementation complexity and requiring fewer distinct pseudo-random objects.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.35/LIPIcs.APPROX-RANDOM.2023.35.pdf
Distinct Elements
Distributed Algorithms
Randomized Algorithms
Expander Graphs
Derandomization
Sketching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
36:1
36:21
10.4230/LIPIcs.APPROX/RANDOM.2023.36
article
Sampling and Certifying Symmetric Functions
Filmus, Yuval
1
https://orcid.org/0000-0002-1739-0872
Leigh, Itai
2
https://orcid.org/0009-0007-4298-6284
Riazanov, Artur
3
https://orcid.org/0000-0001-7892-1502
Sokolov, Dmitry
3
https://orcid.org/0000-0003-2809-3467
Technion - Israel Institute of Technology, Haifa, Israel
Tel-Aviv University, Israel
École Polytechnique Fédérale de Lausanne, Switzerland
A circuit 𝒞 samples a distribution X with an error ε if the statistical distance between the output of 𝒞 on the uniform input and X is ε. We study the hardness of sampling a uniform distribution over the set of n-bit strings of Hamming weight k denoted by Uⁿ_k for decision forests, i.e. every output bit is computed as a decision tree of the inputs. For every k there is an O(log n)-depth decision forest sampling Uⁿ_k with an inverse-polynomial error [Emanuele Viola, 2012; Czumaj, 2015]. We show that for every ε > 0 there exists τ such that for decision depth τ log (n/k) / log log (n/k), the error for sampling U_kⁿ is at least 1-ε. Our result is based on the recent robust sunflower lemma [Ryan Alweiss et al., 2021; Rao, 2019].
Our second result is about matching a set of n-bit strings with the image of a d-local circuit, i.e. such that each output bit depends on at most d input bits. We study the set of all n-bit strings whose Hamming weight is at least n/2. We improve the previously known locality lower bound from Ω(log^* n) [Beyersdorff et al., 2013] to Ω(√log n), leaving only a quartic gap from the best upper bound of O(log² n).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.36/LIPIcs.APPROX-RANDOM.2023.36.pdf
sampling
lower bounds
robust sunflowers
decision trees
switching networks
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
37:1
37:20
10.4230/LIPIcs.APPROX/RANDOM.2023.37
article
Hardness of the (Approximate) Shortest Vector Problem: A Simple Proof via Reed-Solomon Codes
Bennett, Huck
1
Peikert, Chris
2
3
Oregon State University, Corvallis, OR, USA
University of Michigan, Ann Arbor, MI, USA
Algorand, Inc., Boston, MA, USA
We give a simple proof that the (approximate, decisional) Shortest Vector Problem is NP-hard under a randomized reduction. Specifically, we show that for any p ≥ 1 and any constant γ < 2^{1/p}, the γ-approximate problem in the 𝓁_p norm (γ-GapSVP_p) is not in RP unless NP ⊆ RP. Our proof follows an approach pioneered by Ajtai (STOC 1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing hardness of γ-GapSVP_p using locally dense lattices. We construct such lattices simply by applying "Construction A" to Reed-Solomon codes with suitable parameters, and prove their local density via an elementary argument originally used in the context of Craig lattices.
As in all known NP-hardness results for GapSVP_p with p < ∞, our reduction uses randomness. Indeed, it is a notorious open problem to prove NP-hardness via a deterministic reduction. To this end, we additionally discuss potential directions and associated challenges for derandomizing our reduction. In particular, we show that a close deterministic analogue of our local density construction would improve on the state-of-the-art explicit Reed-Solomon list-decoding lower bounds of Guruswami and Rudra (STOC 2005 and IEEE Transactions on Information Theory 2006).
As a related contribution of independent interest, we also give a polynomial-time algorithm for decoding n-dimensional "Construction A Reed-Solomon lattices" (with different parameters than those used in our hardness proof) to a distance within an O(√log n) factor of Minkowski’s bound. This asymptotically matches the best known distance for decoding near Minkowski’s bound, due to Mook and Peikert (IEEE Transactions on Information Theory 2022), whose work we build on with a somewhat simpler construction and analysis.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.37/LIPIcs.APPROX-RANDOM.2023.37.pdf
Lattices
Shortest Vector Problem
Reed-Solomon codes
NP-hardness
derandomization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
38:1
38:18
10.4230/LIPIcs.APPROX/RANDOM.2023.38
article
Perfect Sampling for Hard Spheres from Strong Spatial Mixing
Anand, Konrad
1
Göbel, Andreas
2
Pappik, Marcus
2
Perkins, Will
3
Queen Mary, University of London, UK
Hasso Plattner Institute, University of Potsdam, Germany
School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA
We provide a perfect sampling algorithm for the hard-sphere model on subsets of R^d with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.38/LIPIcs.APPROX-RANDOM.2023.38.pdf
perfect sampling
hard-sphere model
Gibbs point processes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
39:1
39:18
10.4230/LIPIcs.APPROX/RANDOM.2023.39
article
Subset Sum in Time 2^{n/2} / poly(n)
Chen, Xi
1
https://orcid.org/0000-0002-1263-1024
Jin, Yaonan
1
https://orcid.org/0000-0001-6256-7625
Randolph, Tim
1
https://orcid.org/0000-0003-4287-0680
Servedio, Rocco A.
1
https://orcid.org/0000-0003-2407-543X
Columbia University, New York, NY, USA
A major goal in the area of exact exponential algorithms is to give an algorithm for the (worst-case) n-input Subset Sum problem that runs in time 2^{(1/2 - c)n} for some constant c > 0. In this paper we give a Subset Sum algorithm with worst-case running time O(2^{n/2} ⋅ n^{-γ}) for a constant γ > 0.5023 in standard word RAM or circuit RAM models. To the best of our knowledge, this is the first improvement on the classical "meet-in-the-middle" algorithm for worst-case Subset Sum, due to Horowitz and Sahni, which can be implemented in time O(2^{n/2}) in these memory models [Horowitz and Sahni, 1974].
Our algorithm combines a number of different techniques, including the "representation method" introduced by Howgrave-Graham and Joux [Howgrave-Graham and Joux, 2010] and subsequent adaptations of the method in Austrin, Kaski, Koivisto, and Nederlof [Austrin et al., 2016], and Nederlof and Węgrzycki [Jesper Nederlof and Karol Wegrzycki, 2021], and "bit-packing" techniques used in the work of Baran, Demaine, and Pǎtraşcu [Baran et al., 2005] on subquadratic algorithms for 3SUM.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.39/LIPIcs.APPROX-RANDOM.2023.39.pdf
Exact algorithms
subset sum
log shaving
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
40:1
40:14
10.4230/LIPIcs.APPROX/RANDOM.2023.40
article
On Optimization and Counting of Non-Broken Bases of Matroids
Abdolazimi, Dorna
1
Lindberg, Kasper
1
Gharan, Shayan Oveis
1
University of Washington, Seattle, WA, USA
Given a matroid M = (E,I), and a total ordering over the elements E, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in I with no broken circuit. The set of NBC independent sets of any matroid M define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota.
We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.40/LIPIcs.APPROX-RANDOM.2023.40.pdf
Complexity
Hardness
Optimization
Counting
Random walk
Local to Global
Matroids
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
41:1
41:22
10.4230/LIPIcs.APPROX/RANDOM.2023.41
article
Low-Degree Testing over Grids
Amireddy, Prashanth
1
https://orcid.org/0000-0002-2713-8961
Srinivasan, Srikanth
2
https://orcid.org/0000-0001-6491-124X
Sudan, Madhu
1
https://orcid.org/0000-0003-3718-6489
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Department of Computer Science, Aarhus University, Denmark
We study the question of local testability of low (constant) degree functions from a product domain 𝒮_1 × … × 𝒮_n to a field 𝔽, where 𝒮_i ⊆ 𝔽 can be arbitrary constant sized sets. We show that this family is locally testable when the grid is "symmetric". That is, if 𝒮_i = 𝒮 for all i, there is a probabilistic algorithm using constantly many queries that distinguishes whether f has a polynomial representation of degree at most d or is Ω(1)-far from having this property. In contrast, we show that there exist asymmetric grids with |𝒮_1| = ⋯ = |𝒮_n| = 3 for which testing requires ω_n(1) queries, thereby establishing that even in the context of polynomials, local testing depends on the structure of the domain and not just the distance of the underlying code.
The low-degree testing problem has been studied extensively over the years and a wide variety of tools have been applied to propose and analyze tests. Our work introduces yet another new connection in this rich field, by building low-degree tests out of tests for "junta-degrees". A function f:𝒮_1 × ⋯ × 𝒮_n → 𝒢, for an abelian group 𝒢 is said to be a junta-degree-d function if it is a sum of d-juntas. We derive our low-degree test by giving a new local test for junta-degree-d functions. For the analysis of our tests, we deduce a small-set expansion theorem for spherical/hamming noise over large grids, which may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.41/LIPIcs.APPROX-RANDOM.2023.41.pdf
Property testing
Low-degree testing
Small-set expansion
Local testing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
42:1
42:23
10.4230/LIPIcs.APPROX/RANDOM.2023.42
article
Improved Local Computation Algorithms for Constructing Spanners
Arviv, Rubi
1
Chung, Lily
2
Levi, Reut
1
https://orcid.org/0000-0003-3167-1766
Pyne, Edward
2
Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
MIT, Cambridge, MA, USA
A spanner of a graph is a subgraph that preserves lengths of shortest paths up to a multiplicative distortion. For every k, a spanner with size O(n^{1+1/k}) and stretch (2k+1) can be constructed by a simple centralized greedy algorithm, and this is tight assuming Erdős girth conjecture.
In this paper we study the problem of constructing spanners in a local manner, specifically in the Local Computation Model proposed by Rubinfeld et al. (ICS 2011).
We provide a randomized Local Computation Agorithm (LCA) for constructing (2r-1)-spanners with Õ(n^{1+1/r}) edges and probe complexity of Õ(n^{1-1/r}) for r ∈ {2,3}, where n denotes the number of vertices in the input graph. Up to polylogarithmic factors, in both cases, the stretch factor is optimal (for the respective number of edges). In addition, our probe complexity for r = 2, i.e., for constructing a 3-spanner, is optimal up to polylogarithmic factors. Our result improves over the probe complexity of Parter et al. (ITCS 2019) that is Õ(n^{1-1/2r}) for r ∈ {2,3}. Both our algorithms and the algorithms of Parter et al. use a combination of neighbor-probes and pair-probes in the above-mentioned LCAs.
For general k ≥ 1, we provide an LCA for constructing O(k²)-spanners with Õ(n^{1+1/k}) edges using O(n^{2/3}Δ²) neighbor-probes, improving over the Õ(n^{2/3}Δ⁴) algorithm of Parter et al.
By developing a new randomized LCA for graph decomposition, we further improve the probe complexity of the latter task to be O(n^{2/3-(1.5-α)/k}Δ²), for any constant α > 0. This latter LCA may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.42/LIPIcs.APPROX-RANDOM.2023.42.pdf
Local Computation Algorithms
Spanners
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
43:1
43:17
10.4230/LIPIcs.APPROX/RANDOM.2023.43
article
Classical Simulation of One-Query Quantum Distinguishers
Bogdanov, Andrej
1
Cheung, Tsun Ming
2
Dinesh, Krishnamoorthy
3
Lui, John C. S.
4
School of EECS, University of Ottawa, Canada
School of Computer Science, McGill University, Montreal, Canada
Dept. of Computer Science and Engineering, Indian Institute of Technology, Palakkad, India
Dept. of Computer Science and Engineering, Chinese University of Hong Kong, China
We study the relative advantage of classical and quantum distinguishers of bounded query complexity over n-bit strings, focusing on the case of a single quantum query. A construction of Aaronson and Ambainis (STOC 2015) yields a pair of distributions that is ε-distinguishable by a one-query quantum algorithm, but O(ε k/√n)-indistinguishable by any non-adaptive k-query classical algorithm.
We show that every pair of distributions that is ε-distinguishable by a one-query quantum algorithm is distinguishable with k classical queries and (1) advantage min{Ω(ε√{k/n})), Ω(ε²k²/n)} non-adaptively (i.e., in one round), and (2) advantage Ω(ε²k/√{n log n}) in two rounds.
As part of our analysis, we introduce a general method for converting unbiased estimators into distinguishers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.43/LIPIcs.APPROX-RANDOM.2023.43.pdf
Query complexity
quantum algorithms
hypothesis testing
Grothendieck’s inequality
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
44:1
44:22
10.4230/LIPIcs.APPROX/RANDOM.2023.44
article
On the Power of Regular and Permutation Branching Programs
Lee, Chin Ho
1
Pyne, Edward
2
Vadhan, Salil
1
Harvard University, Cambridge, MA, USA
MIT, Cambridge, MA, USA
We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation.
- Regular SOBPs of length n and width ⌊w(n+1)/2⌋ can exactly simulate general SOBPs of length n and width w, and moreover an n/2-o(n) blow-up in width is necessary for such a simulation. Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width poly(n) or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs.
- There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod 2 is average-case hard for arbitrary-order permutation ROBPs of exponential width.
- There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs.
- Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.44/LIPIcs.APPROX-RANDOM.2023.44.pdf
Pseudorandomness
Branching Programs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
45:1
45:24
10.4230/LIPIcs.APPROX/RANDOM.2023.45
article
Private Data Stream Analysis for Universal Symmetric Norm Estimation
Braverman, Vladimir
1
Manning, Joel
2
Wu, Zhiwei Steven
2
Zhou, Samson
3
1
Rice University, Houston, TX, USA
Carnegie Mellon University, Pittsburgh, PA, USA
University of California Berkeley, CA, USA
We study how to release summary statistics on a data stream subject to the constraint of differential privacy. In particular, we focus on releasing the family of symmetric norms, which are invariant under sign-flips and coordinate-wise permutations on an input data stream and include L_p norms, k-support norms, top-k norms, and the box norm as special cases. Although it may be possible to design and analyze a separate mechanism for each symmetric norm, we propose a general parametrizable framework that differentially privately releases a number of sufficient statistics from which the approximation of all symmetric norms can be simultaneously computed. Our framework partitions the coordinates of the underlying frequency vector into different levels based on their magnitude and releases approximate frequencies for the "heavy" coordinates in important levels and releases approximate level sizes for the "light" coordinates in important levels. Surprisingly, our mechanism allows for the release of an arbitrary number of symmetric norm approximations without any overhead or additional loss in privacy. Moreover, our mechanism permits (1+α)-approximation to each of the symmetric norms and can be implemented using sublinear space in the streaming model for many regimes of the accuracy and privacy parameters.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.45/LIPIcs.APPROX-RANDOM.2023.45.pdf
Differential privacy
norm estimation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
46:1
46:18
10.4230/LIPIcs.APPROX/RANDOM.2023.46
article
Testing Versus Estimation of Graph Properties, Revisited
Gishboliner, Lior
1
Kushnir, Nick
2
Shapira, Asaf
2
ETH Zürich, Switzerland
School of Mathematics, Tel Aviv University, Israel
A graph G on n vertices is ε-far from property P if one should add/delete at least ε n² edges to turn G into a graph satisfying P. A distance estimator for P is an algorithm that given G and α, ε > 0 distinguishes between the case that G is (α-ε)-close to 𝒫 and the case that G is α-far from 𝒫. If P has a distance estimator whose query complexity depends only on ε, then P is said to be estimable.
Every estimable property is clearly also testable, since testing corresponds to estimating with α = ε. A central result in the area of property testing is the Fischer-Newman theorem, stating that an inverse statement also holds, that is, that every testable property is in fact estimable. The proof of Fischer and Newmann was highly ineffective, since it incurred a tower-type loss when transforming a testing algorithm for P into a distance estimator. This raised the natural problem, studied recently by Fiat-Ron and by Hoppen-Kohayakawa-Lang-Lefmann-Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results.
- We show that if P is hereditary, then one can turn a tester for P into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et. al., who obtained a transformation with a double exponential loss.
- We show that for every P, one can turn a testing algorithm for P into a distance estimator with a double exponential loss. This improves over the transformation of Fischer-Newman that incurred a tower-type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer-Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze-Kannan Weak Regular partitions that are of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.46/LIPIcs.APPROX-RANDOM.2023.46.pdf
Testing
estimation
weak regularity
randomized algorithms
graph theory
Frieze-Kannan Regularity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
47:1
47:22
10.4230/LIPIcs.APPROX/RANDOM.2023.47
article
Efficient Interactive Proofs for Non-Deterministic Bounded Space
Cook, Joshua
1
https://orcid.org/0000-0002-4851-7573
Rothblum, Ron D.
2
https://orcid.org/0000-0001-5481-7276
Department of Computer Science, University of Texas Austin, TX, USA
Faculty of Computer Science, Technion, Haifa, Israel
The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers.
More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time Õ(n+S²). This improves on the best previous bound of Õ(n+S³) and matches the result for deterministic space bounded algorithms, up to polylog(S) factors.
We further generalize to alternating bounded space algorithms. For any language L decided by a time T, space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time Õ(n + S log(T) + S d) and the prover runs in time 2^O(S). For d = O(log(T)), this matches the best known interactive proofs for deterministic algorithms, up to polylog(S) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log(T). We also improve the best prior verifier time for unbounded alternations by a factor of S.
Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.47/LIPIcs.APPROX-RANDOM.2023.47.pdf
Interactive Proofs
Alternating Algorithms
AC0[2]
Doubly Efficient Proofs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
48:1
48:22
10.4230/LIPIcs.APPROX/RANDOM.2023.48
article
On the Complexity of Triangle Counting Using Emptiness Queries
Bishnu, Arijit
1
Ghosh, Arijit
1
Mishra, Gopinath
2
https://orcid.org/0000-0003-0540-0292
Indian Statistical Institute, Kolkata, India
University of Warwick, Coventry, UK
Beame et al. [ITCS'18 & TALG'20] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC'18 and TOCT'21], Dell, Lapinskas, and Meeks [SODA'20 and SICOMP'22], Bhattacharya et al. [ISAAC'19 & TOCS'21], and Chen et al. [SODA'20]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.’s upper bound result for edge estimation using IS and also showed an almost matching lower bound. Beame et al. in their introductory work asked a few open questions out of which one was on estimating structures of higher order than edges, like triangles and cliques, using BIS queries.
In this work, we almost resolve the query complexity of estimating triangles using BIS oracle. While doing so, we prove a lower bound for an even stronger query oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi, Chakrabarty, and Khanna [ESA'21] to test graph connectivity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.48/LIPIcs.APPROX-RANDOM.2023.48.pdf
Triangle Counting
Emptiness Queries
Bipartite Independent Set Query
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
49:1
49:22
10.4230/LIPIcs.APPROX/RANDOM.2023.49
article
Fine Grained Analysis of High Dimensional Random Walks
Gotlib, Roy
1
Kaufman, Tali
1
Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
One of the most important properties of high dimensional expanders is that high dimensional random walks converge rapidly. This property has proven to be extremely useful in a variety of fields in the theory of computer science from agreement testing to sampling, coding theory and more. In this paper we present a state of the art result in a line of works analyzing the convergence of high dimensional random walks [Tali Kaufman and David Mass, 2017; Irit Dinur and Tali Kaufman, 2017; Tali Kaufman and Izhar Oppenheim, 2018; Vedat Levi Alev and Lap Chi Lau, 2020], by presenting a structured version of the result of [Vedat Levi Alev and Lap Chi Lau, 2020]. While previous works examined the expansion in the viewpoint of the worst possible eigenvalue, in this work we relate the expansion of a function to the entire spectrum of the random walk operator using the structure of the function; We call such a theorem a Fine Grained High Order Random Walk Theorem. In sufficiently structured cases the fine grained result that we present here can be much better than the worst case while in the worst case our result is equivalent to [Vedat Levi Alev and Lap Chi Lau, 2020].
In order to prove the Fine Grained High Order Random Walk Theorem we introduce a way to bootstrap the expansion of random walks on the vertices of a complex into a fine grained understanding of higher order random walks, provided that the expansion is good enough.
In addition, our single bootstrapping theorem can simultaneously yield our Fine Grained High Order Random Walk Theorem as well as the well known Trickling down Theorem. Prior to this work, High order Random walks theorems and Tricking down Theorem have been obtained from different proof methods.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.49/LIPIcs.APPROX-RANDOM.2023.49.pdf
High Dimensional Expanders
High Dimensional Random Walks
Local Spectral Expansion
Local to Global
Trickling Down
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
50:1
50:10
10.4230/LIPIcs.APPROX/RANDOM.2023.50
article
A Deterministic Construction of a Large Distance Code from the Wozencraft Ensemble
Guruswami, Venkatesan
1
https://orcid.org/0000-0001-7926-3396
Li, Shilun
2
https://orcid.org/0000-0001-5765-0432
Department of EECS, University of California, Berkeley, CA, USA
Department of Mathematics, University of California, Berkeley, CA, USA
We present an explicit construction of a sequence of rate 1/2 Wozencraft ensemble codes (over any fixed finite field 𝔽_q) that achieve minimum distance Ω(√k) where k is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of 𝔽_{q^k} where k+1 is prime with q a primitive root modulo k+1. Assuming Artin’s conjecture, there are infinitely many such k for any prime power q.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.50/LIPIcs.APPROX-RANDOM.2023.50.pdf
Algebraic codes
Pseudorandomness
Explicit Construction
Wozencraft Ensemble
Sidon Sets
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
51:1
51:12
10.4230/LIPIcs.APPROX/RANDOM.2023.51
article
NP-Hardness of Almost Coloring Almost 3-Colorable Graphs
Hecht, Yahli
1
https://orcid.org/0009-0000-0596-080X
Minzer, Dor
2
https://orcid.org/0000-0002-8093-1328
Safra, Muli
1
https://orcid.org/0000-0002-5022-7727
School of Computer Science, Tel Aviv University, Israel
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
A graph G = (V,E) is said to be (k,δ) almost colorable if there is a subset of vertices V' ⊆ V of size at least (1-δ)|V| such that the induced subgraph of G on V' is k-colorable. We prove that for all k, there exists δ > 0 such for all ε > 0, given a graph G it is NP-hard (under randomized reductions) to distinguish between:
1) Yes case: G is (3,ε) almost colorable.
2) No case: G is not (k,δ) almost colorable. This improves upon an earlier result of Khot et al. [Irit Dinur et al., 2018], who showed a weaker result wherein in the "yes case" the graph is (4,ε) almost colorable.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.51/LIPIcs.APPROX-RANDOM.2023.51.pdf
PCP
Hardness of approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
52:1
52:22
10.4230/LIPIcs.APPROX/RANDOM.2023.52
article
Extracting Mergers and Projections of Partitions
Kopparty, Swastik
1
https://orcid.org/0000-0003-2704-8808
N, Vishvajeet
2
https://orcid.org/0000-0002-1502-9231
Department of Computer Science and Department of Mathematics, University of Toronto, Canada
School of Informatics, University of Edinburgh,UK
We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer’s lemma on projections.
A somewhere-random source is a tuple (X_1, …, X_t) of (possibly correlated) {0,1}ⁿ-valued random variables X_i where for some unknown i ∈ [t], X_i is guaranteed to be uniformly distributed. An extracting merger is a seeded device that takes a somewhere-random source as input and outputs nearly uniform random bits. We study the seed-length needed for extracting mergers with constant t and constant error.
Since a somewhere-random source has min-entropy at least n, a standard extractor can also serve as an extracting merger. Our goal is to understand whether the further structure of being somewhere-random rather than just having high entropy enables smaller seed-length, and towards this we show:
- Just like in the case of standard extractors, seedless extracting mergers with even just one output bit do not exist.
- Unlike the case of standard extractors, it is possible to have extracting mergers that output a constant number of bits using only constant seed. Furthermore, a random choice of merger does not work for this purpose!
- Nevertheless, just like in the case of standard extractors, an extracting merger which gets most of the entropy out (namely, having Ω(n) output bits) must have Ω(log n) seed. This is the main technical result of our work, and is proved by a second-moment strengthening of the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors.
All this is in contrast to the status for condensing mergers (where the output is only required to have high min-entropy), whose seed-length/output-length tradeoffs can all be fully explained by using standard condensers.
Inspired by such considerations, we also formulate a new and basic class of problems in combinatorics: partition analogues of Shearer’s lemma. We show basic results in this direction; in particular, we prove that in any partition of the 3-dimensional cube [0,1]³ into two parts, one of the parts has an axis parallel 2-dimensional projection of area at least 3/4.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.52/LIPIcs.APPROX-RANDOM.2023.52.pdf
randomness extractors
randomness mergers
extracting mergers
partitions
projections of partitions
covers
projections of covers
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
53:1
53:19
10.4230/LIPIcs.APPROX/RANDOM.2023.53
article
Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case
Blanca, Antonio
1
Zhang, Xusheng
1
https://orcid.org/0000-0001-8751-7958
Pennsylvania State University, University Park, PA, USA
We consider spin systems on general n-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on graphs of maximum degree Δ, where Δ is allowed to grow with n, converges in O((Δ log n)^c) steps where c > 0 is a constant independent of Δ and n. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is O(Δ^c log n) for a constant c > 0 independent of Δ and n. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. This result implies optimal O(log n) mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called k-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.53/LIPIcs.APPROX-RANDOM.2023.53.pdf
Markov chains
Spectral Independence
Potts model
Mixing Time
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
54:1
54:14
10.4230/LIPIcs.APPROX/RANDOM.2023.54
article
The Full Rank Condition for Sparse Random Matrices
Coja-Oghlan, Amin
1
Gao, Jane
2
Hahn-Klimroth, Max
1
Lee, Joon
3
Müller, Noela
4
Rolvien, Maurice
1
Department of Computer Science, TU Dortmund, Germany
Department of Combinatorics and Optimization, University of Waterloo, Canada
Communication Theory Laboratory, École Polytechnique Fédérale de Lausanne, Switzerland
Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.54/LIPIcs.APPROX-RANDOM.2023.54.pdf
random matrices
rank
finite fields
rationals
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
55:1
55:22
10.4230/LIPIcs.APPROX/RANDOM.2023.55
article
Tighter MA/1 Circuit Lower Bounds from Verifier Efficient PCPs for PSPACE
Cook, Joshua
1
https://orcid.org/0000-0002-4851-7573
Moshkovitz, Dana
1
https://orcid.org/0000-0002-4151-568X
Department of Computer Science, University of Texas Austin, TX, USA
We prove that for some constant a > 1, for all k ≤ a, MATIME[n^{k+o(1)}]/1 ⊄ SIZE[O(n^k)], for some specific o(1) function. This is a super linear polynomial circuit lower bound.
Previously, Santhanam [Santhanam, 2007] showed that there exists a constant c > 1 such that for all k > 1: MATIME[n^{ck}]/1 ⊄ SIZE[O(n^k)]. Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c. Using ideas from Murray and Williams [Murray and Williams, 2018], one can show for all k > 1: MATIME [n^{10 k²}]/1 ⊄ SIZE[O(n^k)].
To prove this result, we construct the first PCP for SPACE[n] with quasi-linear verifier time: our PCP has a Õ(n) time verifier, Õ(n) space prover, O(log(n)) queries, and polynomial alphabet size. Prior to this work, PCPs for SPACE[O(n)] had verifiers that run in Ω(n²) time. This PCP also proves that NE has MIP verifiers which run in time Õ(n).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.55/LIPIcs.APPROX-RANDOM.2023.55.pdf
MA
PCP
Circuit Complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
56:1
56:21
10.4230/LIPIcs.APPROX/RANDOM.2023.56
article
Robustness for Space-Bounded Statistical Zero Knowledge
Allender, Eric
1
https://orcid.org/0000-0002-0650-028X
Gray, Jacob
2
Mutreja, Saachi
3
Tirumala, Harsha
1
https://orcid.org/0000-0002-4600-3675
Wang, Pengxiang
4
Rutgers University, Piscataway, NJ, USA
University of Massachusetts, Amherst, MA, USA
University of California, Berkeley, CA, USA
University of Michigan, Ann Arbor, MI, USA
We show that the space-bounded Statistical Zero Knowledge classes SZK_L and NISZK_L are surprisingly robust, in that the power of the verifier and simulator can be strengthened or weakened without affecting the resulting class. Coupled with other recent characterizations of these classes [Eric Allender et al., 2023], this can be viewed as lending support to the conjecture that these classes may coincide with the non-space-bounded classes SZK and NISZK, respectively.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.56/LIPIcs.APPROX-RANDOM.2023.56.pdf
Interactive Proofs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
57:1
57:18
10.4230/LIPIcs.APPROX/RANDOM.2023.57
article
On Constructing Spanners from Random Gaussian Projections
Assadi, Sepehr
1
Kapralov, Michael
2
Yu, Huacheng
3
Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland
Department of Computer Science, Princeton University, NJ, USA
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area.
We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA'21), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.57/LIPIcs.APPROX-RANDOM.2023.57.pdf
sketching algorithm
lower bound
graph spanner
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
58:1
58:23
10.4230/LIPIcs.APPROX/RANDOM.2023.58
article
Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance
Ashvinkumar, Vikrant
1
Assadi, Sepehr
1
2
Deng, Chengyuan
1
Gao, Jie
1
https://orcid.org/0000-0001-5083-6082
Wang, Chen
1
https://orcid.org/0000-0003-4044-9438
Department of Computer Science, Rutgers University, New Brunswick, NJ, USA
Cheriton School of Computer Science, University of Waterloo, Canada
Structural balance theory studies stability in networks. Given a n-vertex complete graph G = (V,E) whose edges are labeled positive or negative, the graph is considered balanced if every triangle either consists of three positive edges (three mutual "friends"), or one positive edge and two negative edges (two "friends" with a common "enemy"). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: (i) detecting whether a given graph is balanced, or (ii) finding a partition that approximates the frustration index, i.e., the minimum number of edge flips that turn the graph balanced.
We study these problems in the streaming model where edges are given one by one and focus on memory efficiency. We provide randomized single-pass algorithms for: (i) determining whether an input graph is balanced with O(log n) memory, and (ii) finding a partition that induces a (1 + ε)-approximation to the frustration index with O(n ⋅ polylog(n)) memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation.
To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen vertices in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from n^O(1/ε²) to O(n²log³n/ε² + n log n ⋅ (1/ε)^O(1/ε⁴)) time for (1+ε)-approximation. These results may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.58/LIPIcs.APPROX-RANDOM.2023.58.pdf
Streaming algorithms
structural balance
pseudo-randomness generator
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
59:1
59:24
10.4230/LIPIcs.APPROX/RANDOM.2023.59
article
How to Make Your Approximation Algorithm Private: A Black-Box Differentially-Private Transformation for Tunable Approximation Algorithms of Functions with Low Sensitivity
Blocki, Jeremiah
1
Grigorescu, Elena
1
Mukherjee, Tamalika
1
Zhou, Samson
2
3
Purdue University, West Lafayette, IN, USA
University of California Berkeley, CA, USA
Rice University, Houston, TX, USA
We develop a framework for efficiently transforming certain approximation algorithms into differentially-private variants, in a black-box manner. Specifically, our results focus on algorithms A that output an approximation to a function f of the form (1-α)f(x)-κ ≤ A(x) ≤ (1+α)f(x)+κ, where κ ∈ ℝ_{≥ 0} denotes additive error and α ∈ [0,1) denotes multiplicative error can be"tuned" to small-enough values while incurring only a polynomial blowup in the running time/space. We show that such algorithms can be made differentially private without sacrificing accuracy, as long as the function f has small "global sensitivity". We achieve these results by applying the "smooth sensitivity" framework developed by Nissim, Raskhodnikova, and Smith (STOC 2007).
Our framework naturally applies to transform non-private FPRAS and FPTAS algorithms into ε-differentially private approximation algorithms where the former case requires an additional postprocessing step. We apply our framework in the context of sublinear-time and sublinear-space algorithms, while preserving the nature of the algorithm in meaningful ranges of the parameters. Our results include the first (to the best of our knowledge) ε-edge differentially-private sublinear-time algorithm for estimating the number of triangles, the number of connected components, and the weight of a minimum spanning tree of a graph whose accuracy holds with high probability. In the area of streaming algorithms, our results include ε-DP algorithms for estimating L_p-norms, distinct elements, and weighted minimum spanning tree for both insertion-only and turnstile streams. Our transformation also provides a private version of the smooth histogram framework, which is commonly used for converting streaming algorithms into sliding window variants, and achieves a multiplicative approximation to many problems, such as estimating L_p-norms, distinct elements, and the length of the longest increasing subsequence.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.59/LIPIcs.APPROX-RANDOM.2023.59.pdf
Differential privacy
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
60:1
60:16
10.4230/LIPIcs.APPROX/RANDOM.2023.60
article
Fast Decoding of Explicit Almost Optimal ε-Balanced q-Ary Codes And Fast Approximation of Expanding k-CSPs
Jeronimo, Fernando Granha
1
Institute for Advanced Study, Princeton, NJ, USA
Good codes over an alphabet of constant size q can approach but not surpass distance 1-1/q. This makes the use of q-ary codes a necessity in some applications, and much work has been devoted to the case of constant alphabet q. In the large distance regime, namely, distance 1-1/q-ε for small ε > 0, the Gilbert-Varshamov (GV) bound asserts that rate Ω_q(ε²) is achievable whereas the q-ary MRRW bound gives a rate upper bound of O_q(ε²log(1/ε)). In this sense, the GV bound is almost optimal in this regime. Prior to this work there was no known explicit and efficiently decodable q-ary codes near the GV bound, in this large distance regime, for any constant q ≥ 3.
We design an Õ_{ε,q}(N) time decoder for explicit (expander based) families of linear codes C_{N,q,ε} ⊆ F_q^N of distance (1-1/q)(1-ε) and rate Ω_q(ε^{2+o(1)}), for any desired ε > 0 and any constant prime q, namely, almost optimal in this regime. These codes are ε-balanced,i.e., for every non-zero codeword, the frequency of each symbol lies in the interval [1/q - ε, 1/q + ε]. A key ingredient of the q-ary decoder is a new near-linear time approximation algorithm for linear equations (k-LIN) over ℤ_q on expanding hypergraphs, in particular, those naturally arising in the decoding of these codes.
We also investigate k-CSPs on expanding hypergraphs in more generality. We show that special trade-offs available for k-LIN over ℤ_q hold for linear equations over a finite group. To handle general finite groups, we design a new matrix version of weak regularity for expanding hypergraphs. We also obtain a near-linear time approximation algorithm for general expanding k-CSPs over q-ary alphabet. This later algorithm runs in time Õ_{k,q}(m + n), where m is the number of constraints and n is the number of variables. This improves the previous best running time of O(n^{Θ_{k,q}(1)}) by a Sum-of-Squares based algorithm of [AJT, 2019] (in the expanding regular case).
We obtain our results by generalizing the framework of [JST, 2021] based on weak regularity decomposition for expanding hypergraphs. This framework was originally designed for binary k-XOR with the goal of providing near-linear time decoder for explicit binary codes, near the GV bound, from the breakthrough work of Ta-Shma [STOC, 2017]. The explicit families of codes over prime F_q are based on suitable instatiations of the Jalan-Moshkovitz (Abelian) generalization of Ta-Shma’s distance amplification procedure.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.60/LIPIcs.APPROX-RANDOM.2023.60.pdf
Decoding
Approximation
GV bound
CSPs
HDXs
Regularity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
61:1
61:18
10.4230/LIPIcs.APPROX/RANDOM.2023.61
article
Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions
Ferreira Pinto Jr., Renato
1
University of Waterloo, Canada
We study the connection between directed isoperimetric inequalities and monotonicity testing. In recent years, this connection has unlocked breakthroughs for testing monotonicity of functions defined on discrete domains. Inspired the rich history of isoperimetric inequalities in continuous settings, we propose that studying the relationship between directed isoperimetry and monotonicity in such settings is essential for understanding the full scope of this connection.
Hence, we ask whether directed isoperimetric inequalities hold for functions f:[0,1]ⁿ → R, and whether this question has implications for monotonicity testing. We answer both questions affirmatively. For Lipschitz functions f:[0,1]ⁿ → ℝ, we show the inequality d^mono₁(f) ≲ 𝔼 [‖∇^- f‖₁], which upper bounds the L¹ distance to monotonicity of f by a measure of its "directed gradient". A key ingredient in our proof is the monotone rearrangement of f, which generalizes the classical "sorting operator" to continuous settings. We use this inequality to give an L¹ monotonicity tester for Lipschitz functions f:[0,1]ⁿ → ℝ, and this framework also implies similar results for testing real-valued functions on the hypergrid.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.61/LIPIcs.APPROX-RANDOM.2023.61.pdf
Monotonicity testing
property testing
isoperimetric inequalities
Poincaré inequalities
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
62:1
62:21
10.4230/LIPIcs.APPROX/RANDOM.2023.62
article
Bias Reduction for Sum Estimation
Eden, Talya
1
https://orcid.org/0000-0001-8470-9508
Houen, Jakob Bæk Tejs
2
Narayanan, Shyam
3
Rosenbaum, Will
4
https://orcid.org/0000-0002-7723-9090
Tětek, Jakub
2
Bar-Ilan University, Ramat Gan, Israel
BARC, University of Copenhagen, Denmark
MIT, Cambridge, MA, USA
Amherst College, MA, USA
In classical statistics and distribution testing, it is often assumed that elements can be sampled exactly from some distribution 𝒫, and that when an element x is sampled, the probability 𝒫(x) of sampling x is also known. In this setting, recent work in distribution testing has shown that many algorithms are robust in the sense that they still produce correct output if the elements are drawn from any distribution 𝒬 that is sufficiently close to 𝒫. This phenomenon raises interesting questions: under what conditions is a "noisy" distribution 𝒬 sufficient, and what is the algorithmic cost of coping with this noise?
In this paper, we investigate these questions for the problem of estimating the sum of a multiset of N real values x_1, …, x_N. This problem is well-studied in the statistical literature in the case 𝒫 = 𝒬, where the Hansen-Hurwitz estimator [Annals of Mathematical Statistics, 1943] is frequently used. We assume that for some (known) distribution 𝒫, values are sampled from a distribution 𝒬 that is pointwise close to 𝒫. That is, there is a parameter γ < 1 such that for all x_i, (1 - γ) 𝒫(i) ≤ 𝒬(i) ≤ (1 + γ) 𝒫(i). For every positive integer k we define an estimator ζ_k for μ = ∑_i x_i whose bias is proportional to γ^k (where our ζ₁ reduces to the classical Hansen-Hurwitz estimator). As a special case, we show that if 𝒬 is pointwise γ-close to uniform and all x_i ∈ {0, 1}, for any ε > 0, we can estimate μ to within additive error ε N using m = Θ(N^{1-1/k}/ε^{2/k}) samples, where k = ⌈lg ε/lg γ⌉. We then show that this sample complexity is essentially optimal. Interestingly, our upper and lower bounds show that the sample complexity need not vary uniformly with the desired error parameter ε: for some values of ε, perturbations in its value have no asymptotic effect on the sample complexity, while for other values, any decrease in its value results in an asymptotically larger sample complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.62/LIPIcs.APPROX-RANDOM.2023.62.pdf
bias reduction
sum estimation
sublinear time algorithms
sample complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
63:1
63:23
10.4230/LIPIcs.APPROX/RANDOM.2023.63
article
On the Composition of Randomized Query Complexity and Approximate Degree
Chakraborty, Sourav
1
Kayal, Chandrima
1
Mittal, Rajat
2
Paraashar, Manaswi
3
Sanyal, Swagato
4
Saurabh, Nitin
5
Indian Statistical Institute, Kolkata, India
Indian Institute of Technology Kanpur, India
Aarhus University, Denmark
Indian Institute of Technology Kharagpur, India
Indian Institute of Technology Hyderabad, India
For any Boolean functions f and g, the question whether R(f∘g) = Θ̃(R(f) ⋅ R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg̃(f∘g) = Θ̃(deg̃(f)⋅deg̃(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily.
It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg̃ compose.
A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g) = Ω(noisyR(f)⋅ R(g)). This implies that composition holds whenever noisyR(f) = Θ̃(R(f)). We show two results:
1. When R(f) = Θ(n), then noisyR(f) = Θ(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full.
2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg̃(f∘g) = Ω(M(f) ⋅ deg̃(g)) (for some non-trivial complexity measure M(⋅)) was known to the best of our knowledge. We prove that deg̃(f∘g) = Ω̃(√{bs(f)} ⋅ deg̃(g)), where bs(f) is the block sensitivity of f. This implies that deg̃ composes when deg̃(f) is asymptotically equal to √{bs(f)}.
It is already known that both R and deg̃ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.63/LIPIcs.APPROX-RANDOM.2023.63.pdf
Approximate degree
Boolean functions
Composition Theorem
Partial functions
Randomized Query Complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
64:1
64:12
10.4230/LIPIcs.APPROX/RANDOM.2023.64
article
Sampling from the Random Cluster Model on Random Regular Graphs at All Temperatures via Glauber Dynamics
Galanis, Andreas
1
Goldberg, Leslie Ann
1
Smolarova, Paulina
1
Department of Computer Science, University of Oxford, UK
We consider the performance of Glauber dynamics for the random cluster model with real parameter q > 1 and temperature β > 0. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random Δ-regular graphs for all sufficiently large q and obtained an efficient sampling algorithm for all temperatures β using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition.
Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large q (with respect to Δ). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures β, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.64/LIPIcs.APPROX-RANDOM.2023.64.pdf
approximate counting
Glauber dynamics
random cluster model
approximate sampling
random regular graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
65:1
65:18
10.4230/LIPIcs.APPROX/RANDOM.2023.65
article
Range Avoidance for Constant Depth Circuits: Hardness and Algorithms
Gajulapalli, Karthik
1
Golovnev, Alexander
1
Nagargoje, Satyajeet
1
Saraogi, Sidhant
1
Georgetown University, Washington, DC, USA
Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit 𝖢: {0,1}ⁿ → {0,1}^m, m > n, the task is to find a y ∈ {0,1}^m outside the range of 𝖢. For an integer k ≥ 2, NC⁰_k-Avoid is a special case of Avoid where each output bit of 𝖢 depends on at most k input bits. While there is a very natural randomized algorithm for Avoid, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to NC⁰₄-Avoid, thus establishing conditional hardness of the NC⁰₄-Avoid problem. On the other hand, NC⁰₂-Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC⁰₃-Avoid open.
We give the first reduction of an explicit construction question to NC⁰₃-Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC⁰₃-Avoid for the case of m = n+n^{2/3} would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits.
We also give deterministic polynomial-time algorithms for all NC⁰_k-Avoid problems for m ≥ n^{k-1}/log(n). Prior work required an NP oracle, and required larger stretch, m ≥ n^{k-1}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.65/LIPIcs.APPROX-RANDOM.2023.65.pdf
Boolean function analysis
Explicit Constructions
Low-depth Circuits
Range Avoidance
Matrix Rigidity
Circuit Lower Bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
275
66:1
66:15
10.4230/LIPIcs.APPROX/RANDOM.2023.66
article
Testing Connectedness of Images
Berman, Piotr
1
https://orcid.org/0000-0002-2363-3535
Murzabulatov, Meiram
2
https://orcid.org/0000-0002-8329-7481
Raskhodnikova, Sofya
3
https://orcid.org/0000-0002-4902-050X
Ristache, Dragos
3
https://orcid.org/0009-0007-4442-4939
Unaffiliated Researcher
Computer Science Department, School of Digital Sciences, Nazarbayev University, Astana, Kazakhstan
Boston University, MA, USA
We investigate algorithms for testing whether an image is connected. Given a proximity parameter ε ∈ (0,1) and query access to a black-and-white image represented by an n×n matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is ε-far from connected. We show that connectedness can be tested nonadaptively with O(1/ε²) queries and adaptively with O(1/ε^{3/2} √{log1/ε}) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity O(1/ε² log 1/ε) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make Ω(1/ε log 1/ε) queries.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.66/LIPIcs.APPROX-RANDOM.2023.66.pdf
Property testing
sublinear-algorithms
lower bounds
connectivity
graphs