eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
0
0
10.4230/LIPIcs.APPROX-RANDOM.2016
article
LIPIcs, Volume 60, APPROX/RANDOM'16, Complete Volume
Jansen, Klaus
Mathieu, Claire
Rolim, José D. P.
Umans, Chris
LIPIcs, Volume 60, APPROX/RANDOM'16, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016/LIPIcs.APPROX-RANDOM.2016.pdf
Theory of Computation, Models of Computation, Modes of Computation – Online Computation, Complexity Measures and Classes, Analysis of Algorithms and Problem Complexity, Numerical Algorithms and Problems – Computations on Matrices, Nonnumerical Algorithms and Problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
0:i
0:xvi
10.4230/LIPIcs.APPROX-RANDOM.2016.0
article
Front Matter, Table of Contents, Preface, Program Committees, External Reviewers, List of Authors
Jansen, Klaus
Mathieu, Claire
Rolim, José D. P.
Umans, Chris
Front Matter, Table of Contents, Preface, Program Committees, External Reviewers, List of Authors
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.0/LIPIcs.APPROX-RANDOM.2016.0.pdf
Front Matter
Table of Contents
Preface
Program Committees
External Reviewers
List of Authors
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
1:1
1:15
10.4230/LIPIcs.APPROX-RANDOM.2016.1
article
Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces
Backurs, Arturs
Sidiropoulos, Anastasios
We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}.
We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.1/LIPIcs.APPROX-RANDOM.2016.1.pdf
metric embeddings
Hausdorff metric
distortion
dimension
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
2:1
2:12
10.4230/LIPIcs.APPROX-RANDOM.2016.2
article
Computing Approximate PSD Factorizations
Basu, Amitabh
Dinitz, Michael
Li, Xin
We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error. The main ingredient is an exact factorization algorithm when the rows and columns of the factors are constrained to lie in a general polyhedron. This strictly generalizes nonnegative matrix factorizations which can be captured by letting this polyhedron to be the nonnegative orthant.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.2/LIPIcs.APPROX-RANDOM.2016.2.pdf
PSD rank
PSD factorizations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
3:1
3:17
10.4230/LIPIcs.APPROX-RANDOM.2016.3
article
Hardness of Approximation for H-Free Edge Modification Problems
Bliznets, Ivan
Cygan, Marek
Komosa, Pawel
Pilipczuk, Michal
The H-free Edge Deletion problem asks, for a given graph G and integer k, whether it is possible to delete at most k edges from G to make it H-free, that is, not containing H as an induced subgraph. The H-free Edge Completion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) for almost all graphs H, with several important exceptions occurring when the class of H-free graphs exhibits some structural properties.
In this work we complement the parameterized study of edge modification problems to H-free graphs by considering their approximability. We prove that whenever H is 3-connected and has at least two non-edges, then both H-free Edge Deletion and H-free Edge Completion are very hard to approximate: they do not admit poly(OPT)-approximation in polynomial time, unless P=NP, or even in time subexponential in OPT, unless the Exponential Time Hypothesis fails. The assumption of the existence of two non-edges appears to be important: we show that whenever H is a complete graph without one edge, then H-free Edge Deletion is tightly connected to the \minhorn problem, whose approximability is still open. Finally, in an attempt to extend our hardness results beyond 3-connected graphs, we consider the cases of H being a path or a cycle, and we achieve an almost complete dichotomy there.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.3/LIPIcs.APPROX-RANDOM.2016.3.pdf
hardness of approximation
parameterized complexity
kernelization
edge modification problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
4:1
4:16
10.4230/LIPIcs.APPROX-RANDOM.2016.4
article
On Approximating Target Set Selection
Charikar, Moses
Naamad, Yonatan
Wirth, Anthony
We study the Target Set Selection (TSS) problem introduced by Kempe, Kleinberg, and Tardos (2003). This problem models the propagation of influence in a network, in a sequence of rounds. A set of nodes is made "active" initially. In each subsequent round, a vertex is activated if at least a certain number of its neighbors are (already) active. In the minimization version, the goal is to activate a small set of vertices initially - a seed, or target, set - so that activation spreads to the entire graph. In the absence of a sublinear-factor algorithm for the general version, we provide a (sublinear) approximation algorithm for the bounded-round version, where the goal is to activate all the vertices in r rounds. Assuming a known conjecture on the hardness of Planted Dense Subgraph, we establish hardness-of-approximation results for the bounded-round version. We show that they translate to general Target Set Selection, leading to a hardness factor of n^(1/2-epsilon) for all epsilon > 0. This is the first polynomial hardness result for Target Set Selection, and the strongest conditional result known for a large class of monotone satisfiability problems. In the maximization version of TSS, the goal is to pick a target set of size k so as to maximize the number of nodes eventually active. We show an n^(1-epsilon) hardness result for the undirected maximization version of the problem, thus establishing that the undirected case is as hard as the directed case. Finally, we demonstrate an SETH lower bound for the exact computation of the optimal seed set.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.4/LIPIcs.APPROX-RANDOM.2016.4.pdf
target set selection
influence propagation
approximation algorithms
hardness of approximation
planted dense subgraph
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
5:1
5:12
10.4230/LIPIcs.APPROX-RANDOM.2016.5
article
Approximation Algorithms for Parallel Machine Scheduling with Speed-up Resources
Chen, Lin
Ye, Deshi
Zhang, Guochuan
We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machines so as to minimize the makespan. In our problem, a job has its original processing time, which could be reduced by utilizing one of the resources. As resources are different, the amount of the time reduced for each job is different depending on the resource it uses. Once a resource is being used by one job, it can not be used simultaneously by any other job until this job is finished, hence the scheduler should take into account the job-to-machine assignment together with the resource-to-job assignment.
We observe that, the classical unrelated machine scheduling problem is actually a special case of our problem when m=c, i.e., the number of resources equals the number of machines. Extending the techniques for the unrelated machine scheduling, we give a 2-approximation algorithm when both m and c are part of the input. We then consider two special cases for the problem, with m or c being a constant, and derive PTASes (Polynomial Time Approximation Schemes) respectively. We also establish the relationship between the two parameters m and c, through which we are able to transform the PTAS for the case when m is constant to the case when c is a constant. The relationship between the two parameters reveals the structure within the problem, and may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.5/LIPIcs.APPROX-RANDOM.2016.5.pdf
approximation algorithms
scheduling
linear programming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
6:1
6:19
10.4230/LIPIcs.APPROX-RANDOM.2016.6
article
The Densest k-Subhypergraph Problem
Chlamtac, Eden
Dinitz, Michael
Konrad, Christian
Kortsarz, Guy
Rabanca, George
The Densest k-Subgraph (DkS) problem, and its corresponding minimization problem Smallest p-Edge Subgraph (SpES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out).
In this paper we generalize both DkS and SpES from graphs to hypergraphs. We consider the Densest k-Subhypergraph problem (given a hypergraph (V, E), find a subset W subseteq V of k vertices so as to maximize the number of hyperedges contained in W) and define the Minimum p-Union problem (given a hypergraph, choose p of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n^{4(4-sqrt{3})/13 + epsilon}) <= O(n^{0.697831+epsilon})-approximation (for arbitrary constant epsilon > 0) for Densest k-Subhypergraph and an ~O(n^{2/5})-approximation for Minimum p-Union. We also give an O(sqrt{m})-approximation for Minimum p-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.6/LIPIcs.APPROX-RANDOM.2016.6.pdf
Hypergraphs
Approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
7:1
7:18
10.4230/LIPIcs.APPROX-RANDOM.2016.7
article
Online Row Sampling
Cohen, Michael B.
Musco, Cameron
Pachocki, Jakub
Finding a small spectral approximation for a tall n x d matrix A is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of A. Row sampling improves interpretability, saves space when A is sparse, and preserves row structure, which is especially important, for example, when A represents a graph.
However, correctly sampling rows from A can be costly when the matrix is large and cannot be stored and processed in memory. Hence, a number of recent publications focus on row sampling in the streaming setting, using little more space than what is required to store the outputted approximation [Kelner Levin 2013] [Kapralov et al. 2014].
Inspired by a growing body of work on online algorithms for machine learning and data analysis, we extend this work to a more restrictive online setting: we read rows of A one by one and immediately decide whether each row should be kept in the spectral approximation or discarded, without ever retracting these decisions. We present an extremely simple algorithm that approximates A up to multiplicative error epsilon and additive error delta using O(d log d log (epsilon ||A||_2^2/delta) / epsilon^2) online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses O(d^2) memory but only requires O(d log (epsilon ||A||_2^2/delta) / epsilon^2) samples, which we show is optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior work, and expose new theoretical properties of leverage score based matrix approximation.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.7/LIPIcs.APPROX-RANDOM.2016.7.pdf
spectral sparsification
leverage score sampling
online sparsification
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
8:1
8:23
10.4230/LIPIcs.APPROX-RANDOM.2016.8
article
Oblivious Rounding and the Integrality Gap
Feige, Uriel
Feldman, Michal
Talgam-Cohen, Inbal
The following paradigm is often used for handling NP-hard combinatorial optimization problems. One first formulates the problem as an integer program, then one relaxes it to a linear program (LP, or more generally, a convex program), then one solves the LP relaxation in polynomial time, and finally one rounds the optimal LP solution, obtaining a feasible solution to the original problem. Many of the commonly used rounding schemes (such as randomized rounding, threshold rounding and others) are "oblivious" in the sense that the rounding is performed based on the LP solution alone, disregarding the objective function. The goal of our work is to better understand in which cases oblivious rounding suffices in order to obtain approximation ratios that match the integrality gap of the underlying LP. Our study is information theoretic - the rounding is restricted to be oblivious but not restricted to run in polynomial time. In this information theoretic setting we characterize the approximation ratio achievable by oblivious rounding. It turns out to equal the integrality gap of the underlying LP on a problem that is the closure of the original combinatorial optimization problem. We apply our findings to the study of the approximation ratios obtainable by oblivious rounding for the maximum welfare problem, showing that when valuation functions are submodular oblivious rounding can match the integrality gap of the configuration LP (though we do not know what this integrality gap is), but when valuation functions are gross substitutes oblivious rounding cannot match the integrality gap (which is 1).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.8/LIPIcs.APPROX-RANDOM.2016.8.pdf
Welfare-maximization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
9:1
9:11
10.4230/LIPIcs.APPROX-RANDOM.2016.9
article
A Deterministic Fully Polynomial Time Approximation Scheme For Counting Integer Knapsack Solutions Made Easy
Halman, Nir
Given n elements with nonnegative integer weights w=(w_1,...,w_n), an integer capacity C and positive integer ranges u=(u_1,...,u_n), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error epsilon in time polynomial in n, log U and 1/epsilon, where U=max_i u_i. More precisely, our algorithm runs in O((n^3 log^2 U)/epsilon) log (n log U)/epsilon) time. This is an improvement of n^2 and 1/epsilon (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.9/LIPIcs.APPROX-RANDOM.2016.9.pdf
Approximate counting
integer knapsack
dynamic programming
bounding constraints
$K$-approximating sets and functions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
10:1
10:15
10.4230/LIPIcs.APPROX-RANDOM.2016.10
article
A Competitive Flow Time Algorithm for Heterogeneous Clusters Under Polytope Constraints
Im, Sungjin
Kulkarni, Janardhan
Moseley, Benjamin
Munagala, Kamesh
Modern data centers consist of a large number of heterogeneous resources such as CPU, memory, network bandwidth, etc. The resources are pooled into clusters for various reasons such as scalability, resource consolidation, and privacy. Clusters are often heterogeneous so that they can better serve jobs with different characteristics submitted from clients. Each job benefits differently depending on how much resource is allocated to the job, which in turn translates to how quickly the job gets completed.
In this paper, we formulate this setting, which we term Multi-Cluster Polytope Scheduling (MCPS). In MCPS, a set of n jobs arrive over time to be executed on m clusters. Each cluster i is associated with a polytope P_i, which constrains how fast one can process jobs assigned to the cluster. For MCPS, we seek to optimize the popular objective of minimizing average weighted flow time of jobs in the online setting. We give a constant competitive algorithm with small constant resource augmentation for a large class of polytopes, which capture many interesting problems that arise in practice. Further, our algorithm is non-clairvoyant. Our algorithm and analysis combine and generalize techniques developed in the recent results for the classical unrelated machines scheduling and the polytope scheduling problem [10,12,11].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.10/LIPIcs.APPROX-RANDOM.2016.10.pdf
Polytope constraints
average flow time
multi-clusters
online scheduling
and competitive analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
11:1
11:12
10.4230/LIPIcs.APPROX-RANDOM.2016.11
article
Revisiting Connected Dominating Sets: An Optimal Local Algorithm?
Khuller, Samir
Yang, Sheng
In this paper we consider the classical Connected Dominating Set (CDS) problem. Twenty years ago, Guha and Khuller developed two algorithms for this problem - a centralized greedy approach with an approximation guarantee of H(D) +2, and a local greedy approach with an approximation guarantee of 2(H(D)+1) (where H() is the harmonic function, and D is the maximum degree in the graph). A local greedy algorithm uses significantly less information about the graph, and can be useful in a variety of contexts. However, a fundamental question remained - can we get a local greedy algorithm with the same performance guarantee as the global greedy algorithm without the penalty of the multiplicative factor of "2" in the approximation factor? In this paper, we answer that question in the affirmative.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.11/LIPIcs.APPROX-RANDOM.2016.11.pdf
graph algorithms
approximation algorithms
dominating sets
local information algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
12:1
12:23
10.4230/LIPIcs.APPROX-RANDOM.2016.12
article
Online Energy Storage Management: an Algorithmic Approach
Kim, Anthony
Liaghat, Vahid
Qin, Junjie
Saberi, Amin
Motivated by the importance of energy storage networks in smart grids, we provide an algorithmic study of the online energy storage management problem in a network setting, the first to the best of our knowledge. Given online power supplies, either entirely renewable supplies or those in combination with traditional supplies, we want to route power from the supplies to demands using storage units subject to a decay factor. Our goal is to maximize the total utility of satisfied demands less the total production cost of routed power. We model renewable supplies with the zero production cost function and traditional supplies with convex production cost functions. For two natural storage unit settings, private and public, we design poly-logarithmic competitive algorithms in the network flow model using the dual fitting and online primal dual methods for convex problems. Furthermore, we show strong hardness results for more general settings of the problem. Our techniques may be of independent interest in other routing and storage management problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.12/LIPIcs.APPROX-RANDOM.2016.12.pdf
Online Algorithms
Competitive Analysis
Routing
Storage
Approximation Algorithms
Power Control
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
13:1
13:16
10.4230/LIPIcs.APPROX-RANDOM.2016.13
article
LP-Relaxations for Tree Augmentation
Kortsarz, Guy
Nutov, Zeev
In the Tree Augmentation Problem (TAP) the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T+F is 2-edge-connected. The best approximation ratio known for TAP is 1.5. In the more general Weighted TAP problem, F should be of minimum weight. Weighted TAP admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for TAP. In this paper we introduce two different LP-relaxations, and for each of them give a simple algorithm that computes a feasible solution for TAP of size at most 7/4 times the optimal LP value. This gives some hope to break the ratio 2 for the weighted case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.13/LIPIcs.APPROX-RANDOM.2016.13.pdf
Tree Augmentation; LP-relaxation; Laminar family; Approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
14:1
14:20
10.4230/LIPIcs.APPROX-RANDOM.2016.14
article
A Bi-Criteria Approximation Algorithm for k-Means
Makarychev, Konstantin
Makarychev, Yury
Sviridenko, Maxim
Ward, Justin
We consider the classical k-means clustering problem in the setting of bi-criteria approximation, in which an algorithm is allowed to output beta*k > k clusters, and must produce a clustering with cost at most alpha times the to the cost of the optimal set of k clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor.
We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee alpha(beta) depending on the number beta*k of clusters that may be opened. Our guarantee alpha(beta) is always at most 9 + epsilon and improves rapidly with beta (for example: alpha(2) < 2.59, and alpha(3) < 1.4). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.14/LIPIcs.APPROX-RANDOM.2016.14.pdf
k-means clustering
bicriteria approximation algorithms
linear programming
local search
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
15:1
15:28
10.4230/LIPIcs.APPROX-RANDOM.2016.15
article
Near-Optimal UGC-hardness of Approximating Max k-CSP_R
Manurangsi, Pasin
Nakkiran, Preetum
Trevisan, Luca
In this paper, we prove an almost-optimal hardness for Max k-CSP_R based on Khot's Unique Games Conjecture (UGC). In Max k-CSP_R, we are given a set of predicates each of which depends on exactly k variables. Each variable can take any value from 1, 2, ..., R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max k-CSP_R to within factor 2^{O(k log k)}(log R)^{k/2}/R^{k - 1} for any k, R.
To the best of our knowledge, this result improves on all the known hardness of approximation results when 3 <= k = o(log R/log log R). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/R^{k-2}) by Chan. When k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP_R by Kindler, Kolla and Trevisan, we provide an Omega(log R/R^{k - 1})-approximation algorithm for Max k-CSP_R. This algorithm implies that our inapproximability result is tight up to a factor of 2^{O(k \log k)}(\log R)^{k/2 - 1}. In comparison, when 3 <= k is a constant, the previously known gap was $O(R)$, which is significantly larger than our gap of O(polylog R).
Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's d-to-1 Conjecture and still get asymptotically the same hardness of approximation.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.15/LIPIcs.APPROX-RANDOM.2016.15.pdf
inapproximability
unique games conjecture
constraint satisfaction problem
invariance principle
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
16:1
16:54
10.4230/LIPIcs.APPROX-RANDOM.2016.16
article
Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs
Marx, Dániel
Salmasi, Ario
Sidiropoulos, Anastasios
In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014].
We show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.16/LIPIcs.APPROX-RANDOM.2016.16.pdf
asymmetric TSP
approximation algorithms
nearly-embeddable graphs
Held-Karp LP
exponential time hypothesis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
17:1
17:12
10.4230/LIPIcs.APPROX-RANDOM.2016.17
article
Planar Matching in Streams Revisited
McGregor, Andrew
Vorotnikova, Sofya
We present data stream algorithms for estimating the size or weight of the maximum matching in low arboricity graphs. A large body of work has focused on improving the constant approximation factor for general graphs when the data stream algorithm is permitted O(n polylog n) space where n is the number of nodes. This space is necessary if the algorithm must return the matching. Recently, Esfandiari et al. (SODA 2015) showed that it was possible to estimate the maximum cardinality of a matching in a planar graph up to a factor of 24+epsilon using O(epsilon^{-2} n^{2/3} polylog n) space. We first present an algorithm (with a simple analysis) that improves this to a factor 5+epsilon using the same space. We also improve upon the previous results for other graphs with bounded arboricity. We then present a factor 12.5 approximation for matching in planar graphs that can be implemented using O(log n) space in the adjacency list data stream model where the stream is a concatenation of the adjacency lists of the graph. The main idea behind our results is finding "local" fractional matchings, i.e., fractional matchings where the value of any edge e is solely determined by the edges sharing an endpoint with e. Our work also improves upon the results for the dynamic data stream model where the stream consists of a sequence of edges being inserted and deleted from the graph. We also extend our results to weighted graphs, improving over the bounds given by Bury and Schwiegelshohn (ESA 2015), via a reduction to the unweighted problem that increases the approximation by at most a factor of two.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.17/LIPIcs.APPROX-RANDOM.2016.17.pdf
data streams
planar graphs
arboricity
matchings
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
18:1
18:16
10.4230/LIPIcs.APPROX-RANDOM.2016.18
article
A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching
Raghvendra, Sharath
We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S|=|R|=n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a "free" server. The matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching.
We present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n-1 + o(1); it is known that no deterministic algorithm can do better than 2n-1. In the random arrival model, our algorithm obtains a competitive ratio of 2H_n - 1 + o(1); where H_n is the n-th Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model.
We use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.18/LIPIcs.APPROX-RANDOM.2016.18.pdf
Online Algorithms
Metric Bipartite Matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
19:1
19:18
10.4230/LIPIcs.APPROX-RANDOM.2016.19
article
Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One
Stephens-Davidowitz, Noah
We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any gamma <= 1 + O(log n/n), we obtain an efficient dimension-preserving reduction from gamma^{O(n/log n)}-SVP to gamma-GapSVP and an efficient dimension-preserving reduction from gamma^{O(n)}-CVP to gamma-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when gamma = 1. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction - we reduce gamma^{O(n/log n)}-SVP to gamma-unique SVP, a potentially easier problem than gamma-GapSVP.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.19/LIPIcs.APPROX-RANDOM.2016.19.pdf
Lattices
SVP
CVP
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
20:1
20:15
10.4230/LIPIcs.APPROX-RANDOM.2016.20
article
Proving Weak Approximability Without Algorithms
Syed, Ridwan
Tulsiani, Madhur
A boolean predicate is said to be strongly approximation resistant if, given a near-satisfiable instance of its maximum constraint satisfaction problem, it is hard to find an assignment such that the fraction of constraints satisfied deviates significantly from the expected fraction of constraints satisfied by a random assignment. A predicate which is not strongly approximation resistant is known as weakly approximable.
We give a new method for proving the weak approximability of predicates, using a simple SDP relaxation, without designing and analyzing new rounding algorithms for each predicate. Instead, we use the recent characterization of strong approximation resistance by Khot et al. [STOC 2014], and show how to prove that for a given predicate, certain necessary conditions for strong resistance derived from their characterization, are violated. By their result, this implies the existence of a good rounding algorithm, proving weak approximability.
We show how this method can be used to obtain simple proofs of (weak approximability analogues of) various known results on approximability, as well as new results on weak approximability of symmetric predicates.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.20/LIPIcs.APPROX-RANDOM.2016.20.pdf
approximability
constraint satisfaction problems
approximation resistance
linear programming
semidefinite programming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
21:1
21:19
10.4230/LIPIcs.APPROX-RANDOM.2016.21
article
Every Property of Outerplanar Graphs is Testable
Babu, Jasine
Khoury, Areej
Newman, Ilan
A D-disc around a vertex v of a graph G=(V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most epsilon n edges, by a set of D-discs around the vertices, for D=D(epsilon) that is independent of the size of the graph. Such a result was already known for planar graphs (and any hyperfinite graph class), in the limited case of bounded degree graphs (that is, their maximum degree is bounded by some fixed constant, independent of |V|). We prove this result with no assumption on the degree of the graph.
A pure combinatorial consequence of this result is that two outerplanar graphs that share the same local views are close to be isomorphic.
We also obtain the following property testing results in the sparse graph model:
* graph isomorphism is testable for outerplanar graphs by poly(log n) queries.
* every graph property is testable for outerplanar graphs by poly(log n) queries.
We note that we can replace outerplanar graphs by a slightly more general family of k-edge-outerplanar graphs. The only previous general testing results, as above, where known for forests (Kusumoto and Yoshida), and for some power-law graphs that are extremely close to be bounded degree hyperfinite (by Ito).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.21/LIPIcs.APPROX-RANDOM.2016.21.pdf
Property testing
Isomorphism
Outerplanar graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
22:1
22:18
10.4230/LIPIcs.APPROX-RANDOM.2016.22
article
The Condensation Phase Transition in the Regular k-SAT Model
Bapst, Victor
Coja-Oghlan, Amin
Much of the recent work on phase transitions in discrete structures has been inspired by ingenious but non-rigorous approaches from physics. The physics predictions typically come in the form of distributional fixed point problems that mimic Belief Propagation, a message passing algorithm. In this paper we show how the Belief Propagation calculation can be turned into a rigorous proof of such a prediction, namely the existence and location of a condensation phase transition in the regular k-SAT model.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.22/LIPIcs.APPROX-RANDOM.2016.22.pdf
random k-SAT
phase transitions
Belief Propagation
condensation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
23:1
23:29
10.4230/LIPIcs.APPROX-RANDOM.2016.23
article
On Higher-Order Fourier Analysis over Non-Prime Fields
Bhattacharyya, Arnab
Bhowmick, Abhishek
Gupta, Chetan
The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character chi, either chi(P) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of chi(P) and its "structure"?
Previously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field F_p. Our technical results are:
1. If P: K^n -> K is a polynomial of degree <= d, and E[chi(P(x))] > |K|^{-s} for some s > 0 and non-trivial additive character chi, then P is a function of O_{d, s}(1) many non-classical polynomials of weight degree < d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work.
2. Suppose K and F are of bounded order, and let H be an affine subspace of K^n. Then, if P: K^n -> K is a polynomial of degree d that is sufficiently regular, then (P(x): x in H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P. Such a theorem was previously known for H an affine subspace over a prime field.
The tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas.
(i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code.
(ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P: K^n -> K can be decomposed as a particular composition of lesser degree polynomials.
(iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f: K^n -> K are testable with one-sided error.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.23/LIPIcs.APPROX-RANDOM.2016.23.pdf
finite fields
higher order fourier analysis
coding theory
property testing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
24:1
24:9
10.4230/LIPIcs.APPROX-RANDOM.2016.24
article
Bounded Independence vs. Moduli
Boppana, Ravi
Håstad, Johan
Lee, Chin Ho
Viola, Emanuele
Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.24/LIPIcs.APPROX-RANDOM.2016.24.pdf
Bounded independence
Modulus
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
25:1
25:19
10.4230/LIPIcs.APPROX-RANDOM.2016.25
article
Approximating Subadditive Hadamard Functions on Implicit Matrices
Braverman, Vladimir
Roytman, Alan
Vorsanger, Gregory
An important challenge in the streaming model is to maintain small-space approximations of entrywise functions performed on a matrix that is generated by the outer product of two vectors given as a stream. In other works, streams typically define matrices in a standard way via a sequence of updates, as in the
work of Woodruff [22] and others. We describe the matrix formed by the outer product, and other matrices that do not fall into this category, as implicit matrices. As such, we consider the general problem of computing over such implicit matrices with Hadamard functions, which are functions applied entrywise on a matrix. In this paper, we apply this generalization to provide new techniques for identifying independence between two data streams. The previous state of the art algorithm of Braverman and Ostrovsky [9] gave a (1 +- epsilon)-approximation for the L_1 distance between the joint and product of the marginal distributions, using space O(log^{1024}(nm) epsilon^{-1024}), where m is the length of the stream and n denotes the size of the universe from which stream elements are drawn. Our general techniques include the L_1 distance as a special case, and we give an improved space bound of O(log^{12}(n) log^{2}({nm}/epsilon) epsilon^{-7}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.25/LIPIcs.APPROX-RANDOM.2016.25.pdf
Streaming Algorithms
Measuring Independence
Hadamard Functions
Implicit Matrices
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
26:1
26:11
10.4230/LIPIcs.APPROX-RANDOM.2016.26
article
Local Convergence and Stability of Tight Bridge-Addable Graph Classes
Chapuy, Guillaume
Perarnau, Guillem
A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1/2}. The constant e^{-1/2} is best possible since it is reached for the class of forests.
In this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1/2}, then G_n is asymptotically close to a uniform forest in some "local" sense. For example, if the probability converges to e^{-1/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the "stable" extremum is not a graph but a graph class.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.26/LIPIcs.APPROX-RANDOM.2016.26.pdf
bridge-addable classes
random graphs
stability
local convergence
random forests
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
27:1
27:15
10.4230/LIPIcs.APPROX-RANDOM.2016.27
article
Belief Propagation on Replica Symmetric Random Factor Graph Models
Coja-Oghlan, Amin
Perkins, Will
According to physics predictions, the free energy of random factor graph models that satisfy a certain "static replica symmetry" condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al. PNAS, 2007]. Here we prove this conjecture for a wide class of random factor graph models. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.27/LIPIcs.APPROX-RANDOM.2016.27.pdf
Gibbs distributions
Belief Propagation
Bethe Free Energy
Random k-SAT
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
28:1
28:12
10.4230/LIPIcs.APPROX-RANDOM.2016.28
article
Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
Dadush, Daniel
Garg, Shashwat
Lovett, Shachar
Nikolov, Aleksandar
An important theorem of Banaszczyk (Random Structures & Algorithms 1998) states that for any sequence of vectors of l_2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R^n, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination.
We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric.
As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/sqrt{log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)-subgaussian signing distributions when the vectors have length O(1/sqrt{log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.28/LIPIcs.APPROX-RANDOM.2016.28.pdf
Discrepancy
Vector Balancing
Convex Geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
29:1
29:10
10.4230/LIPIcs.APPROX-RANDOM.2016.29
article
On the Beck-Fiala Conjecture for Random Set Systems
Ezra, Esther
Lovett, Shachar
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Sigma), where each element x in X lies in t randomly selected sets of Sigma, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |\Sigma| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.29/LIPIcs.APPROX-RANDOM.2016.29.pdf
Discrepancy theory
Beck-Fiala conjecture
Random set systems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
30:1
30:14
10.4230/LIPIcs.APPROX-RANDOM.2016.30
article
The Niceness of Unique Sink Orientations
Gärtner, Bernd
Thomas, Antonis
Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of reachmaps and niceness of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least n^{Omega(2^n)} many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.30/LIPIcs.APPROX-RANDOM.2016.30.pdf
random edge
unique sink orientation
random walk
reachmap
niceness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
31:1
31:26
10.4230/LIPIcs.APPROX-RANDOM.2016.31
article
Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems
Guo, Heng
Lu, Pinyan
For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness in infinite regular trees, the decay of correlations (also known as spatial mixing), and the approximability of the partition function. The uniqueness condition implies spatial mixing, and an FPTAS for the partition function exists based on spatial mixing. On the other hand, non-uniqueness implies some long range correlation, based on which NP-hardness reductions are built. These connections for ferromagnetic 2-spin systems are much less clear, despite their similarities to anti-ferromagnetic systems. The celebrated Jerrum-Sinclair Markov chain [JS93] works even if spatial mixing or uniqueness fails.
We provide some partial answers. We use (β,γ) to denote the (+,+) and (−,−) edge interactions and λ the external field, where βγ>1. If all fields satisfy λ<λ_c (assuming β≤γ), where λ_c=(γ/β)^{(Δ_c+1)/2} and Δ_c=(\sqrt{βγ}+1)/(\sqrt{βγ}−1), then a weaker version of spatial mixing holds in all trees. Moreover, if β≤1, then λ<λ_c is sufficient to guarantee strong spatial mixing and FPTAS. This improves the previous best algorithm, which is an FPRAS based on Markov chains and works for λ<γ/β [LLZ14a]. The bound λ_c is almost optimal. When β≤1, uniqueness holds in all infinite regular trees, if and only if λ≤λ^int_c, where λ^int_c=(γ/β)(⌈Δc⌉+1)/2. If we allow fields λ>λ^int′_c, where λ^int′_c=(γ/β)(⌊Δc⌋+2)/2, then approximating the partition function is #BIS-hard.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.31/LIPIcs.APPROX-RANDOM.2016.31.pdf
Approximate counting; Ising model; Spin systems; Correlation decay
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
32:1
32:14
10.4230/LIPIcs.APPROX-RANDOM.2016.32
article
On Polynomial Approximations to AC^0
Harsha, Prahladh
Srinivasan, Srikanth
We make progress on some questions related to polynomial approximations of AC^0. It is known, from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC^0 circuit of size s and depth d has an epsilon-error probabilistic polynomial over the reals of degree (log (s/epsilon))^{O(d)}. We improve this upper bound to (log s)^{O(d)}* log(1/epsilon), which is much better for small values of epsilon.
We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that (log s)^{O(d)}* log(1/epsilon)-wise independence fools AC^0, improving on Tal's strengthening of Braverman's theorem (J. ACM 2010) that (log (s/epsilon))^{O(d)}-wise independence fools AC^0. Up to the constant implicit in the O(d), our result is tight. As far as we know, this is the first PRG construction for AC^0 that achieves optimal dependence on the error epsilon.
We also prove lower bounds on the best polynomial approximations to AC^0. We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least ~Omega(sqrt{log n}). This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.32/LIPIcs.APPROX-RANDOM.2016.32.pdf
Constant-depth Boolean circuits
Polynomials over reals
pseudo-random generators
k-wise independence
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
33:1
33:18
10.4230/LIPIcs.APPROX-RANDOM.2016.33
article
On the Structure of Quintic Polynomials
Hatami, Pooya
We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let F=F_q be a prime field. Suppose f:F^n to F is a degree five polynomial with bias(f)=delta. We prove the following two structural properties for such f.
1. We have f= sum_{i=1}^{c} G_i H_i + Q, where G_i and H_is are nonconstant polynomials satisfying deg(G_i)+deg(H_i)<= 5 and Q is a degree <5 polynomial. Moreover, c does not depend on n.
2. There exists an Omega_{delta,q}(n) dimensional affine subspace V subseteq F^n such that f|_V is a constant.
Cohen and Tal [Random 2015] proved that biased polynomials of degree at most four are constant on a subspace of dimension Omega(n). Item 2.]extends this to degree five polynomials. A corollary to Item 2. is that any degree five affine disperser for dimension k is also an affine extractor for dimension O(k). We note that Item 2. cannot hold for degrees six or higher.
We obtain our results for degree five polynomials as a special case of structure theorems that we prove for biased degree d polynomials when d<|\F|+4. While the d<|F|+4 assumption seems very restrictive, we note that prior to our work such structure theorems were only known for d<|\F| by Green and Tao [Contrib. Discrete Math. 2009] and Bhowmick and Lovett [arXiv:1506.02047]. Using algorithmic regularity lemmas for polynomials developed by Bhattacharyya, et al. [SODA 2015], we show that whenever such a strong structure exists, it can be found algorithmically in time polynomial in n.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.33/LIPIcs.APPROX-RANDOM.2016.33.pdf
Higher-order Fourier analysis
Structure Theorem
Polynomials
Regularity lemmas
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
34:1
34:11
10.4230/LIPIcs.APPROX-RANDOM.2016.34
article
Lower Bounds on Same-Set Inner Product in Correlated Spaces
Hazla, Jan
Holenstein, Thomas
Mossel, Elchanan
Let P be a probability distribution over a finite alphabet Omega^L with all L marginals equal. Let X^(1), ..., X^(L), where X^(j) = (X_1^(j), ..., X_n^(j)) be random vectors such that for every coordinate i in [n] the tuples (X_i^(1), ..., X_i^(L)) are i.i.d. according to P.
The question we address is: does there exist a function c_P independent of n such that for every f: Omega^n -> [0, 1] with E[f(X^(1))] = m > 0 we have E[f(X^(1)) * ... * f(X^(n))] > c_P(m) > 0?
We settle the question for L=2 and when L>2 and P has bounded correlation smaller than 1.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.34/LIPIcs.APPROX-RANDOM.2016.34.pdf
same set hitting
product spaces
correlation
lower bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
35:1
35:13
10.4230/LIPIcs.APPROX-RANDOM.2016.35
article
Estimating Parameters Associated with Monotone Properties
Hoppen, Carlos
Kohayakawa, Yoshiharu
Lang, Richard
Lefmann, Hanno
Stagni, Henrique
There has been substantial interest in estimating the value of a graph parameter, i.e., of a real function defined on the set of finite graphs, by sampling a randomly chosen substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity q_z=q_z(epsilon) of an estimable parameter z is the size of the random sample required to ensure that the value of z(G) may be estimated within error epsilon with probability at least 2/3. In this paper, we study the sample complexity of estimating two graph parameters associated with a monotone graph property, improving previously known results. To obtain our results, we prove that the vertex set of any graph that satisfies a monotone property P may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of P}. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.35/LIPIcs.APPROX-RANDOM.2016.35.pdf
parameter estimation
parameter testing
edit distance to monotone graph properties
entropy of subgraph classes
speed of subgraph classes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
36:1
36:13
10.4230/LIPIcs.APPROX-RANDOM.2016.36
article
Stable Matching with Evolving Preferences
Kanade, Varun
Leonardos, Nikos
Magniez, Frédéric
We consider the problem of stable matching with dynamic preference lists. At each time-step, the preference list of some player may change by swapping random adjacent members. The goal of a central agency (algorithm) is to maintain an approximately stable matching, in terms of number of blocking pairs, at all time-steps. The changes in the preference lists are not reported to the algorithm, but must instead be probed explicitly. We design an algorithm that in expectation and with high probability maintains a matching that has at most O((log n)^2 blocking pairs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.36/LIPIcs.APPROX-RANDOM.2016.36.pdf
Stable Matching
Dynamic Data
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
37:1
37:7
10.4230/LIPIcs.APPROX-RANDOM.2016.37
article
An ~O(n) Queries Adaptive Tester for Unateness
Khot, Subhash
Shinkar, Igor
We present an adaptive tester for the unateness property of Boolean functions. Given a function f:{0,1}^n -> {0,1} the tester makes O(n log(n)/epsilon) adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 if a function is epsilon-far from being unate.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.37/LIPIcs.APPROX-RANDOM.2016.37.pdf
property testing
boolean functions
unateness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
38:1
38:15
10.4230/LIPIcs.APPROX-RANDOM.2016.38
article
A Local Algorithm for Constructing Spanners in Minor-Free Graphs
Levi, Reut
Ron, Dana
Rubinfeld, Ronitt
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider this problem in the setting of local algorithms: one wants to quickly determine whether a given edge e is in a specific spanning tree, without computing the whole spanning tree, but rather by inspecting the local neighborhood of e. The challenge is to maintain consistency. That is, to answer queries about different edges according to the same spanning tree. Since it is known that this problem cannot be solved without essentially viewing all the graph, we consider the relaxed version of finding a spanning subgraph with (1+c)n edges instead of n-1 edges (where n is the number of vertices and c is a given approximation/sparsity parameter).
It is known that this relaxed problem requires inspecting order of n^{1/2} edges in general graphs (for any constant c), which motivates the study of natural restricted families of graphs. One such family is the family of graphs with an excluded minor (which in particular includes planar graphs). For this family there is an algorithm that achieves constant success probability, and inspects (d/c)^{poly(h)log(1/c)} edges (for each edge it is queried on), where d is the maximum degree in the graph and h is the size of the excluded minor. The distances between pairs of vertices in the spanning subgraph G' are at most a factor of poly(d, 1/c, h) larger than in G.
In this work, we show that for an input graph that is H-minor free for any H of size h, this task can be performed by inspecting only poly(d, 1/c, h) edges in G.
The distances between pairs of vertices in the spanning subgraph G' are at most a factor of h log(d)/c (up to poly-logarithmic factors) larger than in G. Furthermore, the error probability of the new algorithm is significantly improved to order of 1/n. This algorithm can also be easily adapted to yield an efficient algorithm for the distributed (message passing) setting.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.38/LIPIcs.APPROX-RANDOM.2016.38.pdf
spanners
sparse subgraphs
local algorithms
excluded-minor
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
39:1
39:11
10.4230/LIPIcs.APPROX-RANDOM.2016.39
article
Tight Bounds for Sketching the Operator Norm, Schatten Norms, and Subspace Embeddings
Li, Yi
Woodruff, David P.
We consider the following oblivious sketching problem: given epsilon in (0,1/3) and n >= d/epsilon^2, design a distribution D over R^{k * nd} and a function f: R^k * R^{nd} -> R}, so that for any n * d matrix A, Pr_{S sim D} [(1-epsilon) |A|_{op} <= f(S(A),S) <= (1+epsilon)|A|_{op}] >= 2/3, where |A|_{op} = sup_{x:|x|_2 = 1} |Ax|_2 is the operator norm of A and S(A) denotes S * A, interpreting A as a vector in R^{nd}. We show a tight lower bound of k = Omega(d^2/epsilon^2) for this problem. Previously, Nelson and Nguyen (ICALP, 2014) considered the problem of finding a distribution D over R^{k * n} such that for any n * d matrix A, Pr_{S sim D}[forall x, (1-epsilon)|Ax|_2 <= |SAx|_2 <= (1+epsilon)|Ax|_2] >= 2/3, which is called an oblivious subspace embedding (OSE). Our result considerably strengthens theirs, as it (1) applies only to estimating the operator norm, which can be estimated given any OSE, and (2) applies to distributions over general linear operators S which treat A as a vector and compute S(A), rather than the restricted class of linear operators corresponding to matrix multiplication. Our technique also implies the first tight bounds for approximating the Schatten p-norm for even integers p via general linear sketches, improving the previous lower bound from k = Omega(n^{2-6/p}) [Regev, 2014] to k = Omega(n^{2-4/p}). Importantly, for sketching the operator norm up to a factor of alpha, where alpha - 1 = \Omega(1), we obtain a tight k = Omega(n^2/alpha^4) bound, matching the upper bound of Andoni and Nguyen (SODA, 2013), and improving the previous k = Omega(n^2/\alpha^6) lower bound. Finally, we also obtain the first lower bounds for approximating Ky Fan norms.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.39/LIPIcs.APPROX-RANDOM.2016.39.pdf
data streams
sketching
matrix norms
subspace embeddings
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
40:1
40:26
10.4230/LIPIcs.APPROX-RANDOM.2016.40
article
Bounds on the Norms of Uniform Low Degree Graph Matrices
Medarametla, Dhruv
Potechin, Aaron
The Sum Of Squares hierarchy is one of the most powerful tools we know of for solving combinatorial optimization problems. However, its performance is only partially understood. Improving our understanding of the sum of squares hierarchy is a major open problem in computational complexity theory.
A key component of analyzing the sum of squares hierarchy is understanding the behavior of certain matrices whose entries are random but not independent. For these matrices, there is a random input graph and each entry of the matrix is a low degree function of the edges of this input graph. Moreoever, these matrices are generally invariant (as a function of the input graph) when we permute the vertices of the input graph. In this paper, we bound the norms of all such matrices up to a polylogarithmic factor.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.40/LIPIcs.APPROX-RANDOM.2016.40.pdf
sum of squares hierarchy
matrix norm bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
41:1
41:30
10.4230/LIPIcs.APPROX-RANDOM.2016.41
article
Lower Bounds for CSP Refutation by SDP Hierarchies
Mori, Ryuhei
Witmer, David
For a k-ary predicate P, a random instance of CSP(P) with n variables and m constraints is unsatisfiable with high probability when m >= O(n). The natural algorithmic task in this regime is refutation: finding a proof that a given random instance is unsatisfiable. Recent work of Allen et al. suggests that the difficulty of refuting CSP(P) using an SDP is determined by a parameter cmplx(P), the smallest t for which there does not exist a t-wise uniform distribution over satisfying assignments to P. In particular they show that random instances of CSP(P) with m >> n^{cmplx(P)/2} can be refuted efficiently using an SDP.
In this work, we give evidence that n^{cmplx(P)/2} constraints are also necessary for refutation using SDPs. Specifically, we show that if P supports a (t-1)-wise uniform distribution over satisfying assignments, then the Sherali-Adams_+ and Lovasz-Schrijver_+ SDP hierarchies cannot refute a random instance of CSP(P) in polynomial time for any m <= n^{t/2-epsilon}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.41/LIPIcs.APPROX-RANDOM.2016.41.pdf
constraint satisfaction problems
LP and SDP relaxations
average-case complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
42:1
42:29
10.4230/LIPIcs.APPROX-RANDOM.2016.42
article
A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian
Moshkovitz, Dana
Ramnarayan, Govind
Yuen, Henry
In this work we show a barrier towards proving a randomness-efficient parallel repetition, a promising avenue for achieving many tight inapproximability results. Feige and Kilian (STOC'95) proved an impossibility result for randomness-efficient parallel repetition for two prover games with small degree, i.e., when each prover has only few possibilities for the question of the other prover. In recent years, there have been indications that randomness-efficient parallel repetition (also called derandomized parallel repetition) might be possible for games with large degree, circumventing the impossibility result of Feige and Kilian. In particular, Dinur and Meir (CCC'11) construct games with large degree whose repetition can be derandomized using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However, obtaining derandomized parallel repetition theorems that would yield optimal inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by proving a limitation on derandomized parallel repetition. We formalize two properties which we call "fortification-friendliness" and "yields robust embeddings". We show that any proof of derandomized parallel repetition achieving almost-linear blow-up cannot both (a) be fortification-friendly and (b) yield robust embeddings. Unlike Feige and Kilian, we do not require the small degree assumption.
Given that virtually all existing proofs of parallel repetition, including the derandomized parallel repetition result of Dinur and Meir, share these two properties, our no-go theorem highlights a major barrier to achieving almost-linear derandomized parallel repetition.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.42/LIPIcs.APPROX-RANDOM.2016.42.pdf
Derandomization
parallel repetition
Feige-Killian
fortification
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
43:1
43:12
10.4230/LIPIcs.APPROX-RANDOM.2016.43
article
Fast Synchronization of Random Automata
Nicaud, Cyril
A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Cerny conjectured in 1964 that if a $n$-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n-1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization: he proved that, for the uniform distribution on deterministic automata with n states, an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log^{3}n) with high probability. As a consequence, this proves that most automata satisfy the Cerny conjecture.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.43/LIPIcs.APPROX-RANDOM.2016.43.pdf
random automata
synchronization
the Černý conjecture
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
44:1
44:15
10.4230/LIPIcs.APPROX-RANDOM.2016.44
article
A Direct-Sum Theorem for Read-Once Branching Programs
Rao, Anup
Sinha, Makrand
We study a direct-sum question for read-once branching programs. If M(f) denotes the minimum average memory required to compute a function f(x_1,x_2, ..., x_n) how much memory is required to compute f on k independent inputs that arrive in parallel? We show that when the inputs are sampled independently from some domain X and M(f) = Omega(n), then computing the value of f on k streams requires average memory at least Omega(k * M(f)/n).
Our results are obtained by defining new ways to measure the information complexity of read-once branching programs. We define two such measures: the transitional and cumulative information content. We prove that any read-once branching program with transitional information content I can be simulated using average memory O(n(I+1)). On the other hand, if every read-once branching program with cumulative information content I can be simulated with average memory O(I+1), then computing f on k inputs requires average memory at least Omega(k * (M(f)-1)).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.44/LIPIcs.APPROX-RANDOM.2016.44.pdf
Direct-sum
Information complexity
Streaming Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
45:1
45:38
10.4230/LIPIcs.APPROX-RANDOM.2016.45
article
Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels
Shaltiel, Ronen
Silbak, Jad
A stochastic code is a pair of encoding and decoding procedures where Encoding procedure receives a k bit message m, and a d bit uniform string S. The code is (p,L)-list-decodable against a class C of "channel functions" from n bits to n bits, if for every message m and every channel C in C that induces at most $pn$ errors, applying decoding on the "received word" C(Enc(m,S)) produces a list of at most L messages that contain m with high probability (over the choice of uniform S). Note that both the channel C and the decoding algorithm Dec do not receive the random variable S. The rate of a code is the ratio between the message length and the encoding length, and a code is explicit if Enc, Dec run in time poly(n).
Guruswami and Smith (J. ACM, to appear), showed that for every constants 0 < p < 1/2 and c>1 there are Monte-Carlo explicit constructions of stochastic codes with rate R >= 1-H(p)-epsilon that are (p,L=poly(1/epsilon))-list decodable for size n^c channels. Monte-Carlo, means that the encoding and decoding need to share a public uniformly chosen poly(n^c) bit string Y, and the constructed stochastic code is (p,L)-list decodable with high probability over the choice of Y.
Guruswami and Smith pose an open problem to give fully explicit (that is not Monte-Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97).
Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against O(log n)-space online channels. (These are channels that have space O(log n) and are allowed to read the input codeword in one pass). We resolve this open problem.
Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching 1-H(p) for every p <= p_0 for some p_0>0) for channels that are circuits of size 2^{n^{Omega(1/d)}} and depth d. Here, the running time of encoding and decoding is a fixed polynomial (that does not depend on d).
Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the reductions in the proof more efficient, so that we can handle weak classes of channels.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.45/LIPIcs.APPROX-RANDOM.2016.45.pdf
Error Correcting Codes
List Decoding
Pseudorandomness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
46:1
46:29
10.4230/LIPIcs.APPROX-RANDOM.2016.46
article
Counting Hypergraph Matchings up to Uniqueness Threshold
Song, Renjie
Yin, Yitong
Zhao, Jinman
We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter lambda, each matching M is assigned a weight lambda^{|M|}. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings).
For this model, the critical activity lambda_c= (d^d)/(k (d-1)^{d+1}) is the threshold for the uniqueness of Gibbs measures on the infinite (d+1)-uniform (k+1)-regular hypertree. Consider hypergraphs of maximum degree at most k+1 and maximum size of hyperedges at most d+1. We show that when lambda < lambda_c, there is an FPTAS for computing the partition function; and when lambda = lambda_c, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When lambda > 2lambda_c, there is no PRAS for the partition function or the log-partition function unless NP=RP.
Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.46/LIPIcs.APPROX-RANDOM.2016.46.pdf
approximate counting; phase transition; spatial mixing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-09-06
60
47:1
47:22
10.4230/LIPIcs.APPROX-RANDOM.2016.47
article
Sampling in Potts Model on Sparse Random Graphs
Yin, Yitong
Zhang, Chihao
We study the problem of sampling almost uniform proper q-colorings in sparse Erdos-Renyi random graphs G(n,d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [Dyer et al., RANDOM STRUCT ALGOR, 2006]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q>3d+4, improving the current best bound q>5.5d [Efthymiou, SODA, 2014].
Our sampling algorithm works for more generalized models and broader family of sparse graphs. It is an efficient sampler (in the same sense of FPAUS) for anti-ferromagnetic Potts model with activity 0<=b<1 on G(n,d/n) provided q>3(1-b)d+4. We further identify a family of sparse graphs to which all these results can be extended. This family of graphs is characterized by the notion of contraction function, which is a new measure of the average degree in graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol060-approx-random2016/LIPIcs.APPROX-RANDOM.2016.47/LIPIcs.APPROX-RANDOM.2016.47.pdf
Potts model
Sampling
Random Graph
Approximation Algorithm