On the Unique Games Conjecture
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The Strongish Planted Clique Hypothesis and Its Consequences
The Strongish Planted Clique Hypothesis and Its Consequences Pasin Manurangsi Google Research, Mountain View, CA, USA [email protected] Aviad Rubinstein Stanford University, CA, USA [email protected] Tselil Schramm Stanford University, CA, USA [email protected] Abstract We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time nΩ(log n) (so that the state-of-the-art running time of nO(log n) is optimal up to a constant in the exponent). We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best ko(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k. Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis. 2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness; Theory of computation → Fixed parameter tractability Keywords and phrases Planted Clique, Densest k-Subgraph, Hardness of Approximation Digital Object Identifier 10.4230/LIPIcs.ITCS.2021.10 Related Version A full version of the paper is available at https://arxiv.org/abs/2011.05555. -
An Efficient Implementation of Sugiyama's Algorithm for Layered
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 9, no. 3, pp. 305–325 (2005) An Efficient Implementation of Sugiyama’s Algorithm for Layered Graph Drawing Markus Eiglsperger Z¨uhlke Engineering AG 8952 Schlieren, Switzerland Martin Siebenhaller WSI f¨urInformatik, Universit¨at T¨ubingen Sand 13, 72076 T¨ubingen, Germany [email protected] Michael Kaufmann WSI f¨urInformatik, Universit¨at T¨ubingen Sand 13, 72076 T¨ubingen, Germany [email protected] Abstract Sugiyama’s algorithm for layered graph drawing is very popular and commonly used in practical software. The extensive use of dummy vertices to break long edges between non-adjacent layers often leads to unsatis- fying performance. The worst-case running-time of Sugiyama’s approach is O(|V ||E| log |E|) requiring O(|V ||E|) memory, which makes it unusable for the visualization of large graphs. By a conceptually simple new tech- nique we are able to keep the number of dummy vertices and edges linear in the size of the graph without increasing the number of crossings. We reduce the worst-case time complexity of Sugiyama’s approach by an order of magnitude to O((|V | + |E|) log |E|) requiring O(|V | + |E|) space. Article Type Communicated by Submitted Revised Regular paper E. R. Gansner and J. Pach November 2004 July 2005 This work has partially been supported by the DFG-grant Ka512/8-2. It has been performed when the first author was with the Universit¨atT¨ubingen. Eiglsperger et al., Implementing Sugiyama’s Alg., JGAA, 9(3) 305–325 (2005)306 1 Introduction Most approaches for drawing directed graphs used in practice follow the frame- work developed by Sugiyama et al. -
Sorting Heuristics for the Feedback Arc Set Problem
Sorting Heuristics for the Feedback Arc Set Problem Franz J. Brandenburg and Kathrin Hanauer Department of Informatics and Mathematics, University of Passau fbrandenb;[email protected] Technical Report, Number MIP-1104 Department of Informatics and Mathematics University of Passau, Germany February 2011 Sorting Heuristics for the Feedback Arc Set Problem? Franz J. Brandenburg and Kathrin Hanauer University of Passau, Germany fbrandenb,[email protected] Abstract. The feedback arc set problem plays a prominent role in the four-phase framework to draw directed graphs, also known as the Sugiyama algorithm. It is equivalent to the linear arrangement problem where the vertices of a graph are ordered from left to right and the backward arcs form the feedback arc set. In this paper we extend classical sorting algorithms to heuristics for the feedback arc set problem. Established algorithms are considered from this point of view, where the directed arcs between vertices serve as binary comparators. We analyze these algorithms and afterwards design hybrid algorithms by their composition in order to gain further improvements. These algorithms primarily differ in the use of insertion sort and sifting and they are very similar in their performance, which varies by about 0:1%. The differences mainly lie in their run time and their convergence to a local minimum. Our studies extend related work by new algorithms and our experiments are conducted on much larger graphs. Overall we can conclude that sifting performs better than insertion sort. 1 Introduction The feedback arc set problem (FAS) asks for a minimum sized subset of arcs of a directed graph whose removal or reversal makes the graph acyclic. -
Approximating Cumulative Pebbling Cost Is Unique Games Hard
Approximating Cumulative Pebbling Cost Is Unique Games Hard Jeremiah Blocki Department of Computer Science, Purdue University, West Lafayette, IN, USA https://www.cs.purdue.edu/homes/jblocki [email protected] Seunghoon Lee Department of Computer Science, Purdue University, West Lafayette, IN, USA https://www.cs.purdue.edu/homes/lee2856 [email protected] Samson Zhou School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA https://samsonzhou.github.io/ [email protected] Abstract P The cumulative pebbling complexity of a directed acyclic graph G is defined as cc(G) = minP i |Pi|, where the minimum is taken over all legal (parallel) black pebblings of G and |Pi| denotes the number of pebbles on the graph during round i. Intuitively, cc(G) captures the amortized Space-Time complexity of pebbling m copies of G in parallel. The cumulative pebbling complexity of a graph G is of particular interest in the field of cryptography as cc(G) is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) fG,H [7] defined using a constant indegree directed acyclic graph (DAG) G and a random oracle H(·). A secure iMHF should have amortized Space-Time complexity as high as possible, e.g., to deter brute-force password attacker who wants to find x such that fG,H (x) = h. Thus, to analyze the (in)security of a candidate iMHF fG,H , it is crucial to estimate the value cc(G) but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that it is NP-Hard to compute cc(G), but their techniques do not even rule out an efficient (1 + ε)-approximation algorithm for any constant ε > 0. -
A New Point of NP-Hardness for Unique Games
A new point of NP-hardness for Unique Games Ryan O'Donnell∗ John Wrighty September 30, 2012 Abstract 1 3 We show that distinguishing 2 -satisfiable Unique-Games instances from ( 8 + )-satisfiable instances is NP-hard (for all > 0). A consequence is that we match or improve the best known c vs. s NP-hardness result for Unique-Games for all values of c (except for c very close to 0). For these c, ours is the first hardness result showing that it helps to take the alphabet size larger than 2. Our NP-hardness reductions are quasilinear-size and thus show nearly full exponential time is required, assuming the ETH. ∗Department of Computer Science, Carnegie Mellon University. Supported by NSF grants CCF-0747250 and CCF-0915893, and by a Sloan fellowship. Some of this research performed while the author was a von Neumann Fellow at the Institute for Advanced Study, supported by NSF grants DMS-083537 and CCF-0832797. yDepartment of Computer Science, Carnegie Mellon University. 1 Introduction Thanks largely to the groundbreaking work of H˚astad[H˚as01], we have optimal NP-hardness of approximation results for several constraint satisfaction problems (CSPs), including 3Lin(Z2) and 3Sat. But for many others | including most interesting CSPs with 2-variable constraints | we lack matching algorithmic and NP-hardness results. Take the 2Lin(Z2) problem for example, in which there are Boolean variables with constraints of the form \xi = xj" and \xi 6= xj". The largest approximation ratio known to be achievable in polynomial time is roughly :878 [GW95], whereas it 11 is only known that achieving ratios above 12 ≈ :917 is NP-hard [H˚as01, TSSW00]. -
An Exact Method for the Minimum Feedback Arc Set Problem
AN EXACT METHOD FOR THE MINIMUM FEEDBACK ARC SET PROBLEM ALI BAHAREV, HERMANN SCHICHL, ARNOLD NEUMAIER, TOBIAS ACHTERBERG Abstract. Given a directed graph G, a feedback arc set of G is a subset of its edges containing at least one edge of every cycle in G. Finding a feedback arc set of minimum cardinality is the minimum feedback arc set problem. The minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all the simple cycles in G can be enumerated. Unfortunately, even sparse graphs can have W(2n) simple cycles, and such graphs appear in practice. An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion, and extends an incomplete cycle matrix iteratively. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. Numerical results are given on a test set containing computationally challenging sparse graphs, relevant for industrial applications. Key words. minimum feedback arc set, maximum acyclic subgraph, minimum feedback vertex set, linear ordering problem, tearing 1. Introduction. A directed graph G is a pair (V;E) of finite sets, the vertices V and the edges E V V. It is a simple graph if it has no multiple edges or self-loops (edges of ⊆ × the form (v;v)). Let G = (V;E) denote a simple directed graph, and let n = V denote the j j number of vertices (nodes), and m = E denote the number of edges. A subgraph H of the j j graph G is said to be induced if, for every pair of vertices u and v of H, (u;v) is an edge of H if and only if (u;v) is an edge of G. -
Research Statement
RESEARCH STATEMENT SAM HOPKINS Introduction The last decade has seen enormous improvements in the practice, prevalence, and usefulness of machine learning. Deep nets give our phones ears; matrix completion gives our televisions good taste. Data-driven articial intelligence has become capable of the dicult—driving a car on a highway—and the nearly impossible—distinguishing cat pictures without human intervention [18]. This is the stu of science ction. But we are far behind in explaining why many algorithms— even now-commonplace ones for relatively elementary tasks—work so stunningly well as they do in the wild. In general, we can prove very little about the performance of these algorithms; in many cases provable performance guarantees for them appear to be beyond attack by our best proof techniques. This means that our current-best explanations for how such algorithms can be so successful resort eventually to (intelligent) guesswork and heuristic analysis. This is not only a problem for our intellectual honesty. As machine learning shows up in more and more mission- critical applications it is essential that we understand the guarantees and limits of our approaches with the certainty aorded only by rigorous mathematical analysis. The theory-of-computing community is only beginning to equip itself with the tools to make rigorous attacks on learning problems. The sophisticated learning pipelines used in practice are built layer-by-layer from simpler fundamental statistical inference algorithms. We can address these statistical inference problems rigorously, but only by rst landing on tractable problem de- nitions. In their traditional theory-of-computing formulations, many of these inference problems have intractable worst-case instances. -
Limits of Approximation Algorithms: Pcps and Unique Games (DIMACS Tutorial Lecture Notes)1
Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)1 Organisers: Prahladh Harsha & Moses Charikar arXiv:1002.3864v1 [cs.CC] 20 Feb 2010 1Jointly sponsored by the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus on Algorithmic Foundations of the Internet, and the Center for Computational Intractability with support from the National Security Agency and the National Science Foundation. Preface These are the lecture notes for the DIMACS Tutorial Limits of Approximation Algorithms: PCPs and Unique Games held at the DIMACS Center, CoRE Building, Rutgers University on 20-21 July, 2009. This tutorial was jointly sponsored by the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus on Algorithmic Foundations of the Internet, and the Center for Computational Intractability with support from the National Security Agency and the National Science Foundation. The speakers at the tutorial were Matthew Andrews, Sanjeev Arora, Moses Charikar, Prahladh Harsha, Subhash Khot, Dana Moshkovitz and Lisa Zhang. We thank the scribes { Ashkan Aazami, Dev Desai, Igor Gorodezky, Geetha Jagannathan, Alexander S. Kulikov, Darakhshan J. Mir, Alan- tha Newman, Aleksandar Nikolov, David Pritchard and Gwen Spencer for their thorough and meticulous work. Special thanks to Rebecca Wright and Tami Carpenter at DIMACS but for whose organizational support and help, this workshop would have been impossible. We thank Alantha Newman, a phone conversation with whom sparked the idea of this workshop. We thank the Imdadullah Khan and Aleksandar Nikolov for video recording the lectures. The video recordings of the lectures will be posted at the DIMACS tutorial webpage http://dimacs.rutgers.edu/Workshops/Limits/ Any comments on these notes are always appreciated. -
How to Play Unique Games Against a Semi-Random Adversary
How to Play Unique Games against a Semi-Random Adversary Study of Semi-Random Models of Unique Games Alexandra Kolla Konstantin Makarychev Yury Makarychev Microsoft Research IBM Research TTIC Abstract In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an "–fraction of all edges, and finally replaces (“corrupts”) the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1 − ") satisfiable instance, so then the problem is as hard as in the worst case. In our semi-random model, one of the steps is random, and all other steps are adversarial. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games. We present an algorithm that with high probability finds a solution satisfying a (1 − δ) fraction of all constraints in semi-random instances (we require that the average degree of the graph is Ω(log~ k)). To this end, we consider a new non-standard SDP program for Unique Games, which is not a relaxation for the problem, and show how to analyze it. We present a new rounding scheme that simultaneously uses SDP and LP solutions, which we believe is of independent interest. Our result holds only for " less than some absolute constant. We prove that if " ≥ 1=2, then the problem is hard in one of the models, that is, no polynomial–time algorithm can distinguish between the following two cases: (i) the instance is a (1 − ") satisfiable semi–random instance and (ii) the instance is at most δ satisfiable (for every δ > 0); the result assumes the 2–to–2 conjecture. -
Lecture 17: the Strong Exponential Time Hypothesis 1 Introduction 2 ETH, SETH, and the Sparsification Lemma
CS 354: Unfulfilled Algorithmic Fantasies May 29, 2019 Lecture 17: The Strong Exponential Time Hypothesis Lecturer: Joshua Brakensiek Scribe: Weiyun Ma 1 Introduction We start with a recap of where we are. One of the biggest hardness assumption in history is P 6= NP, which came around in the 1970s. This assumption is useful for a number of hardness results, but often people need to make stronger assumptions. For example, in the last few weeks of this course, we have used average-case hardness assumptions (e.g. hardness of random 3-SAT or planted clique) to obtain distributional results. Another kind of results have implications on hardness of approximation. For instance, the PCP Theorem is known to be equivalent to solving some approximate 3-SAT is hard unless P=NP. We have also seen conjectures in this category that are still open, even assuming P 6= NP, such as the Unique Games Conjecture and the 2-to-2 Conjecture. Today, we are going to focus on hardness assumptions on time complexity. For example, for 3-SAT, P 6= NP only implies that it cannot be solved in polynomial time. On the other hand, the best known algorithms run in exponential time in n (the number of variables), which motivates the hardness assumption that better algorithms do not exist. This leads to conjectures known as the Exponential Time Hypothesis (ETH) and a stronger version called the Strong Exponential Time Hypothesis (SETH). Informally, ETH says that 3-SAT cannot be solved in 2o(n) time, and SETH says that k-SAT needs roughly 2n time for large k. -
Lecture 24: Hardness Assumptions 1 Overview 2 NP-Hardness 3 Learning Parity with Noise
A Theorist's Toolkit (CMU 18-859T, Fall 2013) Lecture 24: Hardness Assumptions December 2, 2013 Lecturer: Ryan O'Donnell Scribe: Jeremy Karp 1 Overview This lecture is about hardness and computational problems that seem hard. Almost all of the theory of hardness is based on assumptions. We make assumptions about some problems, then we do reductions from one problem to another. Then, we want to make the minimal number of assumptions necessary to show computational hardness. In fact, all work on computational complexity and hardness is essentially in designing efficient algorithms. 2 NP-Hardness A traditional assumption to make is that 3SAT is not in P or equivalently P 6= NP . From this assumption, we also see that Maximum Independent Set, for example, is not in P through a reduction. Such a reduction takes as input a 3CNF formula '. The reduction itself is an algorithm that runs in polynomial time which outputs a graph G. For the reduction to work as we want it to, we prove that if ' is satisfiable then G has an independent set of size at least some value k and if ' is not satisfiable then the maximum independent set of G has size less than k. This reduction algorithm will be deterministic. This is the basic idea behind proofs of NP -hardness. There are a few downsides to this approach. This only give you a worst-case hardness of a problem. For cryptographic purposes, it would be much better to have average-case hardness. We can think of average-case hardness as having an efficiently sampleable distribution of instances that are hard for polynomial time algorithms. -
A Generalization of the Directed Graph Layering Problem
A Generalization of the Directed Graph Layering Problem Ulf R¨uegg1, Thorsten Ehlers1, Miro Sp¨onemann2, and Reinhard von Hanxleden1 1 Dept. of Computer Science, Kiel University, Kiel, Germany furu,the,[email protected] 2 TypeFox GmbH, Kiel, Germany [email protected] Abstract. The Directed Layering Problem (DLP) solves a step of the widely used layer-based approach to automatically draw directed acyclic graphs. To cater for cyclic graphs, usually a preprocessing step is used that solves the Feedback Arc Set Problem (FASP) to make the graph acyclic before a layering is determined. Here we present the Generalized Layering Problem (GLP), which solves the combination of DLP and FASP simultaneously, allowing general graphs as input. We present an integer programming model and a heuris- tic to solve the NP-complete GLP and perform thorough evaluations on different sets of graphs and with different implementations for the steps of the layer-based approach. We observe that GLP reduces the number of dummy nodes significantly, can produce more compact drawings, and improves on graphs where DLP yields poor aspect ratios. Keywords: layer-based layout, layer assignment, linear arrangement, feedback arc set, integer programming 1 Introduction The layer-based approach is a well-established and widely used method to au- tomatically draw directed graphs. It is based on the idea to assign nodes to subsequent layers that show the inherent direction of the graph, see Fig. 1a for an example. The approach was introduced by Sugiyama et al. [19] and remains arXiv:1608.07809v1 [cs.OH] 28 Aug 2016 a subject of ongoing research.