65 research outputs found

    Hardness Amplification of Optimization Problems

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    In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products. We say that an optimization problem ? is direct product feasible if it is possible to efficiently aggregate any k instances of ? and form one large instance of ? such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem ?, our hardness amplification theorem may be informally stated as follows: If there is a distribution D over instances of ? of size n such that every randomized algorithm running in time t(n) fails to solve ? on 1/?(n) fraction of inputs sampled from D, then, assuming some relationships on ?(n) and t(n), there is a distribution D\u27 over instances of ? of size O(n??(n)) such that every randomized algorithm running in time t(n)/poly(?(n)) fails to solve ? on 99/100 fraction of inputs sampled from D\u27. As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium

    Prophet Secretary for Combinatorial Auctions and Matroids

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    The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/21/2-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/21/2-approximation and obtain (11/e)(1-1/e)-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the writeup on Fixed-Threshold algorithm

    Complexity Theory

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    Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

    Almost Optimal Distribution-Free Sample-Based Testing of k-Modality

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    Towards Hardness of Approximation for Polynomial Time Problems

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    Proving hardness of approximation is a major challenge in the field of fine-grained complexity and conditional lower bounds in P. How well can the Longest Common Subsequence (LCS) or the Edit Distance be approximated by an algorithm that runs in near-linear time? In this paper, we make progress towards answering these questions. We introduce a framework that exhibits barriers for truly subquadratic and deterministic algorithms with good approximation guarantees. Our framework highlights a novel connection between deterministic approximation algorithms for natural problems in P and circuit lower bounds. In particular, we discover a curious connection of the following form: if there exists a delta>0 such that for all eps>0 there is a deterministic (1+eps)-approximation algorithm for LCS on two sequences of length n over an alphabet of size n^{o(1)} that runs in O(n^{2-delta}) time, then a certain plausible hypothesis is refuted, and the class E^NP does not have non-uniform linear size Valiant Series-Parallel circuits. Thus, designing a "truly subquadratic PTAS" for LCS is as hard as resolving an old open question in complexity theory

    Algorithmic Pirogov-Sinai theory

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    We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice Zd\mathbb Z^d and on the torus (Z/nZ)d(\mathbb Z/n \mathbb Z)^d. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of Zd\mathbb Z^d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus (Z/nZ)d(\mathbb Z/n \mathbb Z)^d at sufficiently low temperature

    Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0

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    The threshold degree of a Boolean function f ⁣:{0,1}n{0,1}f\colon\{0,1\}^n\to\{0,1\} is the minimum degree of a real polynomial pp that represents ff in sign: sgn  p(x)=(1)f(x).\mathrm{sgn}\; p(x)=(-1)^{f(x)}. A related notion is sign-rank, defined for a Boolean matrix F=[Fij]F=[F_{ij}] as the minimum rank of a real matrix MM with sgn  Mij=(1)Fij\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC0\text{AC}^{0}) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any ϵ>0,\epsilon>0, we construct an AC0\text{AC}^{0} circuit in nn variables that has threshold degree Ω(n1ϵ)\Omega(n^{1-\epsilon}) and sign-rank exp(Ω(n1ϵ)),\exp(\Omega(n^{1-\epsilon})), improving on the previous best lower bounds of Ω(n)\Omega(\sqrt{n}) and exp(Ω~(n))\exp(\tilde{\Omega}(\sqrt{n})), respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of AC0\text{AC}^{0} circuits of any given depth, with a strict improvement starting at depth 44. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of AC0\text{AC}^{0}, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of AC0\text{AC}^{0}.Comment: 99 page
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