Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > AVERAGE CASE COMPLEXITY:
Reports tagged with average case complexity:
TR98-069 | 7th December 1998
Rüdiger Reischuk, Thomas Zeugmann

#### An Average-Case Optimal One-Variable Pattern Language Learner

A new algorithm for learning one-variable pattern languages from positive data
is proposed and analyzed with respect to its average-case behavior.
We consider the total learning time that takes into account all
operations till convergence to a correct hypothesis is achieved.

For almost all meaningful distributions
defining how ... more >>>

TR99-006 | 10th March 1999
Jin-Yi Cai

#### Some Recent Progress on the Complexity of Lattice Problems

We survey some recent developments in the study of
the complexity of lattice problems. After a discussion of some
problems on lattices which can be algorithmically solved
efficiently, our main focus is the recent progress on complexity
results of intractability. We will discuss Ajtai's worst-case/
average-case connections, NP-hardness and non-NP-hardness,
more >>>

TR03-031 | 8th April 2003
Birgit Schelm

#### Average-Case Complexity Theory of Approximation Problems

Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ... more >>>

TR03-056 | 29th July 2003
Piotr Berman, Marek Karpinski

#### Approximability of Hypergraph Minimum Bisection

We prove that the problems of minimum bisection on k-uniform
hypergraphs are almost exactly as hard to approximate,
up to the factor k/3, as the problem of minimum bisection
on graphs. On a positive side, our argument gives also the
first approximation ... more >>>

TR06-122 | 20th September 2006
Noam Livne

#### All Natural NPC Problems Have Average-Case Complete Versions

Revisions: 1

In 1984 Levin put forward a suggestion for a theory of {\em average
case complexity}. In this theory a problem, called a {\em
distributional problem}, is defined as a pair consisting of a
decision problem and a probability distribution over the instances.
Introducing adequate notions of simple distributions and average
more >>>

TR09-057 | 23rd June 2009
Yonatan Bilu, Nathan Linial

#### Are stable instances easy?

We introduce the notion of a stable instance for a discrete
optimization problem, and argue that in many practical situations
only sufficiently stable instances are of interest. The question
then arises whether stable instances of NP--hard problems are
easier to solve. In particular, whether there exist algorithms
that solve correctly ... more >>>

TR10-019 | 19th February 2010
Andrew Drucker

#### A PCP Characterization of AM

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class $\mathsf{AM}$. This gives a PCP characterization' of $\mathsf{AM}$ analogous to the PCP Theorem for $\mathsf{NP}$. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, ... more >>>

TR10-091 | 14th May 2010
Nikolay Vereshchagin

#### An Encoding Invariant Version of Polynomial Time Computable Distributions

When we represent a decision problem,like CIRCUIT-SAT, as a language over the binary alphabet,
we usually do not specify how to encode instances by binary strings.
This relies on the empirical observation that the truth of a statement of the form CIRCUIT-SAT belongs to a complexity class $C$''
more >>>

TR12-086 | 4th July 2012
Shlomi Dolev, Nova Fandina, Dan Gutfreund

#### Succinct Permanent is NEXP-hard with Many Hard Instances

Finding a problem that is both hard to solve and hard to solve on many instances is a long standing issue
in theoretical computer science.
In this work, we prove that the Succinct Permanent $\bmod \; p$ is $NEXP$
time hard in the worst case (via randomized polynomial time ... more >>>

TR12-120 | 24th September 2012
Boaz Barak

#### Proof vs. Truth in Computational Complexity

Revisions: 1

In this survey, I discuss the general question of what evidence can we use to predict the answer for open questions in computational complexity, as well as the concrete evidence currently known for two conjectures: Khot's Unique Games Conjecture and Feige's Random 3SAT Hypothesis.

more >>>

TR18-092 | 4th May 2018
Marco Carmosino, Russell Impagliazzo, Manuel Sabin

#### Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires $n^{\epsilon k}$ time, for some constant $\epsilon > 1/2$, to count (note that these conjectures are significantly ... more >>>

TR19-125 | 27th August 2019
Elazar Goldenberg, Karthik C. S.

#### Hardness Amplification of Optimization Problems

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 $\Pi$ is direct product feasible if it is possible to efficiently aggregate any $k$ instances of $\Pi$ and form one large instance ... more >>>

TR20-052 | 14th April 2020
Yanyi Liu, Rafael Pass

#### On One-way Functions and Kolmogorov Complexity

Revisions: 2

We prove the equivalence of two fundamental problems in the theory of computation:

- Existence of one-way functions: the existence of one-way functions (which in turn are equivalent to pseudorandom generators, pseudorandom functions, private-key encryption schemes, digital signatures, commitment schemes, and more).

- Mild average-case hardness of $K^{poly}$-complexity: ... more >>>

TR21-055 | 20th April 2021
Yanyi Liu, Rafael Pass

#### Cryptography from Sublinear-Time Average-Case Hardness of Time-Bounded Kolmogorov Complexity

Let $\mktp[s]$ be the set of strings $x$ such that $K^t(x) \leq s(|x|)$, where $K^t(x)$ denotes the $t$-bounded Kolmogorov complexity of the truthtable described by $x$. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every $\varepsilon>0$, polynomial $t(n) \geq (1+\varepsilon)n$, and every nice'' class ... more >>>

TR21-056 | 22nd April 2021
Yanyi Liu, Rafael Pass

#### On the Possibility of Basing Cryptography on $\EXP \neq \BPP$

Liu and Pass (FOCS'20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity---namely, Levin's notion of Kolmogorov Complexity---whose hardness is closely ... more >>>

TR21-059 | 20th April 2021
Yanyi Liu, Rafael Pass

#### On One-way Functions from NP-Complete Problems

Revisions: 2

We present the first natural $\NP$-complete problem whose average-case hardness w.r.t. the uniform distribution over instances implies the existence of one-way functions (OWF). In fact, we prove that the existence of OWFs is \emph{equivalent} to mild average-case hardness of this $\NP$-complete problem. The problem, which originated in the 1960s, is ... more >>>

TR21-092 | 28th June 2021
Yanyi Liu, Rafael Pass

#### A Note on One-way Functions and Sparse Languages

Revisions: 1

We show equivalence between the existence of one-way
functions and the existence of a \emph{sparse} language that is
hard-on-average w.r.t. some efficiently samplable `high-entropy''
distribution.
In more detail, the following are equivalent:
- The existentence of a $S(\cdot)$-sparse language $L$ that is
hard-on-average with respect to some samplable ... more >>>

TR21-127 | 30th August 2021
Ron D. Rothblum, Michael Ezra

#### Small Circuits Imply Efficient Arthur-Merlin Protocols

Revisions: 1

The inner product function $\langle x,y \rangle = \sum_i x_i y_i \bmod 2$ can be easily computed by a (linear-size) ${AC}^0(\oplus)$ circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on ... more >>>

TR21-161 | 16th November 2021
Shuichi Hirahara, Mikito Nanashima

#### On Worst-Case Learning in Relativized Heuristica

A PAC learning model involves two worst-case requirements: a learner must learn all functions in a class on all example distributions. However, basing the hardness of learning on NP-hardness has remained a key challenge for decades. In fact, recent progress in computational complexity suggests the possibility that a weaker assumption ... more >>>

TR21-166 | 21st November 2021
Lijie Chen, Shuichi Hirahara, Neekon Vafa

#### Average-case Hardness of NP and PH from Worst-case Fine-grained Assumptions

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness ... more >>>

TR22-007 | 14th January 2022
Halley Goldberg, Valentine Kabanets

#### A Simpler Proof of the Worst-Case to Average-Case Reduction for Polynomial Hierarchy via Symmetry of Information

We give a simplified proof of Hirahara's STOC'21 result showing that $DistPH \subseteq AvgP$ would imply $PH \subseteq DTIME[2^{O(n/\log n)}]$. The argument relies on a proof of the new result: Symmetry of Information for time-bounded Kolmogorov complexity under the assumption that $NP$ is easy on average, which is interesting in ... more >>>

TR22-177 | 7th December 2022
Vahid Reza Asadi, Alexander Golovnev, Tom Gur, Igor Shinkar, Sathyawageeswar Subramanian

#### Quantum Worst-Case to Average-Case Reductions for All Linear Problems

We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even sub-constant) fraction of their inputs into ones that are correct on all inputs. This stands ... more >>>

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