34 research outputs found
Parameterized Inapproximability Hypothesis under ETH
The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed
parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance,
parameterized by the number of variables, from one where every assignment fails
to satisfy an fraction of constraints for some absolute constant
. PIH plays the role of the PCP theorem in parameterized
complexity. However, PIH has only been established under Gap-ETH, a very strong
assumption with an inherent gap.
In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This
is the first proof of PIH from a gap-free assumption. Our proof is
self-contained and elementary. We identify an ETH-hard CSP whose variables take
vector values, and constraints are either linear or of a special parallel
structure. Both kinds of constraints can be checked with constant soundness via
a "parallel PCP of proximity" based on the Walsh-Hadamard code
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
The study of graph products is a major research topic and typically concerns
the term , e.g., to show that . In this paper, we
study graph products in a non-standard form where is a
"reduction", a transformation of any graph into an instance of an intended
optimization problem. We resolve some open problems as applications.
(1) A tight -approximation hardness for the minimum
consistent deterministic finite automaton (DFA) problem, where is the
sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this
implies the hardness of properly learning DFAs assuming (the
weakest possible assumption).
(2) A tight hardness for the edge-disjoint paths (EDP)
problem on directed acyclic graphs (DAGs), where denotes the number of
vertices.
(3) A tight hardness of packing vertex-disjoint -cycles for large .
(4) An alternative (and perhaps simpler) proof for the hardness of properly
learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004
and J. Comput.Syst.Sci. 2008]
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Complexity Theory
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 developements are related to diverse mathematical ïŹelds such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs
In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a
graph subject to edge insertions and deletions and a source vertex
, and the goal is to maintain the distance for all .
Fine-grained complexity has provided strong lower bounds for exact partially
dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15].
Thus much focus has been directed towards finding efficient partially dynamic
-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14,
FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20].
Despite this rich literature, for directed graphs there are no known
deterministic algorithms for -approximate dynamic SSSP that
perform better than the classic ES-tree [JACM'81]. We present the first such
algorithm.
We present a \emph{deterministic} data structure for incremental SSSP in
weighted digraphs with total update time which is
near-optimal for very dense graphs; here is the ratio of the largest weight
in the graph to the smallest. Our algorithm also improves over the best known
partially dynamic \emph{randomized} algorithm for directed SSSP by Henzinger et
al. [STOC'14, ICALP'15] if .
We also provide improved conditional lower bounds. Henzinger et al. [STOC'15]
showed that under the OMv Hypothesis, the partially dynamic exact -
Shortest Path problem in undirected graphs requires amortized update or query
time , given polynomial preprocessing time. Under a hypothesis
about finding Cliques, we improve the update and query lower bound for
algorithms with polynomial preprocessing time to . Further,
under the -Cycle hypothesis, we show that any partially dynamic SSSP
algorithm with preprocessing time requires amortized update
or query time
Interactive Proof Systems
The report is a compilation of lecture notes that were prepared during the course ``Interactive Proof Systems'' given by the authors at Tata Institute of Fundamental Research, Bombay. These notes were also used for a short course ``Interactive Proof Systems'' given by the second author at MPI, Saarbruecken. The objective of the course was to study the recent developments in complexity theory about interactive proof systems, which led to some surprising consequences on nonapproximability of NP hard problems. We start the course with an introduction to complexity theory and covered some classical results related with circuit complexity, randomizations and counting classes, notions which are either part of the definitions of interactive proof systems or are used in proving the above results. We define arthur merlin games and interactive proof systems, which are equivalent formulations of the notion of interactive proofs and show their equivalence to each other and to the complexity class PSPACE. We introduce probabilistically checkable proofs, which are special forms of interactive proofs and show through sequence of intermediate results that the class NP has probabilistically checkable proofs of very special form and very small complexity. Using this we conclude that several NP hard problems are not even weakly approximable in polynomial time unless P = NP
Hitting Meets Packing: How Hard Can it Be?
We study a general family of problems that form a common generalization of
classic hitting (also referred to as covering or transversal) and packing
problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of
a graph G prevent us from packing vertex-disjoint objects of type X?
This problem captures a spectrum of problems with standard hitting and packing
on opposite ends. Our main motivating question is whether the combination
X-HitPack can be significantly harder than these two base problems. Already for
a particular choice of X, this question can be posed for many different
complexity notions, leading to a large, so-far unexplored domain in the
intersection of the areas of hitting and packing problems.
On a high-level, we present two case studies: (1) X being all cycles, and (2)
X being all copies of a fixed graph H. In each, we explore the classical
complexity, as well as the parameterized complexity with the natural parameters
k+l and treewidth. We observe that the combined problem can be drastically
harder than the base problems: for cycles or for H being a connected graph with
at least 3 vertices, the problem is \Sigma_2^P-complete and requires
double-exponential dependence on the treewidth of the graph (assuming the
Exponential-Time Hypothesis). In contrast, the combined problem admits
qualitatively similar running times as the base problems in some cases,
although significant novel ideas are required. For example, for X being all
cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching
method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover
and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on
graphs of treewidth tw. The key step enabling this running time relies on a
combinatorial bound obtained from an algebraic (linear delta-matroid)
representation of possible matchings