2,374 research outputs found
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
This document collects the lecture notes from my mini-course "Complexity
Theory, Game Theory, and Economics," taught at the Bellairs Research Institute
of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th
McGill Invitational Workshop on Computational Complexity.
The goal of this mini-course is twofold: (i) to explain how complexity theory
has helped illuminate several barriers in economics and game theory; and (ii)
to illustrate how game-theoretic questions have led to new and interesting
complexity theory, including recent several breakthroughs. It consists of two
five-lecture sequences: the Solar Lectures, focusing on the communication and
computational complexity of computing equilibria; and the Lunar Lectures,
focusing on applications of complexity theory in game theory and economics. No
background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some
recent citations to v1 Revised v3 corrects a few typos in v
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input
consists of a set system F over a finite universe U and an integer t; the
question is whether there is a set of t elements that intersects every set in
F. The Hitting Set problem parameterized by the size of the solution is a
well-known W[2]-complete problem in parameterized complexity theory. In this
paper we investigate the complexity of Hitting Set under various structural
parameterizations of the input. Our starting point is the folklore result that
Hitting Set is polynomial-time solvable if there is a tree T on vertex set U
such that the sets in F induce connected subtrees of T. We consider the case
that there is a treelike graph with vertex set U such that the sets in F induce
connected subgraphs; the parameter of the problem is a measure of how treelike
the graph is. Our main positive result is an algorithm that, given a graph G
with cyclomatic number k, a collection P of simple paths in G, and an integer
t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of
size t that hits all paths in P. It is based on a connection to the 2-SAT
problem in multiple valued logic. For other parameterizations we derive
W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic
Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was
corrected in this update.
Arithmetic Expression Construction
When can given numbers be combined using arithmetic operators from a
given subset of to obtain a given target number? We
study three variations of this problem of Arithmetic Expression Construction:
when the expression (1) is unconstrained; (2) has a specified pattern of
parentheses and operators (and only the numbers need to be assigned to blanks);
or (3) must match a specified ordering of the numbers (but the operators and
parenthesization are free). For each of these variants, and many of the subsets
of , we prove the problem NP-complete, sometimes in the
weak sense and sometimes in the strong sense. Most of these proofs make use of
a "rational function framework" which proves equivalence of these problems for
values in rational functions with values in positive integers.Comment: 36 pages, 5 figures. Full version of paper accepted to 31st
International Symposium on Algorithms and Computation (ISAAC 2020
Anticoncentration theorems for schemes showing a quantum speedup
One of the main milestones in quantum information science is to realise
quantum devices that exhibit an exponential computational advantage over
classical ones without being universal quantum computers, a state of affairs
dubbed quantum speedup, or sometimes "quantum computational supremacy". The
known schemes heavily rely on mathematical assumptions that are plausible but
unproven, prominently results on anticoncentration of random prescriptions. In
this work, we aim at closing the gap by proving two anticoncentration theorems
and accompanying hardness results, one for circuit-based schemes, the other for
quantum quench-type schemes for quantum simulations. Compared to the few other
known such results, these results give rise to a number of comparably simple,
physically meaningful and resource-economical schemes showing a quantum speedup
in one and two spatial dimensions. At the heart of the analysis are tools of
unitary designs and random circuits that allow us to conclude that universal
random circuits anticoncentrate as well as an embedding of known circuit-based
schemes in a 2D translation-invariant architecture.Comment: 12+2 pages, added applications sectio
Computational complexity of the landscape I
We study the computational complexity of the physical problem of finding
vacua of string theory which agree with data, such as the cosmological
constant, and show that such problems are typically NP hard. In particular, we
prove that in the Bousso-Polchinski model, the problem is NP complete. We
discuss the issues this raises and the possibility that, even if we were to
find compelling evidence that some vacuum of string theory describes our
universe, we might never be able to find that vacuum explicitly.
In a companion paper, we apply this point of view to the question of how
early cosmology might select a vacuum.Comment: JHEP3 Latex, 53 pp, 2 .eps figure
Motif Clustering and Overlapping Clustering for Social Network Analysis
Motivated by applications in social network community analysis, we introduce
a new clustering paradigm termed motif clustering. Unlike classical clustering,
motif clustering aims to minimize the number of clustering errors associated
with both edges and certain higher order graph structures (motifs) that
represent "atomic units" of social organizations. Our contributions are
two-fold: We first introduce motif correlation clustering, in which the goal is
to agnostically partition the vertices of a weighted complete graph so that
certain predetermined "important" social subgraphs mostly lie within the same
cluster, while "less relevant" social subgraphs are allowed to lie across
clusters. We then proceed to introduce the notion of motif covers, in which the
goal is to cover the vertices of motifs via the smallest number of (near)
cliques in the graph. Motif cover algorithms provide a natural solution for
overlapping clustering and they also play an important role in latent feature
inference of networks. For both motif correlation clustering and its extension
introduced via the covering problem, we provide hardness results, algorithmic
solutions and community detection results for two well-studied social networks
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