36 research outputs found
Computing exact solutions of consensus halving and the Borsuk-Ulam theorem
We study the problem of finding an exact solution to the consensus halving
problem. While recent work has shown that the approximate version of this
problem is PPA-complete, we show that the exact version is much harder.
Specifically, finding a solution with cuts is FIXP-hard, and deciding
whether there exists a solution with fewer than cuts is ETR-complete. We
also give a QPTAS for the case where each agent's valuation is a polynomial.
Along the way, we define a new complexity class BU, which captures all problems
that can be reduced to solving an instance of the Borsuk-Ulam problem exactly.
We show that FIXP BU TFETR and that LinearBU PPA,
where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is
specified by a linear arithmetic circuit
??-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games
We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is ??-complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is ??-complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games
Recognising Multidimensional Euclidean Preferences
Euclidean preferences are a widely studied preference model, in which
decision makers and alternatives are embedded in d-dimensional Euclidean space.
Decision makers prefer those alternatives closer to them. This model, also
known as multidimensional unfolding, has applications in economics,
psychometrics, marketing, and many other fields. We study the problem of
deciding whether a given preference profile is d-Euclidean. For the
one-dimensional case, polynomial-time algorithms are known. We show that, in
contrast, for every other fixed dimension d > 1, the recognition problem is
equivalent to the existential theory of the reals (ETR), and so in particular
NP-hard. We further show that some Euclidean preference profiles require
exponentially many bits in order to specify any Euclidean embedding, and prove
that the domain of d-Euclidean preferences does not admit a finite forbidden
minor characterisation for any d > 1. We also study dichotomous preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
The Computational Complexity of Genetic Diversity
A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition, or whether a single dominant genotype emerges. Classic work by [Kalmus, J. og Genetics, 1945] has established that even in simple diploid species (species with chromosome pairs) diversity can be guaranteed as long as the heterozygous (having different alleles for a gene on two chromosomes) individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g., human blood types) predicting diversity (and its implications) is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus (gene) diploid models, predicting whether diversity survives or not given its fitness landscape is algorithmically intractable.
Our hardness results are structurally robust along several dimensions, e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes. Technically, our results exploit connections between game theory, nonlinear dynamical systems, and complexity theory and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games; finding one Nash equilibrium is easy in these games. In the process we characterize stable fixed points of these dynamics using the notions of Nash equilibrium and negative semidefiniteness. This as well as hardness results for decision problems in coordination games may be of independent interest. Finally, we complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. The full version of this paper is available at http://arxiv.org/abs/1411.6322
Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem
We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete, we show that the exact version is much harder. Specifically, finding a solution with cuts is FIXP-hard, and deciding whether there exists a solution with fewer than cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP BU TFETR and that LinearBU PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit
The Complexity of Recognizing Geometric Hypergraphs
As set systems, hypergraphs are omnipresent and have various representations
ranging from Euler and Venn diagrams to contact representations. In a geometric
representation of a hypergraph , each vertex is associated
with a point and each hyperedge is associated
with a connected set such that for all . We say that a given
hypergraph is representable by some (infinite) family of sets in
, if there exist and such
that is a geometric representation of . For a family F, we define
RECOGNITION(F) as the problem to determine if a given hypergraph is
representable by F. It is known that the RECOGNITION problem is
-hard for halfspaces in . We study the
families of translates of balls and ellipsoids in , as well as of
other convex sets, and show that their RECOGNITION problems are also
-complete. This means that these recognition problems are
equivalent to deciding whether a multivariate system of polynomial equations
with integer coefficients has a real solution.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure
Training Fully Connected Neural Networks is -Complete
We consider the algorithmic problem of finding the optimal weights and biases
for a two-layer fully connected neural network to fit a given set of data
points. This problem is known as empirical risk minimization in the machine
learning community. We show that the problem is -complete.
This complexity class can be defined as the set of algorithmic problems that
are polynomial-time equivalent to finding real roots of a polynomial with
integer coefficients. Our results hold even if the following restrictions are
all added simultaneously.
There are exactly two output neurons.
There are exactly two input neurons.
The data has only 13 different labels.
The number of hidden neurons is a constant fraction of the number
of data points.
The target training error is zero.
The ReLU activation function is used.
This shows that even very simple networks are difficult to train. The result
offers an explanation (though far from a complete understanding) on why only
gradient descent is widely successful in training neural networks in practice.
We generalize a recent result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021].
This result falls into a recent line of research that tries to unveil that a
series of central algorithmic problems from widely different areas of computer
science and mathematics are -complete: This includes the art
gallery problem [JACM/STOC 2018], geometric packing [FOCS 2020], covering
polygons with convex polygons [FOCS 2021], and continuous constraint
satisfaction problems [FOCS 2021].Comment: 38 pages, 18 figure