252 research outputs found
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Î, this problem falls into the regime of LovĂĄsz local lemma when Î âČ qá”. In prior, however, fast approximate counting algorithms exist when Î âČ qá”/Âł, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
âą When q, k â„ 4 are evens and Î â„ 5·qá”/ÂČ, approximating the number of hypergraph colourings is NP-hard.
âą When the input hypergraph is linear and Î âČ qá”/ÂČ, a fast approximate counting algorithm does exist
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Efficient Symbolic Reasoning for Neural-Network Verification
The neural network has become an integral part of modern software systems.
However, they still suffer from various problems, in particular, vulnerability
to adversarial attacks. In this work, we present a novel program reasoning
framework for neural-network verification, which we refer to as symbolic
reasoning. The key components of our framework are the use of the symbolic
domain and the quadratic relation. The symbolic domain has very flexible
semantics, and the quadratic relation is quite expressive. They allow us to
encode many verification problems for neural networks as quadratic programs.
Our scheme then relaxes the quadratic programs to semidefinite programs, which
can be efficiently solved. This framework allows us to verify various
neural-network properties under different scenarios, especially those that
appear challenging for non-symbolic domains. Moreover, it introduces new
representations and perspectives for the verification tasks. We believe that
our framework can bring new theoretical insights and practical tools to
verification problems for neural networks
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs
We consider the question of approximating Max 2-CSP where each variable
appears in at most constraints (but with possibly arbitrarily large
alphabet). There is a simple -approximation algorithm for the
problem. We prove the following results for any sufficiently large :
- Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized
reduction) to approximate this problem to within a factor of .
- It is NP-hard (under randomized reduction) to approximate the problem to
within a factor of .
Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish
the following hardness results for approximating Maximum Independent Set on
-claw-free graphs:
- Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized
reduction) to approximate this problem to within a factor of .
- It is NP-hard (under randomized reduction) to approximate the problem to
within a factor of .
In comparison, known approximation algorithms achieve -approximation in polynomial time [Neuwohner, STACS 2021; Thiery
and Ward, SODA 2023] and -approximation in
quasi-polynomial time [Cygan et al., SODA 2013]
Experimenting with Constraint Programming Techniques in Artificial Intelligence: Automated System Design and Verification of Neural Networks
This thesis focuses on the application of Constraint Satisfaction and Optimization techniques
in two Artificial Intelligence (AI) domains: automated design of elevator systems and
verification of Neural Networks (NNs). The three main areas of interest for my work
are (i) the languages for defining the constraints for the systems, (ii) the algorithms and
encodings that enable solving the problems considered and (iii) the tools that implement
such algorithms.
Given the expressivity of the domain description languages and the availability of effective
tools, several problems in diverse application fields have been solved successfully using
constraint satisfaction techniques. The two case studies herewith presented are no exception,
even if they entail different challenges in the adoption of such techniques. Automated design
of elevator systems not only requires encoding of feasibility (hard) constraints, but should
also take into account design preferences, which can be expressed in terms of cost functions
whose optimal or near-optimal value characterizes âgoodâ design choices versus âpoorâ ones.
Verification of NNs (and other machine-learned implements) requires solving large-scale
constraint problems which may become the main bottlenecks in the overall verification
procedure.
This thesis proposes some ideas for tackling such challenges, including encoding techniques
for automated design problems and new algorithms for handling the optimization
problems arising from verification of NNs. The proposed algorithms and techniques are evaluated
experimentally by developing tools that are made available to the research community
for further evaluation and improvement
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