CORDIS - EU research results
CORDIS

A Unified Theory of Algorithmic Relaxations

Project description

Research investigates constrained optimisation and satisfaction problems

Constrained optimisation and satisfaction problems (CSPs) are a large family of computational problems that are typically classified as either tractable (solvable in polynomial time) or intractable (NP-hard). However, certain CSPs resist classification, including problems in graph theory, combinatorics and mathematical game theory. The ERC-funded AUTAR project aims to bridge this knowledge gap by analysing candidate algorithms that appear to solve all instances and are not limited to describing hard instances or hardness hypotheses. AUTAR’s innovative approach will build on a recent discovery that two methods from different areas – indistinguishability pebble games from mathematical logic and hierarchies of convex relaxations from mathematical programming – match in strength.

Objective

For a large family of computational problems collectively known as constrained optimization and satisfaction problems (CSPs), four decades of research in algorithms and computational complexity have led to a theory that tries to classify them as algorithmically tractable vs. intractable, i.e. polynomial-time solvable vs. NP-hard. However, there remains an important gap in our knowledge in that many CSPs of interest resist classification by this theory. Some such problems of practical relevance include fundamental partition problems in graph theory, isomorphism problems in combinatorics, and strategy-design problems in mathematical game theory. To tackle this gap in our knowledge, the research of the last decade has been driven either by finding hard instances for algorithms that solve tighter and tighter relaxations of the original problem, or by formulating new hardness-hypotheses that are stronger but admittedly less robust than NP-hardness.

The ultimate goal of this project is closing the gap between the partial progress that these approaches represent and the original classification project into tractable vs. intractable problems. Our thesis is that the field has reached a point where, in many cases of interest, the analysis of the current candidate algorithms that appear to solve all instances could suffice to classify the problem one way or the other, without the need for alternative hardness-hypotheses. The novelty in our approach is a program to develop our recent discovery that, in some cases of interest, two methods from different areas match in strength: indistinguishability pebble games from mathematical logic, and hierarchies of convex relaxations from mathematical programming. Thus, we aim at making significant advances in the status of important algorithmic problems by looking for a general theory that unifies and goes beyond the current understanding of its components.

Host institution

UNIVERSITAT POLITECNICA DE CATALUNYA
Net EU contribution
€ 1 725 656,00
Address
CALLE JORDI GIRONA 31
08034 Barcelona
Spain

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Region
Este Cataluña Barcelona
Activity type
Higher or Secondary Education Establishments
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Total cost
€ 1 725 656,00

Beneficiaries (1)