# Tractable Triangles

@inproceedings{Cooper2011TractableT, title={Tractable Triangles}, author={Martin C. Cooper and Stanislav Živn{\'y}}, booktitle={CP}, year={2011} }

We study the computational complexity of binary valued constraint satisfaction problems (VCSP) given by allowing only certain types of costs in every triangle of variable-value assignments to three distinct variables.We show that for several computational problems, including CSP, Max-CSP, finite-valued VCSP, and general-valued VCSP, the only non-trivial tractable classes are the well known maximum matching problem and the recently discovered joint-winner property [9].

## 5 Citations

### Tractable Triangles and Cross-Free Convexity in Discrete Optimisation

- Mathematics, Computer ScienceJ. Artif. Intell. Res.
- 2012

It is shown that for several computational problems, the only non- trivial tractable classes are the well known maximum matching problem and the recently discovered joint-winner property, and it is proved that the conjunction of the two gives rise to a novel tractable class satisfying the cross-free convexity property.

### A Characterisation of the Complexity of Forbidding Subproblems in Binary Max-CSP

- Computer ScienceCP
- 2012

This paper characterises the complexity of all classes of binary Max-CSP instances defined by forbidding a single subproblem and concludes that the only non-trivial tractable class defined by a forbidden subproblem corresponds to the set of instances satisfying the so-called joint-winner property.

### Generating Difficult CNF Instances in Unexplored Constrainedness Regions

- Computer ScienceACM J. Exp. Algorithmics
- 2020

This article introduces the No-Triangle CNF algorithm, a CNF instance generator based on the cluster coefficient graph statistic, and empirically compares the two algorithms, finding that the hardest instances produced by each method belong to different constrainedness regions.

### Generating Difficult SAT Instances by Preventing Triangles

- Computer ScienceArXiv
- 2019

This paper introduces the No-Triangle SAT algorithm, a SAT instance generator based on the cluster coefficient graph statistic, and empirically compare the two algorithms by fixing the arity and the number of variables, but varying thenumber of clauses.

## References

SHOWING 1-10 OF 34 REFERENCES

### Hierarchically Nested Convex VCSP

- Computer ScienceCP
- 2011

It is shown that the class of VCSP instances satisfying the hierarchically nested convexity property is tractable and that, over Boolean domains, it is possible to determine in polynomial time whether there exists some subset of the constraints such that the VCSP satisfies the hierarchic nested conveXity property after renaming the variables in these constraints.

### Valued Constraint Satisfaction Problems: Hard and Easy Problems

- Computer ScienceIJCAI
- 1995

A simple algebraic framework is considered, related to Partial Constraint Satisfaction, which subsumes most of these proposals and is used to characterize existing proposals in terms of rationality and computational complexity.

### Complexity classifications of Boolean constraint satisfaction problems

- Computer ScienceSIAM monographs on discrete mathematics and applications
- 2001

Theorems for Optimization Problems and the Complexity of the Meta-Problems are discussed, as well as some examples of how classification theorems can be applied to optimization problems.

### On the NP-completeness of the k-colorability problem for triangle-free graphs

- MathematicsDiscret. Math.
- 1996

### Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2008

### The Node-Deletion Problem for Hereditary Properties is NP-Complete

- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 1980

### Network-based heuristics for constraint satisfaction problems

- Computer Science, Business
- 1988

This paper identifies classes of problems that lend themselves to easy solutions, and develops algorithms that solve these problems optimally by generating heuristic advice to guide the order of value assignments based on both the sparseness found in the constraint network and the simplicity of tree-structured CSPs.

### A New Classs of Binary CSPs for which Arc-Constistency Is a Decision Procedure

- Computer ScienceCP
- 2003

This new hybrid class includes infinitely many instances whose tractability is not assured by any tractable language or structural restriction, and strongly motivates the search for a unifying principle for the tractable constraint classes decided by arc-consistency.