Approximating the existential theory of the reals

Deligkas, Argyrios; Fearnley, John; Melissourgos, Themistoklis; Spirakis, Paul G.

doi:10.1016/j.jcss.2021.11.002

Cited by 6 publications

(4 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following lemma intuitively states the following: if an ETR-SQUARE formula Φ has something which is "almost a solution", with an error of at most 2 −2 𝑂( |Φ| ) , then Φ also admits an actual solution. Similar results were established in [24].…”

Section: Approximate Solutionssupporting

confidence: 88%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

View full text Add to dashboard Cite

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical conditions. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical conditions) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

show abstract

Section: Approximate Solutionssupporting

confidence: 88%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

View full text Add to dashboard Cite

show abstract

“…So far, no algorithm is known for solving ε-Consensus-Halving (with n cuts) for some constant ε < 1, even for piecewise constant valuations with positive value on two intervals only. The only known upper bounds are for very special cases [Deligkas et al, 2022;Filos-Ratsikas et al, 2020] or with additional cuts [Alon and Graur, 2021].…”

Section: Discussionmentioning

confidence: 99%

Constant Inapproximability for PPA

Deligkas¹,

Fearnley²,

Hollender³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

In the ε-Consensus-Halving problem, we are given n probability measures v 1 , . . . , v n on the interval R = [0, 1], and the goal is to partition R into two parts R + and R − using at most n cuts, so thatThis fundamental fair division problem was the first natural problem shown to be complete for the class PPA, and all subsequent PPA-completeness results for other natural problems have been obtained by reducing from it.We show that ε-Consensus-Halving is PPA-complete even when the parameter ε is a constant. In fact, we prove that this holds for any constant ε < 1/5. As a result, we obtain constant inapproximability results for all known natural PPA-complete problems, including Necklace-Splitting, the Discrete-Ham-Sandwich problem, two variants of the pizza sharing problem, and for finding fair independent sets in cycles and paths.

show abstract

“…Concerning quasi-polynomial algorithms, in addition to the QPTAS of [27], three new QPTASs have been obtained, which contain the original result of [27] as a special case: [5] gave a refined, parameterized, approximation scheme; [3] gave a QPTAS that can be applied to multi-player games as well; [19] gave a more general approach for approximation schemes for the existential theory of the reals. More recently, more negative results for ε-NE were derived: [26] gave an unconditional lower bound, based on the sum of squares hierarchy; [6] proved PPAD-hardness in the smoothed analysis setting; [8,20,2] gave quasi-polynomial time lower bounds for constrained ε-NE, under the exponential time hypothesis.…”

Section: Further Related Workmentioning

confidence: 99%

A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games

Deligkas

Fasoulakis

Markakis

2023

ACM Trans. Algorithms

Self Cite

View full text Add to dashboard Cite

Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [38], with an approximation guarantee of (0.3393 + δ ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a $(\frac{1}{3}+\delta) $ -Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [38], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.

show abstract

Approximating the existential theory of the reals

Cited by 6 publications

References 43 publications

On Classifying Continuous Constraint Satisfaction Problems

On Classifying Continuous Constraint Satisfaction Problems

Constant Inapproximability for PPA

A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games

Contact Info

Product

Resources

About