Non-Promise Version of Unique Sink Orientations (2408.17283v1)

Abstract: A unique sink orientation (USO) is an orientation of the edges of a hypercube such that each face has a unique sink. Many optimization problems like linear programs reduce to USOs, in the sense that each vertex corresponds to a possible solution, and the global sink corresponds to the optimal solution. People have been studying intensively the problem of find the sink of a USO using vertex evaluations, i.e., queries which return the orientation of the edges around a vertex. This problem is a so called promise problem, as it assumes that the orientation it receives is a USO. In this paper, we analyze a non-promise version of the USO problem, in which we try to either find a sink or an efficiently verifiable violation of the USO property. This problem is worth investigating, because some problems which reduce to USO are also promise problems (and so we can also define a non-promise version for them), and it would be interesting to discover where USO lies in the hierarchy of subclasses of $\texttt{TFNP}^\texttt{dt}$, and for this a total search problem is required (which is the case for the non-promise version). We adapt many known properties and algorithms from the promise version to the non-promise one, including known algorithms for small dimensions and lower and upper bounds, like the Fibonacci Seesaw Algorithm. Furthermore, we present an efficient resolution proof of the problem, which shows it is in the search complexity class $\texttt{PLS}^\texttt{dt}$ (although this fact was already known via reductions). Finally, although initially the only allowed violations consist of $2$ vertices, we generalize them to more vertices, and provide a full categorization of violations with $4$ vertices, showing that they are also efficiently verifiable.

• The paper introduces the sink-or-clash framework to extend USO algorithms to non-promise settings.
• It establishes deterministic lower bounds and efficient randomized upper bounds for query complexity across hypercube dimensions.
• The work offers an efficient resolution proof linking Sink-or-Clash to PLS^dt and identifies minimal verifiable non-completable certificates.

Insightful Overview of "Non-Promise Version of Unique Sink Orientations"

The paper "Non-Promise Version of Unique Sink Orientations" by Tiago Oliveira Marques addresses a critical extension in the paper of unique sink orientations (USOs) on hypercubes, which are essential in understanding several optimization problems. Marques investigates the non-promise version of the USO problem, where instead of assuming the given orientation is a USO, the algorithm seeks either the sink or an efficiently verifiable violation of the USO property.

Summary and Contributions

Problem Formulation and Motivation

USOs are particularly relevant due to their reduction in various optimization problems, notably linear programs. Traditional studies focus on finding the sink of a USO given the promise that the input orientation is indeed a USO. Marques explores the scenario where this assumption does not hold, contributing significantly to theoretical computer science by addressing whether a given orientation is not a USO.

The paper formalizes Sink-or-Clash, the non-promise problem where the goal is to return either the global sink or identify a clash verifying the non-USO status. This approach provides a total search problem, critical when certain problems reducing to USO are themselves promise problems.

Main Results

Marques adapts existing properties and algorithms from the promise version to the non-promise version, demonstrating that many known results still hold:

1. Lower Bound: The query complexity lower bound for deterministic algorithms remains $t(n) \in \Omega(n^2/\log(n))$, akin to the promise version.
2. Upper Bound: Adapting the Fibonacci Seesaw algorithm and other strategies, the paper establishes an upper bound of $t(n) \leq O(1.61^n)$, confirming the higher number of queries required.
3. Four-Dimensional Case: Deterministically, for hypercube dimensions up to four, non-promise algorithms align in query complexity with their promise counterparts. The exact value is $t(4) = 7$, validated computationally.
4. Randomized Setting: A divergence in the randomized query complexity demonstrates $\tilde{t}(1) = 2$ and $\tilde{t}(2) = 46/17 \approx 2.706$, higher than for the promise version. For $n=3$, $\tilde{t}(3) \approx 3.591333$, leading to an efficient upper bound $\tilde{t}(n) \in O(1.531^n)$.

Theoretical Implications

Efficient Resolution Proof

Marques constructs an efficient resolution proof for the CNF formula associated with the Sink-or-Clash problem. This proof explicitly demonstrates the membership of Sink-or-Clash in the complexity class $PLS^dt$, thereby confirming its status as having an efficient resolution proof. This connection between the search problem and its proof complexity outlines that each non-promise USO issue can be efficiently reasoned about within this framework.

Certificates and Completeness

The paper dives into certificates of non-completability, minimal subsets indicating an orientation's failure to extend to any USO. Through exhaustive analysis, the paper shows that:

• 2-Vertex Clashes: Minimal non-completable subsets (2-certificates) are precisely clashes.
• 3-Vertex Completeness: Verifying the completable nature of all outmaps involving three vertices.
• 4-Vertex Complexity: Introducing configurations showcasing non-completable partial outmaps with four vertices that do not clash.

Marques generalizes these results and provides boundaries for existing configurations, setting the stage for future work in identifying efficiently verifiable certificates beyond clashes.

Future Directions

The paper's findings prompt several future research avenues:

1. Comparative Analysis: Further upper bounds for Sink-or-Clash might reveal more about the problem's relationship with the promise versions.
2. Resolution for Complex Classes: Extending the constructed resolution proofs to intuitively weaker systems or more complex hierarchies.
3. Generalizing Certificates: Identifying efficiently verifiable non-completable configurations beyond the presented certificates can fine-tune non-promise algorithms further.

In essence, Marques' paper lays foundational work for a comprehensive understanding of non-promise USO problems, balancing algorithmic complexity and theoretical depth. The presented proofs and algorithms offer a robust platform for future explorations in higher dimensions and broader applications within combinatorial optimization and computational complexity theory.