Lower Bounds for Existential Pebble Games and k-Consistency Tests

TL;DR

This paper proves an unconditional lower bound on the computational complexity of deciding the winner in existential pebble games and the establishment of strong k-consistency in CSPs, showing no algorithms faster than a certain bound exist.

Contribution

It establishes a new unconditional lower bound on the time complexity for existential pebble games and k-consistency tests, independent of complexity assumptions.

Findings

No O(n^{(k-3)/12})-time algorithm exists for deciding existential pebble game winners.

The same lower bound applies to deciding strong k-consistency in CSPs.

The results are unconditional and do not rely on complexity-theoretic assumptions.

Abstract

The existential k-pebble game characterizes the expressive power of the existential-positive k-variable fragment of first-order logic on finite structures. The winner of the existential k-pebble game on two given finite structures can be determined in time O(n2k) by dynamic programming on the graph of game configurations. We show that there is no O(n(k-3)/12)-time algorithm that decides which player can win the existential k-pebble game on two given structures. This lower bound is unconditional and does not rely on any complexity-theoretic assumptions. Establishing strong k-consistency is a well-known heuristic for solving the constraint satisfaction problem (CSP). By the game characterization of Kolaitis and Vardi our result implies that there is no O(n(k-3)/12)-time algorithm that decides if strong k-consistency can be established for a given CSP-instance.