[Paper Review] Hard satisfiable formulas for DPLL-type algorithms

Sergey Nikolenko|ArXiv.org|Jan 15, 2003

Constraint Satisfaction and Optimization9 references3 citations

TL;DR

This paper constructs explicitly defined, provably satisfiable formulas in k-CNF that force DPLL-type algorithms—specifically GUC and Randomized GUC—to exhibit exponential running time. The construction exploits structural asymmetry to mislead the heuristic choice mechanism, resulting in exponentially many recursive calls before finding a satisfying assignment, thus establishing the first known exponential lower bounds for satisfiable instances in DPLL-type algorithms with these heuristics.

ABSTRACT

We address lower bounds on the time complexity of algorithms solving the propositional satisfiability problem. Namely, we consider two DPLL-type algorithms, enhanced with the unit clause and pure literal heuristics. Exponential lower bounds for solving satisfiability on provably satisfiable formulas are proven.

Motivation & Objective

• To close the gap in the literature by proving exponential lower bounds for DPLL-type algorithms on satisfiable formulas, as opposed to only unsatisfiable ones.
• To construct explicit, provably satisfiable formulas that force GUC and Randomized GUC algorithms into exponentially many recursive calls.
• To demonstrate that even with pure literal and unit clause heuristics, DPLL-style algorithms can be systematically misled into exploring large portions of the search tree on satisfiable instances.
• To extend known lower bounds from unsatisfiable to satisfiable formulas, showing that hardness is not exclusive to contradiction-finding.

Proposed method

• The formula is built as a conjunction of a hard k-CNF formula G (from prior work) and a set of 3-clauses that enforce a specific assignment on M variables.
• The structure ensures that the only satisfying assignment sets all x₁,…,xₘ to true, while the remaining clauses form symmetric 3-clause clusters that force backtracking if incorrect choices are made.
• The GUC algorithm is misled into exploring the wrong subtree due to its preference for shortest clauses, which are the 3-clauses, and the absence of pure literals initially.
• Randomized GUC is analyzed probabilistically: at each step, it randomly selects a literal from the shortest clause, with a small but non-zero chance of selecting the correct path.
• The analysis computes the success probability of Randomized GUC as (1/2) × (1/4)^{M/3} = (1/2) × 2^{-(2/3)M}, leading to an expected exponential number of trials.
• The time complexity is bounded below by poly(M) × 2^{(2c_{k-1}/(2+3c_{k-1}))N} for formulas with N variables and O(N²) clauses, where c_k = O(1/k^{1/8}).

Experimental results

Research questions

• RQ1 Can exponential lower bounds be established for DPLL-type algorithms on satisfiable formulas, despite the existence of satisfying assignments?
• RQ2 Why do standard heuristics like GUC and Randomized GUC fail to efficiently find satisfying assignments on certain structured, yet satisfiable, formulas?
• RQ3 What structural properties of a formula can systematically mislead DPLL-style algorithms into exponential search, even when a solution exists?
• RQ4 Can such hard satisfiable instances be constructed with linear clause counts rather than quadratic, to better reflect practical SAT formula sizes?

Key findings

• The paper constructs a family of satisfiable k-CNF formulas F^k_N with N variables and at most aN² clauses, such that the Randomized GUC algorithm requires at least poly(N) × 2^{(2c_{k-1}/(2+3c_{k-1}))N} steps to find a satisfying assignment.
• For k ≥ 4, the lower bound is exponential in N, with the exponent depending on c_{k-1} = O(1/(k-1)^{1/8}), showing that hardness increases with k.
• The success probability of Randomized GUC on the constructed formula is (1/2) × 2^{-(2/3)M}, which decays exponentially with M, the number of variables in the outer clause group.
• The GUC algorithm is systematically misled by the formula structure, as it repeatedly selects literals from the 3-clause clusters before reaching the correct path.
• The formula is designed so that setting any x_i to false results in a subformula G that is hard to refute, forcing the algorithm to spend exponential time on unsatisfiable subproblems.
• The results establish the first known exponential lower bounds for DPLL-type algorithms on satisfiable formulas under the GUC and Randomized GUC heuristics, resolving an open question about the existence of such hard instances.