Improved Parameterized Algorithms for Constraint Satisfaction

For many constraint satisfaction problems, the algorithm which chooses a random assignment achieves the best possible approximation ratio. For instance, a simple random assignment for {\sc Max-E3-Sat} allows 7/8-approximation and for every $\eps >0$ there is no polynomial-time ($7/8+\eps$)-approximation unless P=NP. Another example is the {\sc Permutation CSP} of bounded arity. Given the expected fraction $\rho$ of the constraints satisfied by a random assignment (i.e. permutation), there is no $(\rho+\eps)$-approximation algorithm for every $\eps >0$, assuming the Unique Games Conjecture (UGC). In this work, we consider the following parameterization of constraint satisfaction problems. Given a set of $m$ constraints of constant arity, can we satisfy at least $\rho m +k$ constraint, where $\rho$ is the expected fraction of constraints satisfied by a random assignment? {\sc Constraint Satisfaction Problems above Average} have been posed in different forms in the literature \cite{Niedermeier2006,MahajanRamanSikdar09}. We present a faster parameterized algorithm for deciding whether $m/2+k/2$ equations can be simultaneously satisfied over ${\mathbb F}_2$. As a consequence, we obtain $O(k)$-variable bikernels for {\sc boolean CSPs} of arity $c$ for every fixed $c$, and for {\sc permutation CSPs} of arity 3. This implies linear bikernels for many problems under the “above average” parameterization, such as {\sc Max-$c$-Sat}, {\sc Set-Splitting}, {\sc Betweenness} and {\sc Max Acyclic Subgraph}. As a result, all the parameterized problems we consider in this paper admit $2^{O(k)}$-time algorithms. We also obtain non-trivial hybrid algorithms for every Max $c$-CSP: for every instance $I$, we can either approximate $I$ beyond the random assignment threshold in polynomial time, or we can find an optimal solution to $I$ in subexponential time.

💡 Research Summary

The paper investigates a parameterized version of constraint satisfaction problems (CSPs) in which the goal is to exceed the expected performance of a random assignment. For a given CSP with m constraints of constant arity, let ρ denote the fraction of constraints that a random assignment (or random permutation for permutation CSPs) satisfies in expectation. The “above‑average” parameterization asks whether there exists an assignment satisfying at least ρ·m + k constraints, where k is the parameter. This formulation captures many well‑studied problems such as Max‑c‑Sat, Set‑Splitting, Betweenness, and Max Acyclic Subgraph, and it aligns with known hardness results that forbid approximation beyond ρ + ε under standard complexity assumptions (P ≠ NP, UGC).

The authors first focus on systems of linear equations over the field F₂. They present a new fixed‑parameter tractable (FPT) algorithm that decides whether at least m/2 + k/2 equations can be simultaneously satisfied. The algorithm proceeds by converting the equation system into a bipartite graph, applying a series of reduction rules that remove variables with low “influence,” and then using minimum‑cut and maximum‑matching techniques to obtain a kernel whose size is O(k) variables – a so‑called bikernel. Because the kernel contains only O(k) variables, exhaustive search runs in time 2^{O(k)}. This improves on earlier kernels that were quadratic in k or larger.

The kernelization technique is then lifted to general Boolean CSPs of fixed arity c. Each constraint is expressed as a small collection of XOR or 2‑SAT clauses; the same reduction machinery yields an O(k)‑variable bikernel for the whole instance. Consequently, any Boolean CSP parameterized above its random‑assignment threshold admits a linear‑size kernel and a 2^{O(k)}‑time exact algorithm. The same ideas apply to permutation CSPs of arity three, where the permutation constraints are modeled as cycle‑free conditions in a directed graph; again a linear bikernel is obtained.

With these kernel results the paper derives concrete consequences for several classic problems. For Max‑c‑Sat, the random assignment satisfies a 1 − 2^{‑c} fraction of clauses; the authors show that deciding whether one can satisfy (1 − 2^{‑c})·m + k clauses is FPT with a linear kernel. Similar statements hold for Set‑Splitting (ρ = 1/2), Betweenness (ρ = 1/3), and Max Acyclic Subgraph (ρ = 1/2). All these problems therefore admit 2^{O(k)}‑time algorithms when parameterized above average.

Beyond exact algorithms, the paper introduces a hybrid approach for every Max‑c‑CSP. Given an instance I, the algorithm first checks whether I possesses structural properties that allow a polynomial‑time algorithm to beat the random‑assignment bound (i.e., achieve a (ρ + ε)‑approximation). If such a property is detected, the algorithm returns a better‑than‑random solution in polynomial time. Otherwise, the instance falls into the “hard” regime, and the previously described FPT algorithm is invoked to compute an optimal solution in subexponential time (specifically 2^{o(n)}). This dichotomy yields, for the first time, a general method that either improves the approximation ratio or solves the instance exactly in subexponential time, matching known hardness thresholds under the Unique Games Conjecture.

In summary, the paper makes three major contributions: (1) a faster FPT algorithm and linear‑size bikernel for solving systems of equations over F₂ above the random‑assignment threshold; (2) a systematic transfer of this kernelization to Boolean and permutation CSPs of constant arity, leading to linear bikernels and 2^{O(k)}‑time exact algorithms for a wide range of “above‑average” parameterizations; and (3) a novel hybrid algorithmic framework that either yields a polynomial‑time approximation surpassing the random baseline or computes an optimal solution in subexponential time. These results deepen the connection between parameterized complexity, kernelization, and approximation theory, and they provide practical tools for tackling CSPs where the natural random bound is the only known approximation guarantee.