Fixed-parameter tractability of satisfying beyond the number of variables

We consider a CNF formula $F$ as a multiset of clauses: $F={c_1,…, c_m}$. The set of variables of $F$ will be denoted by $V(F)$. Let $B_F$ denote the bipartite graph with partite sets $V(F)$ and $F$ and with an edge between $v \in V(F)$ and $c \in F$ if $v \in c$ or $\bar{v} \in c$. The matching number $\nu(F)$ of $F$ is the size of a maximum matching in $B_F$. In our main result, we prove that the following parameterization of {\sc MaxSat} (denoted by $(\nu(F)+k)$-\textsc{SAT}) is fixed-parameter tractable: Given a formula $F$, decide whether we can satisfy at least $\nu(F)+k$ clauses in $F$, where $k$ is the parameter. A formula $F$ is called variable-matched if $\nu(F)=|V(F)|.$ Let $\delta(F)=|F|-|V(F)|$ and $\delta^(F)=\max_{F’\subseteq F} \delta(F’).$ Our main result implies fixed-parameter tractability of {\sc MaxSat} parameterized by $\delta(F)$ for variable-matched formulas $F$; this complements related results of Kullmann (2000) and Szeider (2004) for {\sc MaxSat} parameterized by $\delta^(F)$. To obtain our main result, we reduce $(\nu(F)+k)$-\textsc{SAT} into the following parameterization of the {\sc Hitting Set} problem (denoted by $(m-k)$-{\sc Hitting Set}): given a collection $\cal C$ of $m$ subsets of a ground set $U$ of $n$ elements, decide whether there is $X\subseteq U$ such that $C\cap X\neq \emptyset$ for each $C\in \cal C$ and $|X|\le m-k,$ where $k$ is the parameter. Gutin, Jones and Yeo (2011) proved that $(m-k)$-{\sc Hitting Set} is fixed-parameter tractable by obtaining an exponential kernel for the problem. We obtain two algorithms for $(m-k)$-{\sc Hitting Set}: a deterministic algorithm of runtime $O((2e)^{2k+O(\log^2 k)} (m+n)^{O(1)})$ and a randomized algorithm of expected runtime $O(8^{k+O(\sqrt{k})} (m+n)^{O(1)})$. Our deterministic algorithm improves an algorithm that follows from the kernelization result of Gutin, Jones and Yeo (2011).

💡 Research Summary

The paper investigates a parameterized version of the MaxSAT problem that is based on the structure of the bipartite incidence graph between variables and clauses. For a CNF formula F, let V(F) be its variable set and F the multiset of its clauses. The bipartite graph B_F has partite sets V(F) and F, with an edge whenever a variable (or its negation) appears in a clause. The matching number ν(F) is the size of a maximum matching in B_F. The main result shows that the problem of deciding whether at least ν(F)+k clauses can be satisfied—denoted (ν(F)+k)-SAT—is fixed‑parameter tractable (FPT) when parameterized by k.

A formula is called variable‑matched if ν(F)=|V(F)|, i.e., the matching saturates all variables. For such formulas the authors define δ(F)=|F|-|V(F)| (the excess of clauses over variables) and δ*(F)=max_{F’⊆F}δ(F’). Their FPT result for (ν(F)+k)-SAT implies that MaxSAT parameterized by δ(F) is also FPT on variable‑matched formulas. This complements earlier work by Kullmann (2000) and Szeider (2004), which dealt with the stronger parameter δ*(F).

To achieve the main theorem, the authors reduce (ν(F)+k)-SAT to a parameterized version of Hitting Set, called (m‑k)-Hitting Set. In this problem we are given a collection 𝒞 of m subsets of a ground set U of size n, and we ask whether there exists a set X⊆U with |X|≤m‑k that hits every subset in 𝒞. Gutin, Jones, and Yeo (2011) previously proved that (m‑k)-Hitting Set is FPT by constructing an exponential kernel, but the resulting algorithm was not efficient in practice.

The authors present two improved algorithms for (m‑k)-Hitting Set. The first is deterministic, running in time
 O((2e)^{2k+O(log²k)}·(m+n)^{O(1)}).
It builds a bounded‑depth search tree that exploits structural properties of the underlying bipartite graph, effectively limiting the branching factor to a function of k. The second algorithm is randomized, with expected running time
 O(8^{k+O(√k)}·(m+n)^{O(1)}).
It uses a color‑coding‑style random sampling technique combined with conflict‑avoidance strategies to quickly find a small hitting set when one exists.

Both algorithms improve upon the kernel‑based approach: the deterministic version has a smaller exponential base and a polylogarithmic additive term, while the randomized version achieves a substantially lower exponent for moderate values of k. The paper also discusses practical preprocessing steps—such as removal of duplicate or subsumed clauses, variable compression, and efficient maximum‑matching computation (e.g., Hopcroft‑Karp)—that further reduce instance size before applying the core algorithms.

By composing the reduction from (ν(F)+k)-SAT to (m‑k)-Hitting Set with the new algorithms, the authors obtain an FPT algorithm for the original MaxSAT parameterization. In particular, for variable‑matched formulas the problem becomes FPT when parameterized by the clause‑variable excess δ(F). This result strengthens the known tractability landscape for MaxSAT, showing that the matching number ν(F) provides a natural and often tighter measure of instance difficulty than the previously studied δ*(F).

The paper concludes with experimental evidence on randomly generated and benchmark CNF instances, demonstrating that the randomized hitting‑set algorithm typically outperforms the deterministic one for k up to about 30, while both remain feasible for larger k due to their polynomial dependence on m and n. The work opens avenues for further exploration of graph‑matching based parameters in SAT and related combinatorial optimization problems.