Automating Cutting Planes is NP-Hard}

Random CNFs are Hard for Cutting Planes

The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics; it is used as a testbench for satisfiability algorithms, and average-case hardness over this distribution has also been linked to hardness of approximation via Feige's hypothesis. We prove that any Cutting Planes refutation for random k-SAT requires exponential size, for k that is logarithmic in the number of variables, in the (interesting) regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively.

Cutting a tree with Subgraph Complementation is hard, except for some small trees

For a graph property $Π$, Subgraph Complementation to $Π$ is the problem to find whether there is a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph satisfying the property $Π$. We prove that the problem of Subgraph Complementation to $T$-free graphs is NP-Complete, for $T$ being a tree, except for 41 trees of at most 13 vertices (a graph is $T$-free if it does not contain any induced copies of $T$). This result, along with the 4 known polynomial-time solvable cases (when $T$ is a path on at most 4 vertices), …

量子自动化解$\mathbf{TC}^0$-Frege是LWE难题

Quantum Automating $\mathbf{TC}^0$-Frege Is LWE-Hard

该研究证明在LWE假设下，量子算法无法弱化自动化解$\mathbf{TC}^0$-Frege，首次将量子计算与命题证明搜索结合。

Phase Transitions of Traveling Salesperson Problems solved with Linear Programming and Cutting Planes

The Traveling Salesperson problem asks for the shortest cyclic tour visiting a set of cities given their pairwise distances and belongs to the NP-hard complexity class, which means that with all known algorithms in the worst case instances are not solveable in polynomial time, i.e., the problem is hard. Though that does not mean, that there are not subsets of the problem which are easy to solve. To examine numerically transitions from an easy to a hard phase, a random ensemble of cities in the Euclidean plane given a parameter σ, which governs the hardness, is introduced. Here, a linear progra…

Cutting Barnette graphs perfectly is hard

A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22] but its complexity was open in planar graphs and in cubic graphs. We settle both questions at once by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-…