arXiv · 0709.2689
This paper has been withdrawn by the author, due an error in claim 1.
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Omar Kettani. 2007-09-21. An algorithm for solving the Independent Set problem. https://arxiv.org/abs/0709.2689
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For distinct integers $a$ and $b$, an $\{a,b\}$-edge-weighting assigns $a$ or $b$ to each edge and labels each vertex by the sum of its incident weights. Such a weighting is proper if adjacent vertices receive distinct labels. We prove that, for every fixed pair of distinct integers, deciding whether a proper weighting exists is NP-complete even for simple cubic planar graphs. On planar multigraphs with $m$ edges, we give an exact $2^{O(\sqrt m)}$-time algorithm and, assuming the Exponential Time Hypothesis (ETH), exclude $2^{o(\sqrt m)}$-time algorithms even for simple cubic planar graphs. As a consequence, locally irregular $2$-edge-coloring is NP-complete on simple cubic planar graphs, admits a deterministic $2^{O(\sqrt n)}$-time algorithm on $n$-vertex graphs in this class, and admits no $2^{o(\sqrt n)}$-time algorithm under ETH. For maximizing the number of edges joining vertices with distinct labels, we give a deterministic efficient polynomial-time approximation scheme (EPTAS) on planar multigraphs, a polynomial-time $1/2$-approximation on multigraphs, and APX-completeness even on simple cubic graphs. Extending a partial $\{a,b\}$-edge-weighting to a proper one is NP-complete for every fixed pair even on simple cubic planar bipartite graphs, while it is polynomial-time solvable on trees. The hardness persists even when the prescribed edges form disjoint paths of length $6$ and all edges of each path have the same prescribed weight.
In the field of constraint satisfaction problems (CSP), a clause is called redundant if its satisfaction is implied by satisfying all other clauses. An instance of CSP$(P)$ is called non-redundant if it does not contain any redundant clause. The non-redundancy (NRD) of a predicate $P$ is the maximum number of clauses in a non-redundant instance of CSP$(P)$, as a function of the number of variables $n$. Recent progress has shown that non-redundancy is crucially linked to many other important questions in computer science and mathematics including sparsification, kernelization, query complexity, universal algebra, and extremal combinatorics. Given that non-redundancy is a nexus for many of these important problems, the central goal of this paper is to more deeply understand non-redundancy. Our first main result shows that for every rational number $r \ge 1$, there exists a finite CSP predicate $P$ such that the non-redundancy of $P$ is $Θ(n^r)$. Our second main result explores the concept of conditional non-redundancy first coined by Brakensiek and Guruswami [STOC 2025]. We completely classify the conditional non-redundancy of all binary predicates (i.e., constraints on two variables) by connecting these non-redundancy problems to the structure of high-girth graphs in extremal combinatorics. Inspired by these concrete results, we build off the work of Carbonnel [CP 2022] to develop an algebraic theory of conditional non-redundancy. As an application of this algebraic theory, we revisit the notion of Mal'tsev embeddings, which is the most general technique known to date for establishing that a predicate has linear non-redundancy. For example, we provide the first example of predicate with a Mal'tsev embedding that cannot be attributed to the structure of an Abelian group, but rather to the structure of the quantum Pauli group.
The Zero Forcing (or corruption) of a graph is the problem of finding a minimum-size ``corrupting'' set. It corresponds to a subset of its vertices that can corrupt the whole graph by iterating the following rule: if a corrupted vertex has exactly one neighbor that is not yet corrupted, the neighbor gets corrupted. The iteration of this process comes from the fact that the corruption of a vertex might enable new corruptions (from itself or some of its neighbors). For this reason, one can consider a step of corruption, where all the possible instances of the corruption rule are applied at once. This paper investigates Zero Forcing on temporal graphs, where the topology of the graph evolves throughout the experiment. At each time step (or snapshot) of the graph, a step of corruption is resolved wherever possible. We study the problem of finding a minimum-size corrupting set such that the whole (temporal) graph is corrupted at the end of the experiment. We present a panorama of results, including NP-hardness in some not-so-restrictive scenarios, polynomial algorithms, and a solution to an open question when the whole graph must be corrupted in a single step.