Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1004.2908

The Dichotomy of List Homomorphisms for Digraphs

Pavol Hell·Arash Rafiey
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Dichotomy Conjecture for constraint satisfaction problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction between the tractable and intractable cases. Such structural characterization is absent in Bulatov's classification, and Bulatov asked whether one can be found. We provide an answer in the case of digraphs; the technique will apply in a broader context. The key concept we introduce is that of a digraph asteroidal triple (DAT). The dichotomy then takes the following form. If a digraph H has a DAT, then the list homomorphism problem for H is NP-complete; and a DAT-free digraph H has a polynomial time solvable list homomorphism problem. DAT-free graphs can be recognized in polynomial time.

Explore related subjects

cs.CCmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pavol Hell, Arash Rafiey. 2010-04-20. The Dichotomy of List Homomorphisms for Digraphs. https://arxiv.org/abs/1004.2908

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Solving 2-MAXSAT in Polynomial Time: a Proof of $\textit{P}$ = $\textit{NP}$

By the MAXSAT problem, we are given a set $V$ of $m$ variables and a collection $C$ of $n$ clauses over $V$, i.e., a conjunctive normal form ($\textit{CNF}$) formula. We will seek a truth assignment to maximize the number of satisfied clauses in $C$. This problem is $\textit{NP}$-complete even for its restricted version, the 2-maxsat problem, by which every clause contains at most 2 literals. In this paper, we discuss an efficient algorithm to solve this problem. Its main idea is to transform the 2-maxsat problem into a related problem of maximizing satisfied conjunctions of a formula $D$ in disjunctive normal form ($\textit{DNF}$). We then represent all those truth assignments for a conjunction $d$ as a graph (called a $p$*-graph), under each of which $d$ evaluates to $\textit{true}$. In this way, our task becomes finding a maximum number of $p$*-graphs, which have a same $\textit{root-to-leaf}$ path. For this purpose, we organize all the $p$*-graphs for the conjunctions in $D$ into a trie-like structure. By exploring the structure and recursively its substructures (with each corresponding to a subgraph dynamically built up by integrating some $p$*-subgraphs), the algorithm can find a maximum set of satisfied conjunctions in $D$ in polynomial time. Its worst-case time complexity is bounded by O($n^2m^4$). This provides in fact a proof of $P$ = $\textit{NP}$.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗
arXiv2026

Euclidean SVP is NP-hard for Cyclic Lattices

We prove that exact Euclidean SVP is NP-hard under deterministic polynomial-time many-one reductions for full-rank cyclic integer lattices, equivalently full-rank ideals of $R_N:=\mathbb{Z}[X]/(X^N-1)$ in the coefficient norm. Hardness holds with $N=q-1$ for a varying odd prime $q$. As an application, we prove the same hardness for the algebraic class of NTRU-form lattices $\{(x,z)\in R_N^2:Hx\equiv z\pmod{QR_N}\}$, where $H,Q$ are unrestricted inputs. The decision problems are NP-complete, and the exact search problems are NP-hard under polynomial-time Turing reductions. No hardness claim is made for cryptographic NTRU parameter subclasses or key-generation distributions.

cs.CC↗