Dichotomy for tree-structured trigraph list homomorphism problems

Trigraph list homomorphism problems (also known as list matrix partition problems) have generated recent interest, partly because there are concrete problems that are not known to be polynomial time solvable or NP-complete. Thus while digraph list homomorphism problems enjoy dichotomy (each problem is NP-complete or polynomial time solvable), such dichotomy is not necessarily expected for trigraph list homomorphism problems. However, in this paper, we identify a large class of trigraphs for which list homomorphism problems do exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and, in particular, include all trigraphs whose underlying graphs are trees. In fact, we show that for these tree-like trigraphs, the trigraph list homomorphism problem is polynomially equivalent to a related digraph list homomorphism problem. We also describe a few examples illustrating that our conditions defining tree-like trigraphs are not unnatural, as relaxing them may lead to harder problems

Matrix partitions of perfect graphs

AbstractGiven a symmetric m by m matrix M over 0,1,*, the M-partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i=j) are non-adjacent if M(i,j)=0, and adjacent if M(i,j)=1. These matrix partition problems generalize graph colourings and homomorphisms, and arise frequently in the study of perfect graphs; example problems include split graphs, clique and skew cutsets, homogeneous sets, and joins.In this paper we study M-partitions restricted to perfect graphs. We identify a natural class of ‘normal’ matrices M for which M-partitionability of perfect graphs can be characterized by a finite family of forbidden induced subgraphs (and hence admits polynomial time algorithms for perfect graphs). We further classify normal matrices into two classes: for the first class, the size of the forbidden subgraphs is linear in the size of M; for the second class we only prove exponential bounds on the size of forbidden subgraphs. (We exhibit normal matrices of the second class for which linear bounds do not hold.)We present evidence that matrices M which are not normal yield badly behaved M-partition problems: there are polynomial time solvable M-partition problems that do not have finite forbidden subgraph characterizations for perfect graphs. There are M-partition problems that are NP-complete for perfect graphs. There are classes of matrices M for which even proving ‘dichotomy’ of the corresponding M-partition problems for perfect graphs—i.e., proving that these problems are all polynomial or NP-complete—is likely to be difficult

List homomorphism problems for signed graphs

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, of allowed images, to a fixed target signed graph $(H,\pi)$. The complexity of the similar homomorphism problem without lists (corresponding to all lists being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when $H$ is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and we illustrate this by classifying the complexity of irreflexive signed graphs in which the unicoloured edges form some simple structures, namely paths or cycles. The structure of the signed graphs in the polynomial cases is interesting, suggesting they may constitute a nice class of signed graphs analogous to the so-called bi-arc graphs (which characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable graphs based on a new conference submission (split possible in future

Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymorphisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and self-contained way

Post-intervention Status in Patients With Refractory Myasthenia Gravis Treated With Eculizumab During REGAIN and Its Open-Label Extension

OBJECTIVE: To evaluate whether eculizumab helps patients with anti-acetylcholine receptor-positive (AChR+) refractory generalized myasthenia gravis (gMG) achieve the Myasthenia Gravis Foundation of America (MGFA) post-intervention status of minimal manifestations (MM), we assessed patients' status throughout REGAIN (Safety and Efficacy of Eculizumab in AChR+ Refractory Generalized Myasthenia Gravis) and its open-label extension. METHODS: Patients who completed the REGAIN randomized controlled trial and continued into the open-label extension were included in this tertiary endpoint analysis. Patients were assessed for the MGFA post-intervention status of improved, unchanged, worse, MM, and pharmacologic remission at defined time points during REGAIN and through week 130 of the open-label study. RESULTS: A total of 117 patients completed REGAIN and continued into the open-label study (eculizumab/eculizumab: 56; placebo/eculizumab: 61). At week 26 of REGAIN, more eculizumab-treated patients than placebo-treated patients achieved a status of improved (60.7% vs 41.7%) or MM (25.0% vs 13.3%; common OR: 2.3; 95% CI: 1.1-4.5). After 130 weeks of eculizumab treatment, 88.0% of patients achieved improved status and 57.3% of patients achieved MM status. The safety profile of eculizumab was consistent with its known profile and no new safety signals were detected. CONCLUSION: Eculizumab led to rapid and sustained achievement of MM in patients with AChR+ refractory gMG. These findings support the use of eculizumab in this previously difficult-to-treat patient population. CLINICALTRIALSGOV IDENTIFIER: REGAIN, NCT01997229; REGAIN open-label extension, NCT02301624. CLASSIFICATION OF EVIDENCE: This study provides Class II evidence that, after 26 weeks of eculizumab treatment, 25.0% of adults with AChR+ refractory gMG achieved MM, compared with 13.3% who received placebo

Minimal Symptom Expression' in Patients With Acetylcholine Receptor Antibody-Positive Refractory Generalized Myasthenia Gravis Treated With Eculizumab

The efficacy and tolerability of eculizumab were assessed in REGAIN, a 26-week, phase 3, randomized, double-blind, placebo-controlled study in anti-acetylcholine receptor antibody-positive (AChR+) refractory generalized myasthenia gravis (gMG), and its open-label extension