THE REDUCTS OF THE HOMOGENEOUS BINARY BRANCHING C -RELATION

ACM Trans. Comput. Logic

Jönsson

Pham

2017

Self Cite

We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains for example the rooted triple consistency problem, forbidden subtree problems, the quartet consistency problem, and many other problems studied in the bioinformatics literature. The studied problems can be described as constraint satisfaction problems where the constraints have a first-order definition over the rooted triple relation. We show that every such phylogeny problem can be solved in polynomial time or is NP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithm of Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithm repeatedly solves linear equation systems to construct a solution in polynomial time. We then show that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Our classification establishes a dichotomy for a large class of infinite structures that we believe is of independent interest in universal algebra, model theory, and topology. The proof of our main result combines results and techniques from various research areas: a recent classification of the model-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb's Ramsey theorem for rooted trees, and universal algebra.

Section: Affine Tree Operationsmentioning

confidence: 72%

mentioning

confidence: 97%

The Complexity of Phylogeny Constraint Satisfaction Problems

ACM Trans. Comput. Logic

Jönsson

Pham

2017

Self Cite

“…We write (L 2 ; C) for the structure induced in (S 2 ; C) by any maximal antichain of (S 2 ; ≤). It is straightforward to verify that (L 2 ; C) satisfies the axioms C1-C8 given in [BJP16], and hence is isomorphic to the homogeneous binary branching C-relation on leaves which is also denoted by (L 2 ; C) in [BJP16] (see Lemma 3.8 in [BJP16]). The reducts of (L 2 ; C) were classified in [BJP16].…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…The reducts of (L 2 ; C) have been classified in [BJP16]. Each reduct of (L 2 ; C) is interdefinable with either…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

The universal homogeneous binary tree

Journal of Logic and Computation

Bradley-Williams

Pinsker

et al. 2018

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A partial order is called semilinear if the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by (S2; ≤). We study the reducts of (S2; ≤), that is, the relational structures with domain S2, all of whose relations are first-order definable in (S2; ≤). Our main result is a classification of the model-complete cores of the reducts of S2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all subgroups of the full symmetric group on S2 that contain the automorphism group of (S2; ≤) and are closed with respect to the pointwise convergence topology.

“…Furthermore, all polynomial-time tractable first-order expansions of (Q; <) are known (see [13]) and there exist tractable expansions which are not convex. One such structure is (Q; =, ≤, R mi ) with Let C be the binary branching homogeneous C-relation on a countably infinite set [10]. The computational complexity of the CSPs of all first-order expansions of this structure has been fully classified in [11].…”

Section: Examplesmentioning

confidence: 99%

Complexity of Combinations of Qualitative Constraint Satisfaction Problems

Greiner

2018

Automated Reasoning

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The CSP of a first-order theory T is the problem of deciding for a given finite set S of atomic formulas whether T ∪ S is satisfiable. Let T 1 and T 2 be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomialtime decidability) of CSP(T 1 ∪ T 2 ) under the assumption that CSP(T 1 ) and CSP(T 2 ) are decidable (or polynomial-time decidable). We show that for a large class of ω-categorical theories T 1 , T 2 the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of CSP(T 1 ∪ T 2 ) (unless P=NP).