Full Dichotomy Research Articles

Complexity Results on the Decomposition of a Digraph into Directed Linear Forests and Out-Stars

We consider two decomposition problems in directed graphs. We say that a digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its connected components contains at most $k$ arcs. For the first problem, a directed linear forest is a collection of vertex-disjoint directed paths and we consider the problem of decomposing a given digraph into a $k$-bounded and an $\ell$-bounded directed linear forest for some fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. We give a full dichotomy for this problem by showing that it can be solved in polynomial time if $k+\ell \leq 3$ and is NP-complete otherwise. This answers a question of Campbell, Hörsch, and Moore. For the second problem, we say that an out-galaxy is a vertex-disjoint collection of out-stars. Again, we give a full dichotomy of when a given digraph can be edge-decomposed into a $k$-bounded and an $\ell$-bounded out-galaxy for fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. More precisely, we show that the problem can be solved in polynomial time if $\min\{k,\ell\}\in \{1,\infty\}$ and is NP-complete otherwise.

The Complexity of Quantified Constraints: Collapsibility, Switchability, and the Algebraic Formulation

Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [ 1 ] and Zhuk [ 2 ], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture . This result becomes more significant now that the original Chen Conjecture (see [ 3 ]) is known to be false [ 4 ]. Switchability was introduced by Chen [ 1 ] as a generalization of the already-known collapsibility [ 5 ]. There, an algebra 𝔸 :=({ 0,1,2}; r ) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.

Revising Johnson’s table for the 21st century

What does it mean today to study a problem from a computational point of view? We focus on parameterized complexity and on Column 16 “Graph Restrictions and Their Effect” of D.S. Johnson’s Ongoing guide, where several puzzles were proposed in a summary table with 30 graph classes as rows and 11 problems as columns. Several of the 330 entries remain unclassified into Polynomial or NP-complete after 35 years. We provide a full dichotomy for the Steiner Tree column by proving that the problem is NP-complete when restricted to Undirected Path graphs. We revise Johnson’s summary table according to the granularity provided by the parameterized complexity for NP-complete problems.

A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation

Holant problems are a family of counting problems parameterised by sets of algebraic-complex valued constraint functions, and defined on graphs. They arise from the theory of holographic algorithms, which was originally inspired by concepts from quantum computation. Here, we employ quantum information theory to explain existing results about holant problems in a concise way and to derive two new dichotomies: one for a new family of problems, which we call Holant$^+$, and, building on this, a full dichotomy for Holant$^c$. These two families of holant problems assume the availability of certain unary constraint functions -- the two pinning functions in the case of Holant$^c$, and four functions in the case of Holant$^+$ -- and allow arbitrary sets of algebraic-complex valued constraint functions otherwise. The dichotomy for Holant$^+$ also applies when inputs are restricted to instances defined on planar graphs. In proving these complexity classifications, we derive an original result about entangled quantum states.

On small bases which admit countably many expansions

Let q∈(1,2) and x∈[0,1q−1]. We say that a sequence (ϵi)i=1∞∈{0,1}N is an expansion of x in base q (or a q-expansion) ifx=∑i=1∞ϵiq−i. Let Bℵ0 denote the set of q for which there exists x with exactly ℵ0 expansions in base q. In [5] it was shown that min⁡Bℵ0=1+52. In this paper we show that the smallest element of Bℵ0 strictly greater than 1+52 is qℵ0≈1.64541, the appropriate root of x6=x4+x3+2x2+x+1. This leads to a full dichotomy for the number of possible q-expansions for q∈(1+52,qℵ0). We also prove some general results regarding Bℵ0∩[1+52,qf], where qf≈1.75488 is the appropriate root of x3=2x2−x+1. Moreover, the techniques developed in this paper imply that if x∈[0,1q−1] has uncountably many q-expansions then the set of q-expansions for x has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case.

Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules

The Possible Winner problem asks, given an election where the votersʼ preferences over the candidates are specified only partially, whether a designated candidate can become a winner by suitably extending all the votes. Betzler and Dorn (2010) [2] proved a result that is only one step away from a full dichotomy of this problem for the important class of pure scoring rules in the case of unweighted votes and an unbounded number of candidates: Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule with vector ( 2 , 1 , … , 1 , 0 ) , but is solvable in polynomial time for plurality and veto. We take the final step to a full dichotomy by showing that Possible Winner is NP-complete also for the scoring rule with vector ( 2 , 1 , … , 1 , 0 ) .

Complexity dichotomy on partial grid recognition

Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even trivial. A natural step, outstanding thus far, was to provide a broad classification of graphs that make for polynomial or NP-complete instances. We provide such a classification based on the set of allowed vertex degrees in the input graphs, yielding a full dichotomy on the complexity of the problem. As byproducts, the previous NP-completeness result for binary trees was strengthened to strictly binary trees, and the three-dimensional version of the problem was for the first time proven to be NP-complete. Our results were made possible by introducing the concepts of consistent orientations and robust gadgets, and by showing how the former allows NP-completeness proofs by local replacement even in the absence of the latter.

The P versus NP–complete dichotomy of some challenging problems in graph theory

The Clay Mathematics Institute has selected seven Millennium Problems to motivate research on important classic questions that have resisted solution over the years. Among them is the central problem in theoretical computer science: the P versus NP problem, which aims to classify the possible existence of efficient solutions to combinatorial and optimization problems. The main goal is to determine whether there are questions whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. In this context, it is important to determine precisely what facet of a problem makes it NP–complete. We shall discuss classes of problems for which dichotomy results do exist: every problem in the class is classified into polynomial or NP–complete. We shall discuss our contribution through the classification of some long-standing problems in important areas of graph theory: perfect graphs, intersection graphs, and structural characterization of graph classes. More precisely, we have shown that Chvátal’s skew partition is polynomial and that Roberts–Spencer’s clique graph is NP–complete. We have also solved the dichotomy for Golumbic–Kaplan–Shamir’s sandwich problem. We shall describe two examples where we can determine the full dichotomy: the edge-colouring problem for graphs with no cycle with a unique chord and the three nonempty part sandwich problem. Some open problems are discussed: the stubborn problem for list partition, the chromatic index of chordal graphs, and the recognition of split clique graphs.

The complexity of the minimum cost homomorphism problem for semicomplete digraphs with possible loops

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM( H ). An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. If each vertex u ∈ V ( D ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( D ) c f ( u ) ( u ) . For each fixed digraph H , we have the minimum cost homomorphism problem for H and denote it as MinHOM( H ). The problem is to decide, for an input graph D with costs c i ( u ) , u ∈ V ( D ) , i ∈ V ( H ) , whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM( H ) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM( H ) were proved when H is a semicomplete digraph Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12,11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in Gutin and Kim (2008) [9].

Classification of Bipartite Boolean Constraint Satisfaction through Delta-Matroid Intersection

Matroid intersection has a known polynomial time algorithm using an oracle. We generalize this result to delta-matroids that do not have equality as a restriction and give a polynomial time algorithm for delta-matroid intersection on delta-matroids without equality using an oracle. We note that when equality is present, delta-matroid intersection is as general as delta-matroid parity. We also obtain algorithms using an oracle for delta-matroid parity on delta-matroids without inequality, and for delta-matroid intersection where one delta-matroid does not contain either equality or inequality, and the second delta-matroid is arbitrary. Both these results also generalize matroid intersection. The results imply a dichotomy for bipartite Boolean constraint satisfaction problems using an oracle when one of the two sides does not contain equality, leaving open cases of delta-matroid parity when both sides have equality; the results also imply a full dichotomy for $k$-partite Boolean constraint satisfaction problems for $k\geq 3$. We then discuss polynomial cases of Boolean constraint satisfaction problems with two occurrences per variable through delta-matroid parity that cannot be obtained using the oracle approach.