arXiv subjects

Arash Rafiey

Publications and source records attributed to Arash Rafiey.

24 records · Page 2Linked to original sources

arXiv2013

Space complexity of list H-colouring: a dichotomy

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is hard for NL. Our algorithm operates by reducing the lists in a complex manner based on a novel decomposition of an auxiliary digraph, combined with repeated applications of Reingold's algorithm for undirected reachability (2005). We also prove an algebraic version of this dichotomy: the digraphs without a circular N are precisely those that admit a finite chain of polymorphisms satisfying the Hagemann-Mitschke identities. This confirms a conjecture of Larose and Tesson (2007) for LHOM(H). Moreover, we show that the presence of a circular N can be decided in time polynomial in the size of H.

cs.CC↗

arXiv2010

The Dichotomy of List Homomorphisms for Digraphs

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.

cs.CC↗

arXiv2009

Duality for Min-Max Orderings and Dichotomy for Min Cost Homomorphisms

Min-Max orderings correspond to conservative lattice polymorphisms. Digraphs with Min-Max orderings have polynomial time solvable minimum cost homomorphism problems. They can also be viewed as digraph analogues of proper interval graphs and bigraphs. We give a forbidden structure characterization of digraphs with a Min-Max ordering which implies a polynomial time recognition algorithm. We also similarly characterize digraphs with an extended Min-Max ordering, and we apply this characterization to prove a conjectured form of dichotomy for minimum cost homomorphism problems.

cs.DM↗

arXiv2007

Minimum Cost Homomorphisms to Reflexive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$ such that $uv\in A(G)$ implies $f(u)f(v)\in A(H)$. If moreover each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of a homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph $H$, the {\em minimum cost homomorphism problem} for $H$, denoted MinHOM($H$), is the following problem. Given an input digraph $G$, together with costs $c_i(u)$, $u\in V(G)$, $i\in V(H)$, and an integer $k$, decide if $G$ admits a homomorphism to $H$ of cost not exceeding $k$. We focus on the minimum cost homomorphism problem for {\em reflexive} digraphs $H$ (every vertex of $H$ has a loop). It is known that the problem MinHOM($H$) is polynomial time solvable if the digraph $H$ has a {\em Min-Max ordering}, i.e., if its vertices can be linearly ordered by $<$ so that $i<j, s<r$ and $ir, js \in A(H)$ imply that $is \in A(H)$ and $jr \in A(H)$. We give a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering; our characterization implies a polynomial time test for the existence of a Min-Max ordering. Using this characterization, we show that for a reflexive digraph $H$ which does not admit a Min-Max ordering, the minimum cost homomorphism problem is NP-complete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs.

cs.DM↗

arXiv2006

Hamilton Cycles in Digraphs of Unitary Matrices

A set $S\subseteq V$ is called an {\em $q^+$-set} ({\em $q^-$-set}, respectively) if $S$ has at least two vertices and, for every $u\in S$, there exists $v\in S, v\neq u$ such that $N^+(u)\cap N^+(v)\neq \emptyset$ ($N^-(u)\cap N^-(v)\neq \emptyset$, respectively). A digraph $D$ is called {\em s-quadrangular} if, for every $q^+$-set $S$, we have $|\cup \{N^+(u)\cap N^+(v): u\neq v, u,v\in S\}|\ge |S|$ and, for every $q^-$-set $S$, we have $|\cup \{N^-(u)\cap N^-(v): u,v\in S)\}\ge |S|$. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.

math.CO↗

arXiv2006

Mediated Digraphs and Quantum Nonlocality

A digraph D=(V,A) is mediated if, for each pair x,y of distinct vertices of D, either xy belongs to A or yx belongs to A or there is a vertex z such that both xz,yz belong to A. For a digraph D, DELTA(D) is the maximum in-degree of a vertex in D. The "nth mediation number" mu(n) is the minimum of DELTA(D) over all mediated digraphs on n vertices. Mediated digraphs and mu(n) are of interest in the study of quantum nonlocality. We obtain a lower bound f(n) for mu(n) and determine infinite sequences of values of n for which mu(n)=f(n) and mu(n)>f(n), respectively. We derive upper bounds for mu(n) and prove that mu(n)=f(n)(1+o(1)). We conjecture that there is a constant c such that mu(n)=<f(n)+c. Methods and results of graph theory, design theory and number theory are used.

math.CO↗