20 research outputs found
No distributed quantum advantage for approximate graph coloring
We give an almost complete characterization of the hardness of -coloring
-chromatic graphs with distributed algorithms, for a wide range of models
of distributed computing. In particular, we show that these problems do not
admit any distributed quantum advantage. To do that: 1) We give a new
distributed algorithm that finds a -coloring in -chromatic graphs in
rounds, with . 2) We prove that any distributed
algorithm for this problem requires rounds.
Our upper bound holds in the classical, deterministic LOCAL model, while the
near-matching lower bound holds in the non-signaling model. This model,
introduced by Arfaoui and Fraigniaud in 2014, captures all models of
distributed graph algorithms that obey physical causality; this includes not
only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL,
even with a pre-shared quantum state.
We also show that similar arguments can be used to prove that, e.g.,
3-coloring 2-dimensional grids or -coloring trees remain hard problems even
for the non-signaling model, and in particular do not admit any quantum
advantage. Our lower-bound arguments are purely graph-theoretic at heart; no
background on quantum information theory is needed to establish the proofs
The Complexity of Distributed Edge Coloring with Small Palettes
The complexity of distributed edge coloring depends heavily on the palette
size as a function of the maximum degree . In this paper we explore the
complexity of edge coloring in the LOCAL model in different palette size
regimes.
1. We simplify the \emph{round elimination} technique of Brandt et al. and
prove that -edge coloring requires
time w.h.p. and time deterministically, even on trees.
The simplified technique is based on two ideas: the notion of an irregular
running time and some general observations that transform weak lower bounds
into stronger ones.
2. We give a randomized edge coloring algorithm that can use palette sizes as
small as , which is a natural barrier for
randomized approaches. The running time of the algorithm is at most
, where is the complexity of a
permissive version of the constructive Lovasz local lemma.
3. We develop a new distributed Lovasz local lemma algorithm for
tree-structured dependency graphs, which leads to a -edge
coloring algorithm for trees running in time. This algorithm
arises from two new results: a deterministic -time LLL algorithm for
tree-structured instances, and a randomized -time graph
shattering method for breaking the dependency graph into independent -size LLL instances.
4. A natural approach to computing -edge colorings (Vizing's
theorem) is to extend partial colorings by iteratively re-coloring parts of the
graph. We prove that this approach may be viable, but in the worst case
requires recoloring subgraphs of diameter . This stands
in contrast to distributed algorithms for Brooks' theorem, which exploit the
existence of -length augmenting paths
Approximating the Orthogonality Dimension of Graphs and Hypergraphs
A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) = Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming
Linear Index Coding via Semidefinite Programming
In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the
goal is to broadcast an n bit word to n receivers (one bit per receiver), where
the receivers have side information represented by a graph G. The objective is
to minimize the length of a codeword sent to all receivers which allows each
receiver to learn its bit. For linear index coding, the minimum possible length
is known to be equal to a graph parameter called minrank (Bar-Yossef et al.,
FOCS, 2006).
We show a polynomial time algorithm that, given an n vertex graph G with
minrank k, finds a linear index code for G of length ,
where f(k) depends only on k. For example, for k=3 we obtain f(3) ~ 0.2574. Our
algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani
and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to
Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a
relaxation of the minimization problem we consider, a crucial component of our
analysis is an upper bound on the objective value of the SDP in terms of the
minrank.
At the heart of our analysis lies a combinatorial result which may be of
independent interest. Namely, we show an exact expression for the maximum
possible value of the Lovasz theta-function of a graph with minrank k. This
yields a tight gap between two classical upper bounds on the Shannon capacity
of a graph.Comment: 24 page
Conditional Hardness for Approximate Coloring
We study the coloring problem: Given a graph G, decide whether
or , where c(G) is the chromatic number of G. We derive conditional
hardness for this problem for any constant . For , our
result is based on Khot's 2-to-1 conjecture [Khot'02]. For , we base our
hardness result on a certain `fish shaped' variant of his conjecture.
We also prove that the problem almost coloring is hard for any constant
\eps>0, assuming Khot's Unique Games conjecture. This is the problem of
deciding for a given graph, between the case where one can 3-color all but a
\eps fraction of the vertices without monochromatic edges, and the case where
the graph contains no independent set of relative size at least \eps.
Our result is based on bounding various generalized noise-stability
quantities using the invariance principle of Mossel et al [MOO'05]
Online edge coloring algorithms via the nibble method
Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that an online (1 + o(1))Δ-edge-coloring algorithm exists for n-node graphs of maximum degree Δ = ω(log n). This conjecture remains open in general, though it was recently proven for bipartite graphs under one-sided vertex arrivals by Cohen et al. [FOCS'19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model.
Our main result is in the random-order online model. For this model, known results fall short of the Bar-Noy et al. conjecture, either in the degree bound [Aggarwal et al. FOCS'03], or number of colors used [Bahmani et al. SODA'10]. We achieve the best of both worlds, thus resolving the Bar-Noy et al. conjecture in the affirmative for this model.
Our second result is in the adversarial online (and dynamic) model with recourse. A recent algorithm of Duan et al. [SODA'19] yields a (1 + ϵ) Δ-edge-coloring with poly(logn/ϵ) recourse. We achieve the same with poly(1/ϵ) recourse, thus removing all dependence on n.
Underlying our results is one common offline algorithm, which we show how to implement in these two online models. Our algorithm, based on the Rödl Nibble Method, is an adaptation of the distributed algorithm of Dubhashi et al. [TCS'98]. The Nibble Method has proven successful for distributed edge coloring. We display its usefulness in the context of online algorithms