The class QMA(k), introduced by , consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
Third, Blier and Tapp @cite_17 recently (and independently of us) gave an interesting @math protocol for an @math -complete problem, namely @math . In this protocol, Arthur verifies that an @math -vertex graph @math is @math -colorable, using two unentangled witnesses with only @math qubits each. There is a crucial caveat, though: if @math is @math -colorable, then Arthur can only detect this with probability @math rather than constant probability. Indeed, if the soundness gap were constant rather than @math , then Blier and Tapp's protocol could presumably be scaled up by an exponential to show @math !
@cite_17
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We study a one-parameter family of discrete dynamical systems called the @math -color firefly cellular automata (FCAs), which were introduced recently by the author. At each discrete time @math , each vertex in a graph has a state in @math , and a special state @math is designated as the blinking' state. At step @math , simultaneously for all vertices, the state of a vertex increments from @math to @math unless @math and at least one of its neighbors is in the state @math . A central question about this system is that on what class of network topologies synchrony is guaranteed to emerge. In a previous work, we have shown that for @math , every @math -coloring on a finite tree synchronizes iff the maximum degree is less than @math , and asked whether this behavior holds for all @math . In this paper, we answer the question positively for @math and negatively for all @math by constructing counterexamples on trees with maximum degree at most @math . Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. In this paper, we provide a detailed comparison between a fully randomized protocol for rumor spreading on a complete graph and a quasirandom protocol introduced by Doerr, Friedrich, and Sauerwald [Quasirandom rumor spreading, in Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2008, pp. 773-781]. In the former, initially there is one vertex which holds a piece of information, and during each round every one of the informed vertices chooses uniformly at random and independently one of its neighbors and informs it. In the quasirandom version of this method (cf. Doerr, Friedrich, and Sauerwald) each vertex has a cyclic list of its neighbors. Once a vertex has been informed, it chooses uniformly at random only one neighbor. In the following round, it informs this neighbor, and at each subsequent round it picks the next neighbor from its list and informs it. We give a precise analysis of the evolution of the quasirandom protocol on the complete graph with @math vertices and show that it evolves essentially in the same way as the randomized protocol. In particular, if @math denotes the number of rounds that are needed until all vertices are informed, we show that for any slowly growing function @math , we have @math , with probability @math . Suppose in a graph G vertices can be either red or blue. Let k be odd. At each time step, each vertex v in G polls k random neighbours and takes the majority colour. If it does not have k neighbours, it simply polls all of them, or all less one if the degree of v is even. We study this protocol on graphs of a given degree sequence, in the following setting: initially each vertex of G is red independently with probability α < 1 2 , and is otherwise blue. We show that if α is sufficiently biased, then with high probability consensus is reached on the initial global majority within O ( log k log k n ) steps if 5 ? k ? d , and O ( log d log d n ) steps if k d . Here, d ? 5 is the effective minimum degree, the smallest integer which occurs ? ( n ) times in the degree sequence. We further show that on such graphs, any local protocol in which a vertex does not change colour if all its neighbours have that same colour, takes time at least ? ( log d log d n ) , with high probability. Additionally, we demonstrate how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erd?s-Renyi random graphs in the connected regime. We give a new proof showing that it is NP-hard to color a 3-colorable graph using just 4 colors. This result is already known , [S. Khanna, N. Linial, and S. Safra, Combinatorica, 20 (2000), pp. 393--415], but our proof is novel because it does not rely on the PCP theorem, while the known one does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor @math hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [M. Bellare, O. Goldreich, and M. Sudan, SIAM J. Comput., 27 (1998), pp. 805--915]. Another aspect in which our proof is novel is in its use of the PCP theorem to show that 4-coloring of 3-colorable graphs remains NP-hard even on bounded-degree graphs (this hardness result does not seem to follow from the earlier reduction of Khanna, Linial, and Safra). We point out that such graphs can always be colored using O(1) colors by a simple greedy algorithm, while the best known algorithm for coloring (general) 3-colorable graphs requires @math colors. Our proof technique also shows that there is an @math such that it is NP-hard to legally 4-color even a @math fraction of the edges of a 3-colorable graph. We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph @math on @math vertices has a rainbow cycle on at least @math vertices, by showing that @math has a rainbow cycle on at least @math vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least @math different colors appear. For large @math , this is an improvement of the previous best known lower bound of @math of Andersen. When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph @math . In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class @math . Builder wins the game @math if a monochromatic copy of @math can be forced. The on-line degree Ramsey number @math is the least @math such that Builder wins @math when @math is the class of graphs with maximum degree at most @math . Our results include: 1) @math if and only if @math is a linear forest or each component lies inside @math . 2) @math , where @math . 3) @math for a tree @math , where @math and @math are two largest vertex degrees. 4) @math , with @math except for finitely many odd values of @math . 5) @math when @math . The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently". Consider an n-vertex graph G = (V, E) of maximum degree Δ, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, that is, it proceeds in discrete rounds. In the distributed vertex coloring problem, the objective is to color G with Δ + 1, or slightly more than Δ + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years [ 1986; Luby 1986]. Specifically, these algorithms produce a (Δ + 1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires polylogarithmic time employs O(Δ2) colors. This algorithm was devised in a seminal FOCS’87 paper by Linial [1987]. Its running time is O(log* n). In the same article, Linial asked whether one can color with significantly less than Δ2 colors in deterministic polylogarithmic time. By now, this question of Linial became one of the most central long-standing open questions in this area. In this article, we answer this question in the affirmative, and devise a deterministic algorithm that employs Δ1+o(1) colors, and runs in polylogarithmic time. Specifically, the running time of our algorithm is O(f(Δ)log Δ log n), for an arbitrarily slow-growing function f(Δ) = ω(1). We can also produce an O(Δ1+η)-coloring in O(log Δ log n)-time, for an arbitrarily small constant η > 0, and an O(Δ)-coloring in O(Δe log n) time, for an arbitrarily small constant e > 0. Our results are, in fact, far more general than this. In particular, for a graph of arboricity a, our algorithm produces an O(a1+η)-coloring, for an arbitrarily small constant η > 0, in time O(log a log n). For a 3-colourable graph G, the 3-colour graph of G, denoted C3(G), is the graph with node set the proper vertex 3-colourings of G, and two nodes adjacent whenever the corresponding colourings differ on precisely one vertex of G. We consider the following question : given G, how easily can we decide whether or not C3(G) is connected?We show that the 3-colour graph of a 3-chromatic graph is never connected, and characterise the bipartite graphs for which C3(G) is connected. We also show that the problem of deciding the connectedness of the 3-colour graph of a bipartite graph is coNP-complete, but that restricted to planar bipartite graphs, the question is answerable in polynomial time. We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of @math -dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties. The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family @math of forbidden vertex-ordered graphs, and any @math , there exist @math and @math so that any vertex-ordered graph which is @math -far from being @math -free contains at least @math copies of some @math (with the correct vertex order) where @math . The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for @math -partite graphs with "undesirable" edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph. We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with @math colours, by prior work it is known that we can find a proper 3-colouring in @math communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many @math the time complexity is precisely @math communication rounds. Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p @math c n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p @math polylog(n) n [J. Algorithms, 19 (1995), pp. 204--234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c. We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property @math is called a @math -graph . If @math satisfies certain properties, then an n-node m-edge @math -graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most @math bits for any continuous superadditive function @math so that there are at most @math distinct @math -node @math -graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such @math include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) @math is directed or undirected; (3) @math is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from @math for @math ; (5) the edges of G are labeled with labels from @math for @math ; and (6) each node (respectively, edge) of G has at most @math self-loops (respectively, @math multiple edges). Moreover, @math and @math are not required to be O(1) for the cases of @math being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes. We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol. Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random @math -SAT or @math -NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph @math -coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of @math -colorings, allows us to establish a substantially improved lower bound on the @math -colorability threshold. The new lower bound is within an additive @math of a simple first-moment upper bound and within @math of the physics conjecture. By comparison, the best previous lower bound left a gap of about @math , unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. In this paper we describe and analyze an improved algorithm for deciding the 3-Colourability problem. If G is a simple graph on n vertices then we will show that this algorithm tests a graph for 3-Colourability, i.e. an assignment of three colours to the vertices of G such that two adjacent vertices obtain different colours, in less than O(1.415n) steps. We consider the problem of @math -colouring a random @math -uniform hypergraph with @math vertices and @math edges, where @math , @math , @math remain constant as @math tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case @math , must have one of two easily computable values as @math tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs. We provide the first two-party protocol allowing Alice and Bob to evaluate privately even against active adversaries any completely positive, trace-preserving map @math @math , given as a quantum circuit, upon their joint quantum input state @math . Our protocol leaks no more to any active adversary than an ideal functionality for @math provided Alice and Bob have the cryptographic resources for active secure two-party classical computation. Our protocol is constructed from the protocol for the same task secure against specious adversaries presented in [4]. Consider the following probabilistic one-player game: The board is a graph with @math vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of @math available colors to each edge immediately, where @math is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph @math has been created, and Painter's goal is to “survive” for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are chosen not randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number @math , the ratio of edges to vertices in all subgraphs of the evolving board ne... Given a 3-colorable graph G together with two proper vertex 3-colorings α and β of G, consider the following question: is it possible to transform α into β by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, α, β where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolorings does exist, the algorithm uses O(|V(G)|2) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances G, α, β that require Ω(|V(G)|2) recoloring steps. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 69-82, 2011 © 2011 Wiley Periodicals, Inc.