4,142 research outputs found

    Optimal Collision/Conflict-free Distance-2 Coloring in Synchronous Broadcast/Receive Tree Networks

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    This article is on message-passing systems where communication is (a) synchronous and (b) based on the "broadcast/receive" pair of communication operations. "Synchronous" means that time is discrete and appears as a sequence of time slots (or rounds) such that each message is received in the very same round in which it is sent. "Broadcast/receive" means that during a round a process can either broadcast a message to its neighbors or receive a message from one of them. In such a communication model, no two neighbors of the same process, nor a process and any of its neighbors, must be allowed to broadcast during the same time slot (thereby preventing message collisions in the first case, and message conflicts in the second case). From a graph theory point of view, the allocation of slots to processes is know as the distance-2 coloring problem: a color must be associated with each process (defining the time slots in which it will be allowed to broadcast) in such a way that any two processes at distance at most 2 obtain different colors, while the total number of colors is "as small as possible". The paper presents a parallel message-passing distance-2 coloring algorithm suited to trees, whose roots are dynamically defined. This algorithm, which is itself collision-free and conflict-free, uses Δ+1\Delta + 1 colors where Δ\Delta is the maximal degree of the graph (hence the algorithm is color-optimal). It does not require all processes to have different initial identities, and its time complexity is O(dΔ)O(d \Delta), where d is the depth of the tree. As far as we know, this is the first distributed distance-2 coloring algorithm designed for the broadcast/receive round-based communication model, which owns all the previous properties.Comment: 19 pages including one appendix. One Figur

    On the Complexity of Distributed Splitting Problems

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    One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, polylog⁥n\text{poly}\log n-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph G=(V,E)G=(V,E) has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient polylog⁥n\text{poly}\log n-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if ÎŽ\delta and Δ\Delta are the minimum and maximum degrees of GG and if ÎŽ=Ω(log⁥n)\delta=\Omega(\log n), weak splitting can be solved deterministically in time O(Δή⋅poly(log⁥n))O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log n)\big). Further, if ÎŽ=Ω(log⁥log⁥n)\delta = \Omega(\log\log n) and Δ≀2ΔΎ\Delta\leq 2^{\varepsilon\delta}, there is a randomized algorithm with time complexity O(Δή⋅poly(log⁥log⁥n))O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log\log n)\big)

    Tomescu\u27s Graph Coloring Conjecture for -Connected Graphs

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    Let PG(k) be the number of proper k-colorings of a finite simple graph G. Tomescu\u27s conjecture, which was recently solved by Fox, He, and Manners, states that PG(k)k!(k-1)(n – k) for all connected graphs G on n vertices with chromatic number k≄4. In this paper, we study the same problem with the additional constraint that G is ℓ-connected. For 2-connected graphs G, we prove a tight bound PG(k)≀(k – 1)!((k – 1)(n – k+1) + ( - 1)n – k) and show that equality is only achieved if G is a k-clique with an ear attached. For ℓ≄3, we prove an asymptotically tight upper bound PG(k)≀k!(k-1)n-l-k+1+O((k – 2)n ) and provide a matching lower bound construction. For the ranges k≄ℓ or ℓ ≄ (k-2)(k-1)+ 1 we further find the unique graph maximizing . We also consider generalizing ℓ-connected graphs to connected graphs with minimum degree ÎŽ

    Problems in extremal graph theory

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    We consider a variety of problems in extremal graph and set theory. The {\em chromatic number} of GG, χ(G)\chi(G), is the smallest integer kk such that GG is kk-colorable. The {\it square} of GG, written G2G^2, is the supergraph of GG in which also vertices within distance 2 of each other in GG are adjacent. A graph HH is a {\it minor} of GG if HH can be obtained from a subgraph of GG by contracting edges. We show that the upper bound for χ(G2)\chi(G^2) conjectured by Wegner (1977) for planar graphs holds when GG is a K4K_4-minor-free graph. We also show that χ(G2)\chi(G^2) is equal to the bound only when G2G^2 contains a complete graph of that order. One of the central problems of extremal hypergraph theory is finding the maximum number of edges in a hypergraph that does not contain a specific forbidden structure. We consider as a forbidden structure a fixed number of members that have empty common intersection as well as small union. We obtain a sharp upper bound on the size of uniform hypergraphs that do not contain this structure, when the number of vertices is sufficiently large. Our result is strong enough to imply the same sharp upper bound for several other interesting forbidden structures such as the so-called strong simplices and clusters. The {\em nn-dimensional hypercube}, QnQ_n, is the graph whose vertex set is {0,1}n\{0,1\}^n and whose edge set consists of the vertex pairs differing in exactly one coordinate. The generalized Tur\'an problem asks for the maximum number of edges in a subgraph of a graph GG that does not contain a forbidden subgraph HH. We consider the Tur\'an problem where GG is QnQ_n and HH is a cycle of length 4k+24k+2 with k≄3k\geq 3. Confirming a conjecture of Erd{\H o}s (1984), we show that the ratio of the size of such a subgraph of QnQ_n over the number of edges of QnQ_n is o(1)o(1), i.e. in the limit this ratio approaches 0 as nn approaches infinity

    Bicoloring Random Hypergraphs

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    We study the problem of bicoloring random hypergraphs, both numerically and analytically. We apply the zero-temperature cavity method to find analytical results for the phase transitions (dynamic and static) in the 1RSB approximation. These points appear to be in agreement with the results of the numerical algorithm. In the second part, we implement and test the Survey Propagation algorithm for specific bicoloring instances in the so called HARD-SAT phase.Comment: 14 pages, 10 figure

    Streaming and Massively Parallel Algorithms for Edge Coloring

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    A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: - Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018]. - Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors
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