Recoloring Interval Graphs with Limited Recourse Budget

We consider the problem of coloring an interval graph dynamically. Intervals arrive one after the other and have to be colored immediately such that no two intervals of the same color overlap. In each step only a limited number of intervals may be recolored to maintain a proper coloring (thus interpolating between the well-studied online and offline settings). The number of allowed recolorings per step is the so-called recourse budget. Our main aim is to prove both upper and lower bounds on the required recourse budget for interval graphs, given a bound on the allowed number of colors. For general interval graphs with n vertices and chromatic number k it is known that some recoloring is needed even if we have 2k colors available. We give an algorithm that maintains a 2k-coloring with an amortized recourse budget of $\u1d4aa(log n)$. For maintaining a k-coloring with k ≤ n, we give an amortized upper bound of \u1d4aa(k⋅ k! ⋅ √n), and a lower bound of $Ω(k) for k ∈ \u1d4aa(√n)$, which can be as large as $Ω(√n$). For unit interval graphs it is known that some recoloring is needed even if we have k+1 colors available. We give an algorithm that maintains a (k+1)-coloring with at most $\u1d4aa(k²)$ recolorings per step in the worst case. We also give a lower bound of $Ω(log n)$ on the amortized recourse budget needed to maintain a k-coloring. Additionally, for general interval graphs we show that if one does not insist on maintaining an explicit coloring, one can have a k-coloring algorithm which does not incur a factor of $\u1d4aa(k ⋅ k! ⋅ √n)$ in the running time. For this we provide a data structure, which allows for adding intervals in $\u1d4aa(k² log³ n)$ amortized time per update and querying for the color of a particular interval in $\u1d4aa(log n) time$. Between any two updates, the data structure answers consistently with some optimal coloring. The data structure maintains the coloring implicitly, so the notion of recourse budget does not apply to it

BILANGAN KETERHUBUNGAN PELANGI PADA PEWARNAAN-SISI GRAF

Let  be a graph. An edge-coloring of  is a function , where  is a set of colors. Respect to  a subgraph  of  is called a rainbow subgraph if all edges of  get different colors. Graph  is called rainbow connected if for every two distinct vertices of  is joined by a rainbow path. The rainbow connection number of , denoted by , is the minimum number of colors needed in coloring all edges of  such that  is a rainbow connected. The main problem considered in this thesis is determining the rainbow connection number of graph. In this thesis, we determine the exact value of the rainbow connection number of some classes of graphs such as Cycles, Complete graph, and Tree. We also determining the lower bound and upper bound for the rainbow connection number of graph. Keywords: Rainbow Connection Number, Graph, Edge-Coloring on Graph. &nbsp

Distributed ∆-Coloring Plays Hide-and-Seek

We prove several new tight or near-tight distributed lower bounds for classic symmetry breaking problems in graphs. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a ∆-coloring on ∆-regular trees requires Omega(log_∆ n) rounds and any randomized such algorithm requires Omega(log_∆ log n) rounds. We prove this by showing that a natural relaxation of the ∆-coloring problem is a fixed point in the round elimination framework. As a first application, we show that our ∆-coloring lower bound proof directly extends to arbdefective colorings. An arbdefective c-coloring of a graph G=(V,E) is given by a c-coloring of V and an orientation of E, where the arbdefect of a color i is the maximum number of monochromatic outgoing edges of any node of color i. We exactly characterize which variants of the arbdefective coloring problem can be solved in O(f(∆) + log* n) rounds, for some function f, and which of them instead require Omega(log_∆ n) rounds for deterministic algorithms and Omega(log_∆ log n) rounds for randomized ones. As a second application, which we see as our main contribution, we use the structure of the fixed point as a building block to prove lower bounds as a function of ∆ for problems that, in some sense, are much easier than ∆-coloring, as they can be solved in O(log* n) deterministic rounds in bounded-degree graphs. More specifically, we prove lower bounds as a function of ∆ for a large class of distributed symmetry breaking problems, which can all be solved by a simple sequential greedy algorithm. For example, we obtain novel results for the fundamental problem of computing a (2,β)-ruling set, i.e., for computing an independent set S ⊆ V such that every node v ∈ V is within distance ≤ β of some node in S. We in particular show that Omega(β∆^{1/β}) rounds are needed even if initially an O(∆)-coloring of the graph is given. With an initial O(∆)-coloring, this lower bound is tight and without, it still nearly matches the existing O(β∆^{2/(β+1)}+log* n) upper bound. The new (2,β)-ruling set lower bound is an exponential improvement over the best existing lower bound for the problem, which was proven in [FOCS '20]. As a special case of the lower bound, we also obtain a tight linear-in-∆ lower bound for computing a maximal independent set (MIS) in trees. While such an MIS lower bound was known for general graphs, the best previous MIS lower bounds for trees was Omega(log ∆). Our lower bound even applies to a much more general family of problems that allows for almost arbitrary combinations of natural constraints from coloring problems, orientation problems, and independent set problems, and provides a single unified proof for known and new lower bound results for these types of problems. All of our lower bounds as a function of ∆ also imply substantial lower bounds as a function of n. For instance, we obtain that the maximal independent set problem, on trees, requires Omega(log n / log log n) rounds for deterministic algorithms, which is tight

Partitioning the power set of [n][n] into CkC_k-free parts

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that $A\cap B$, $A\cap C$, and $B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.Comment: 12 page

Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

We introduce the fully-dynamic conflict-free coloring problem for a set S of intervals in R^1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. A coloring is conflict-free if for each point p contained in some interval, p is contained in an interval whose color is not shared with any other interval containing p. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Omega(log n/log log n), and that any strategy using O(1/epsilon) colors needs Omega(epsilon n^epsilon) recolorings; - a coloring strategy that uses O(log n) colors at the cost of O(log n) recolorings, and another strategy that uses O(1/epsilon) colors at the cost of O(n^epsilon/epsilon) recolorings; - stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight

Fractional Path Coloring in Bounded Degree Trees with Applications

OPTx-editorial-board=yes, OPTx-proceedings=yes, OPTx-international-audience=yesInternational audienceThis paper studies the natural linear programming relaxation of the path coloring problem. We prove constructively that finding an optimal fractional path coloring is Fixed Parameter Tractable (FPT), with the degree of the tree as parameter: the fractional coloring of paths in a bounded degree trees can be done in a time which is linear in the size of the tree, quadratic in the load of the set of paths, while exponential in the degree of the tree. We give an algorithm based on the generation of an efficient polynomial size linear program. Our algorithm is able to explore in polynomial time the exponential number of different fractional colorings, thanks to the notion of trace of a coloring that we introduce. We further give an upper bound on the cost of such a coloring in binary trees and extend this algorithm to bounded degree graphs with bounded treewidth. Finally, we also show some relationships between the integral and fractional problems, and derive a (1 + 5/3e) ~= 1.61 approximation algorithm for the path coloring problem in bounded degree trees, improving on existing results. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (WDM) optical networks