COMP Online Algorithms. Online Graph Problems. Shahin Kamali. Lecture 23 - Nov. 28th, 2017 University of Manitoba
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1 COMP Online Algorithms Online Graph Problems Shahin Kamali Lecture 23 - Nov. 28th, 2017 University of Manitoba COMP Online Algorithms Online Graph Problems 1 / 13
2 Review & Plan COMP Online Algorithms Online Graph Problems 1 / 13
3 Today s plan Online Edge-coloring Online bipartite matching (marriage problem) Assignment 3 & logistics COMP Online Algorithms Online Graph Problems 2 / 13
4 Online Edge Coloring in Graphs COMP Online Algorithms Online Graph Problems 2 / 13
5 Problem Definition In edge coloring, the goal is to color edges of a graph with minimum number of colors No two adjacent edges (edges sharing an endpoint) should have the same color In the offline setting, the problem is NP-hard! For a graph of degree, at least and at most + 1 colors are required (Vizing theorem) This implies that cost(opt) COMP Online Algorithms Online Graph Problems 3 / 13
6 Problem Definition In the online setting, edges arrive one by one, and an algorithm should take an irrevocable decision on coloring the edges without any knowledge about future edges (or how graph looks). For example, Greedy family of algorithms maintain a set of colors and use them, if possible, before requesting a new coloring cost of Opt is 3 Cost of Greedy is 4, which is not optimal COMP Online Algorithms Online Graph Problems 4 / 13
7 Greedy algorithm Theorem Greedy has a competitive ratio of at most 2. For any graph of degree, cost of Opt is at least. Cost of greedy is at most 2 1. COMP Online Algorithms Online Graph Problems 5 / 13
8 Lower Bound Theorem No deterministic online algorithm can have a competitive ratio better than 2. Adversary forms a graph of degree which can be colored using colors In doing so, any online algorithm needs to use at least 2 colors COMP Online Algorithms Online Graph Problems 6 / 13
9 Lower Bound The input starts by sending star of degree 1 Recall that a star of degree d is a tree formed by a center vertex connected to d leaves. There are at most K = ( 2 1) ways to color a star using 2 colors. If we have K + 1 stars, at least two of them have the same coloring (pigeonhole principle). If we have 2K + 1 stars, at least three of them have the same coloring If we have ( + 1)K + 1 stars, at least of them will have the same coloring COMP Online Algorithms Online Graph Problems 7 / 13
10 Lower Bound After sending + 1)K + 1 stars, at least stars have the same coloring. Adversary reveals edges forming another star, of degree, connected to centers of these stars. Any of these new edges requires a color other than the 1 colors in the old star. In total ( 1) + = 2 1 colors are used by the algorithm COMP Online Algorithms Online Graph Problems 8 / 13
11 Lower Bound Opt colors edges adjacent to the new center (at the bottom) using colors Other edges form stars connected to only of the colored edges; each star can be colored using the remaining 1 colors Opt uses colors COMP Online Algorithms Online Graph Problems 9 / 13
12 Lower Bound Summary For any given online algorithm, adversary created a graph G so that: G has degree Opt uses colors. the online algorithm uses 2 1 colors Competitive ratio of the algorithm is at least COMP Online Algorithms Online Graph Problems 10 / 13
13 Lower Bound Theorem No deterministic online algorithm can have a competitive ratio better than 2. This implies that greedy algorithms are the best deterministic algorithm COMP Online Algorithms Online Graph Problems 11 / 13
14 Online Bipartite Matching (Marriage problem) COMP Online Algorithms Online Graph Problems 11 / 13
15 Online Bipartite Matching Given a bipartite graph, the goal is to create a matching (creating pair of non-adjacent edges) with maximum size In the online setting, vertices in one side of the graph are given, and vertices in the other side arrive one by one Upon arrival of vertex x, all edges connecting x to its neighors on the left are revealed an online algorithm should match x with another vertex, if possible, without any information about future vertices Greedy algorithm: match with any vertex on the right if possible! COMP Online Algorithms Online Graph Problems 12 / 13
16 Competitive ratio for max. problems In this example, Opt has a benefit of 4 (a perfect matching) while greedy has a benefit of 3! Matching is a maximization problem: we would like to maximize the benefit instead of minimizing the cost Competitive ratio is oftend defined as the maximim value of Benefit(Opt) benefit(alg) Greedy algorithm always creates a maximal matching For any mistake match, it blocks to possible matches The benefit of greedy is no less than twice that of Opt Theorem Greedy has a competitive ratio of at most 2 (details in the next class). COMP Online Algorithms Online Graph Problems 13 / 13
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