A General Approach to Online Network Optimization Problems Seffi Naor Computer Science Dept. Technion Haifa, Israel Joint work: Noga Alon, Yossi Azar, Baruch Awerbuch, and Niv Buchbinder

The Set Cover Problem Input: X = 1, 2,...,n – ground set of elements. S – family of subsets of X. c – cost function on S. Goal: A min cost collection of sets from S that cover X. Classic: greedy algorithm is an O(logn)- approximation.

The Online Set Cover Problem An adversary gives the elements one-by-one to the algorithm. When a new element arrives, the algorithm must cover it by a set from S. X’ – Elements given by adversary ( ). Competitive Factor:

Example (1) The sets are servers and the elements are potential clients. Each server can provide the service to a subset of the clients. There is a setup cost for activating a server. Clients arrive one-by-one.

Example (2) Input: X = {1,2, …,n} – a ground set. S – All subsets of X of size. Game: Adversary gives uncovered element at each step. Online algorithm picks a set. Termination: All elements are covered: Performance: Competitive ratio is at least

Example (2) (contd.) Good news or bad news? Not so bad … Competitive Ratio is O(log m). Depends on both n and m (unlike offline case).

Graphical Representation r e a babcedc Request for element a: Purchase a path from r to a leaf labeled a.

Network Optimization Problems Network = Weighted graph, directed or undirected Demands: Disjoint sets of vertices D i = (S i, T i ) Problems: Connectivity - Connect the sets by “picking” edges such that there is path from a vertex in S i to a vertex in T i.

Network Optimization Problems (contd.) Problems (contd.): Cuts - Disconnect the sets by “removing” edges such that each vertex in S i is disconnected from each vertex in T i. Goal: Minimize the total cost of picked or removed edges.

Online Network Optimization Problems Network and weight function are known in advance to the online algorithm. The demands D i = (S i, T i ) are given one-by-one. Each demand is satisfied upon arrival by purchasing edges. Competitive factor is ratio between: cost of edges purchased by the online algorithm and cost of optimal solution.

Connectivity Problems - Examples Online (Non-Metric) Facility Location: There are potential locations of facilities. Each location has a “setup cost”. Clients arrive one-by-one. Each client may connect to each facility by paying a “connection cost”. Goal: Decide which facilities to open to minimize the total cost: “Total Opening Cost” + “Total Connection Cost”

Non-Metric Facility location - Demo ∞

Connectivity Problems - Examples Online Multicast Problem: A family of arbitrary rooted trees, where the tree edges have costs. Each tree leaf is associated with a subset of the clients. Clients arrive one-by-one. Upon arrival of a client: a path from a leaf (associated with the client) to a root has to be purchased. Goal: Minimize cost of purchased edges.

The Multicast Problem

Connectivity Problems - Examples Online Group Steiner problem in trees: Same as the multicast problem – but now there is a single arbitrary rooted tree. This means that paths from leaves associated with the same client to the root are not necessarily disjoint. Goal: Minimize total cost of purchased edges.

Online Group Steiner in Trees Problem

Cuts Problem - Example Online Multicut Problem: General weighted undirected Graph Demands: pairs of vertices D i = (s i, t i ) Goal: Disconnect each pair D i = (s i, t i ) by removing edges from the graph. Minimize the total cost of edges.

Online Multi-cut Problem S3S3 T3T3 S1S1 T1T1 S2S2 T2T2

Fractional Network Problems For each demand (S,T): Connectivity Problems: Give fractional weights to edges s. t. maximum flow from S to T is at least 1. Minimize  c(e) w(e) Cut Problems: Give fractional weights to edges s. t. distance from S from T (closest vertices) is at least 1. Minimize  c(e) w(e)

General Approach to Online Optimization Problems Two Steps: Generate in an online fashion a fractional solution such that: Cost of online fractional solution is close to cost of optimal fractional solution. Round the fractional solution online into an integral solution such that: Cost of integral solution is close to cost of fractional solution.

First Part: Online Fractional Solution Connectivity Problems: Optimal Cost – W* Cost of edges – [1, 2m 2 ] (m = num. of edges) Initially: Give each edge weight = 1/(2m 3 ) Total initial weight: m edges Maximal cost of edge – 2m 2  Total initial cost at most 1

Algorithm – Online connectivity If maximum flow from S to T is at least 1: Do nothing Else: While the flow is less than 1: 1.Compute minimum cut C between S and T 2.For each edge e in the cut: w(e)  w(e)[1+ 1/c(e)] New demand D = (S, T):

The Algorithm - Analysis Lemma: The total number of weight increments during the algorithm is O(W* logm) Proof: Potential function:

Analysis – cont. Initial value of the potential function is: -2W* log 2 (2m) Initial weights of edges: w e = 1/(2m 3 ). The potential function never exceeds: 2W* The weight of each edge is at most 2. Each time weights are increased, the potential function increases by at least 1.

Analysis – cont. Proof of third fact: First inequality – (c e ≥ 1) Second inequality – OPT is feasible.

The Algorithm – Competitive Ratio Theorem: The algorithm is O(log m) competitive. Proof: 1.The initial value of the solution is at most 1. 2.Each time the algorithm increases weights, the cost it pays increases by:  c(e) w(e)/c(e) =  w(e) ≤ 1 (The minimum cut is at most 1) 3.There are O(W* log m) weight increments in the algorithm.

Online Multicut - Algorithm If the shortest path from S to T is at least 1 Do nothing Else: While the distance is less than 1: 1.Compute a shortest path P between S and T 2.For each edge e in the path P : w(e)  w(e)[1+ 1/c(e)] New demand D = (S, T):

The Algorithm – Competitive Ratio Theorem: The algorithm for generating a fractional multicut online is O(log m) competitive. Proof: Similar Analysis

Lower Bounds Lemma: Any deterministic (and randomized) online algorithm for the fractional connectivity and fractional cuts problem has a competitive ratio of at least Ω(log m) Remarks: 1.Holds even with respect to the optimal integral solution.

Rounding the Fractional Solution The rounding is problem specific. Results: 1.Set cover, non-metric facility location and multicast – O(logn logm)- competitive algorithm. m – number of possible facilities. n – number of clients. Remark: Lower bound for deterministic algorithm for online set cover – almost tight.

Rounding the Fractional Solution (cont.) Results (cont.): 2.Online group Steiner Problem: 1.Trees: O(logk log N logn) 2.General Graphs: O(logk log N log 2 n) n – number of vertices in the graph k – number of clients N – maximal size of a group ( at most n) Remark: General Graphs via HST’s

Rounding the Fractional Solution Example: Online Set Cover Problem. Offline case: Classic “randomized rounding”: Choose each set S with probability O(w(S)logn): Elements are covered with high probability. Expected cost is fractional cost x O(logn).

Rounding the Fractional Solution Online case: randomized rounding on the “increments” of the fractional increase. In each weight augmentation: w(S)  w(S)[1+1/c(S)] Repeat O(logn) times: Choose Set S with probability w(S)/c(S). Surprisingly, this can be de-randomized online using a suitable potential function [AAABN, STOC ’03].

Rounding the Fractional Solution Example: Multicast problem on trees For each tree: choose 2logn’ r. v. uniformly in [0,1]. (n’ – # terminals so far) Threshold of a tree: minimum r. v. Online Rounding: Take an edge if weight exceeds tree threshold. (Weights on a path – monotone non-increasing) Open: Can it be de-randomized? (even for facility location.)

Online Multicut Problem Techniques: 1.Raecke’s hierarchical decomposition of a graph into a tree. (Harrelson, Hidrum, Rao). 2.Ratio of Minimum Cut / Maximum multi- commodity Flow in trees is at most 2. 3.Simple online primal-dual algorithm on trees.

Online Multicut Problem Results: Deterministic online algorithm for the multicut problem with competitive ratio: O(log 3 n loglogn) for general graphs. O(log 2 n loglogn) for planar graphs. O(log 2 n) for trees. n – number of vertices

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