a general approach to online network optimization problems
DESCRIPTION
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 . - PowerPoint PPT PresentationTRANSCRIPT
A General Approach to Online Network
Optimization Problems
Seffi NaorComputer Science Dept.
TechnionHaifa, Israel
Joint work: Noga Alon, Yossi Azar, Baruch Awerbuch, and Niv Buchbinder
The Set Cover ProblemInput:• 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:
'X X
Cost of sets used by online algorithm
Cost of optimal solution of X'
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
n
n
n
| ' |X n
n
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).
log logn
n nn
Graphical Representation
r
ea
50 100 150
bab c ed c
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 Di = (Si, Ti)
Problems:
Connectivity - Connect the sets by “picking” edges such that there is path from a vertex in Si to a vertex in Ti.
Network Optimization Problems (contd.)
Problems (contd.):
Cuts - Disconnect the sets by “removing” edges such that each vertex in Si is disconnected from each vertex in Ti.
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 Di = (Si, Ti) 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”
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 Di = (si, ti)
Goal:• Disconnect each pair Di = (si, ti) by
removing edges from the graph.
• Minimize the total cost of edges.
Online Multi-cut Problem
S3
T3
S1 T1
S2
T2
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, 2m2] (m = num. of
edges)Initially: Give each edge weight = 1/(2m3)
Total initial weight:• m edges• Maximal cost of edge – 2m2
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:)(log2
*ee
Eee wwc
Algin edge of weight -
OPTin edge ofweight *
e
e
w
w
Analysis – cont.
• Initial value of the potential function is: -2W* log2(2m)
Initial weights of edges: we = 1/(2m3).
• 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.
)(log2*
eeEe
e wwc
Analysis – cont.
Proof of third fact:
Cee
ee
Cee
eeCe
ee
eeCe
e
wc
wc
wwcc
wwc
11
1log
log1
1log
*2
*
2*
2*
)(log2*
eeEe
e wwc
• First inequality – (ce ≥ 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 T2. 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 SolutionThe 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 log2n)
n – number of vertices in the graphk – number of clientsN – 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 SolutionExample: 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(log3n loglogn) for general graphs.• O(log2n loglogn) for planar graphs.• O(log2n) for trees.
n – number of vertices
Thank you!