Algorithms for Budget-Constrained Survivable Topology Design Nikhil Garg, Rahul Simha and Wenxun Xing Presented by Lillian Tseng

Outline Introduction Formulation of 2-PMOP Greedy Algorithms Path-based Greedy Algorithm Largest-Overlap-Reduction Greedy Algorithm Local Search Simulated Annealing Conclusion 2019/1/18 ICC 2002

Introduction Complete augmentation where the goal is to ensure two edge-disjoint (or node-disjoint) paths for every pairs of nodes. Practical networking constraints: It’s impossible to build a link between every pair of nodes. The costs of each potential edges may vary drastically. The requirement that complete edge-disjoint paths be found may decrease the acceptable solution space sharply. Penalty to overlapped paths. Only a subset of S-D pairs may be included. Optimization with a limited budget. 2. Cost to lay a new optical fiber or lease a new line 4. Different connections has different protection requirements 2019/1/18 ICC 2002

Introduction Characteristics of the topology augmentation problem: Both costs and topology structure play important roles in determining an effective solution  knapsack-like problem. Finding the paths between S-D pairs  Hamiltonian-path problem. So it’s hard to perform single-parameter greedy heuristic. 2019/1/18 ICC 2002

Formulation of 2-PMOP G=(V, E), a connected, undirected network graph, V is the vertex set and E is the set of links, for each of which has distance d(i,j). Collection of source-destination required pairs Set of potential edges For each associate a positive cost of building an edge c(i,j). 2019/1/18 ICC 2002

Formulation of 2-PMOP Define the overlap distance of Ps,d and P’s,d (two paths from s to d in G): Here we need 2 paths for each , so let to be the total network overlap distance. 2-PMOP: for a given budget B, find 2019/1/18 ICC 2002

Formulation of 2-PMOP Decision Variables: 2019/1/18 ICC 2002

t 2 4 6 s 1 3 5 2-PMOP is NP-hard Show 2-PMOP is NP-hard by reducing the Partition problem: Given any instance of PARTITION, a connected communication graph G=(V,E) can be constructed, where V={s,t,1,2,…,2n+2} and E={(s,1),(s,2),(1,2),(1,3),(3,4),(3,5),…,(2n+1,2n+2),(2n+1,t),(2n+2,t)}. 2019/1/18 ICC 2002

2-PMOP is NP-hard The distance of each edge is defined as The cost of the potential edge is defined as t 2 c1 4 c2 6 s 1 c1 3 c2 5 2019/1/18 ICC 2002

2-PMOP is NP-hard Budget: . Transformation: Given a connected communication graph G=(V,E), distances , costs is there a solution of 2-PMOP such that and the total cost of the added edges is with the given budget? 2019/1/18 ICC 2002

2-PMOP is NP-hard The common shared edges of 2-PMOP are (2i-1,2(i+1)-1),i=1,2,…,n. When removing one edge (2i-1,2(i+1)-1) of distance ci out of the shared edge list, an edge (2i,2(i+1)) with cost ci must be added. Derive the relationship between the solutions of PARTITION and 2-PMOP: The objective value satisfies If i is select to an element of S, then edge (2i,2i+2) is add to the graph t 2 c1 4 c2 6 s 1 c1 3 c2 5 2019/1/18 ICC 2002

Relaxed Problem of 2-PMOP Given a selected set of edges. Budget constraint is relaxed by a given set T 2019/1/18 ICC 2002

Some Propositions k pairs O(n2) at most (n-1) edges total O(kn3) 2019/1/18 ICC 2002

Restate the Problem The set T is constructed and varied in each heuristic: Greedy Algorithms Path-based Greedy Algorithm Largest-Overlap-Reduction Greedy Algorithm Local Search Simulated Annealing 2019/1/18 ICC 2002

Greedy Algorithm Input: 2019/1/18 ICC 2002

Greedy Algorithm Time Complexity: For each iteration |F|*O(kn3) At most |F| edges can be added |F|2*O(kn3)=O(|F|2kn3) Greedy Algorithm adds one edge at a time The order of edges is determined, knapsack-like, by the per-unit-cost effectiveness in reducing the objective function 2019/1/18 ICC 2002

Greedy Algorithm 2019/1/18 ICC 2002

Path-based Greedy Algorithm Sometimes, the overlap on a certain pat is best reduced by a secondary path. Each new path may impact different S-D pairs in various ways. Use the “profit” function f for a path pi, containing both length of the critical edge and partial length of the critical edges, whose secondary paths share edges with pi. The frequency of an edge being used decides the edge’s importance. New path only reduce overlap for some S-D pairs 2019/1/18 ICC 2002

Path-based Greedy Algorithm 2019/1/18 ICC 2002

Path-based Greedy Algorithm 2019/1/18 ICC 2002

Largest-Overlap-Reduction Greedy Algorithm Stitch together some of alternate paths for articulated edges in to more compact paths by weighting commonly occurring edges more heavily. The path-based heuristic can only select edges from secondary paths, i.e. shortest paths. A profitability function for edges rather than paths. Retain the “path” part by assigning profitability based on overlap reduction for individual paths. If the primary path contains several articulated edges, which means they’re also contained in the alternate paths, it’s difficult to select edges in the alternate paths for these critical edges. We assign the reduction profit for a path rather than total overlap 2019/1/18 ICC 2002

Largest-Overlap-Reduction Greedy Algorithm A B d s 10 The profit function g will assign the same high “profit” to all the edges in the secondary path for the largest overlap reduction, thus retaining the path-based approach, but with an edge based greedy solution. 2019/1/18 ICC 2002

Local Search Number of the neighbors of Y is 1+|F|+|F|(|F|-1)/2. i.e. (no change) + (one edge of different status with Y) + (two edges of different status with Y). We can also solve 2-PMOP by providing a configuration of potential edges: Thus, we can obtain a set T of added edges by , denoted it as T(y). Compute Z2(T(y)) by solving the relaxation problem in O(kn3). Define the neighborhood of Y as 2019/1/18 ICC 2002

Local Search 2019/1/18 ICC 2002

Simulated Annealing It can provide good solutions for many NP-hard problems. Essential components: Initial solution. A method for selecting a random neighbor in solution space. A cooling schedule. A stopping criterion. 2019/1/18 ICC 2002

Computational Results Overlaps remained after adding potential edges from F In all cases, one of the three Greedy Algorithms gets the result as good as annealing Local search got the best result?? Path-based is the worst?? No “time graph” attached Although “Local Search” perform better results than “Greedy Algorithms”, “Greedy Algorithms” outperform “Local Search” in terms of execution time. 2019/1/18 ICC 2002

Computational Results For smaller graphs, it’s useful to use some exhaustive strategy like simulation annealing to obtain solutions for the given problem. The problem depends heavily on the costs of potential edges. For unit-cost edges cases, the solution obtained by all heuristic were identical. 2019/1/18 ICC 2002

Conclusions Class of augmentation problems that form an important area in survivable network design is concerned. The formulation, inspired by some practical considerations, was shown to be NP-hard. The paper presented three polynomial-time heuristics to solve the 2-PMOP problem. At least one of the heuristics presented can provide a near-optimal solution. 2019/1/18 ICC 2002

The End

Summary Survivability: Link connectivity. Objective: Finding the fewest links to add to a topology to make the topology 2-edge-connected(or minimizing the overlaps) under some constraints. Analysis method: Quantitative methods. Non-linear programming Solved by Greedy algorithms, Local Search and Simulated Annealing. 2019/1/18 ICC 2002