A Graph-theoretic QoS-aware Vulnerability Assessment for Network Topologies

Size: px
Start display at page:

Download "A Graph-theoretic QoS-aware Vulnerability Assessment for Network Topologies"

Transcription

1 A Graph-theoretic QoS-aware Vulnerability Assessment for Network Topologies March 24, 2010

2 Table of contents 1 Problem Definition and Contributions 2 3 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks 4

3 System Model given a networks with a set of s t paths; each path satisfies some of the constraints, thus has a satisfactory score by assigning each constraint a score. as long as there exists a path with score higher the some threshold, the network is still operational. goal: at least home many node/link failures will make the network not operational, i.e. no such satisfiable path exists, and QoS routing service is not available from s to t.

4 Graph Model directed graph G(V, E), source s, destination t; ( ) additive weight vector for each edge e j = (u, v) E: w j 1,wj 2,,wj m. ( weight vector for each path: ej p wj 1, ej p wj 2,, ej p m) wj ; constraint threshold vector (c 1,c 2,,c m); priority vector (λ 1, λ 2,, λ m); we define a satisfactory score φ(p) for path p as: where p i iff w p i c i φ(p) = λ j j:p i

5 Graph Model A (1,2) B (3,7) C (1,2) (2,2) (1,1) D (2,2) E (2,1) F (2,2) (1,1) (2,3) G (2,2) (3,2) H I Given a score threshold ρ, find the minimum number of edges whose removal makes φ(p) = max{φ(p) s t path p in G} ρ

6 Integer Program Variables: X e = 1 if edge e is NOT removed in the optimal solution, 0 otherwise; Y pi = 1 if a s t path p has w p i c i, 0 otherwise; a large constant ǫ = max i { e E we i }.

7 Integer Program min s.t. (1 X e) e E m i=1 Y piλ i ρ, s t path p e p we i ǫ Y pi ǫ + e p (1 Xe)ǫ + c i, i,p e p we i > 1 Y pi ǫ e p (1 Xe)ǫ + c i, i,p X e [0,1] Y pi [0,1] Figure: Integer Programming Formulation

8 Main Contributions provide the first graph-theoretic QoS-ware vulnerability assessment method; abstract the assessment problem as a graph optimization problem and study its hardness; present exact solutions to the problem in two practical cases and several heuristics for general cases.

9 decide version Given a set of edges deleted, does there exist a path P satisfying φ(p) ρ in the remaining graph. Definition (QoS-SP) Given a graph with an m-dimension constraint vector, find a path P such that φ(p) = max φ(p) p=[s,,t] G Definition (MCP) Given m constraints C i, the problem is to find a path P from a source node s to a destination node t such that w i (P) w i (u, v) C i (u,v) P for all i [1, m]

10 NP-hard Lemma QoS-SP problem is NP-Complete. Proof. Consider the decision version of QoS-SP: given a positive number k, decide if there exists a path P with φ(p) k. This proof is straightforwardly by reduction from MCP problem by letting k = m i=1 λ i. Then if there exists a solution for the QoS-SP problem, then the path is a feasible path to MCP, otherwise, MCP has no solution. Since MCP is NP-hard, the proof completes.

11 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Exact Solution Enumerate all (up to 2 m ) satisfiable constraint states; Employ FringeMCSP [5] to find the shortest path that satisfies a specific set of constraints; Use revised Edmonds-Karp algorithm to remove all satisfiable augmenting path, w.r.t a set of constraints. Find a minimum set of edge cut.

12 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Exact Solution Exact Algorithm S ; for any maximal subset ss of M with c i ss λ i < ρ do S S {M \ ss} while S do ss extracted from S; for each edge (i,j) E do Set f (i,j) = f (j,i) = 0; Set c f (i,j) = 1 and c f (j, i) = 0; while shortest path q that satisfies all constraints in ss can be found using FringeMCSP do for each edge (u, v) q do c f (q) = min{c f (u, v) : (u, v) q}; f (u, v) = f (u, v) + c f (q); f (v,u) = f (u, v); c f (u, v) = c(u, v) f (u, v); c f (v, u) = c(v, u) f (v,u); all the vertices reachable from s on the residual network induces a cut T. Return the minimum cut among the cuts derived.

13 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Exact Solution Enumeration of satisfiable constraint states does not work. Exact Solution: Employ an existing all-path dynamic programming algorithm label-correcting [3] to discovery satisfiable paths; Use revised Edmonds-Karp algorithm [6] to return the cut.

14 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Label-Correcting Label-Correcting to find satisfiable paths S ; labelset(s) {(0, 0,, 0, )}; Q {s}; while Q do u extracted from the end of Q; for all outgoing edges (u, v) do labellist (v) Merge(labelList(v), labellist(u), (u, v)); /*extend tuples of u to v by adding the weights of (u, v), eliminate all dominated tuples.*/ if labellist (v) labellist(v) then labellist(v) labellist (v), Q Q {v}; S all the PO paths obtained from labellist(t); Return q = max p{φ(p) p S}.

15 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Strategies Adopt a Relax-SAT test for estimate the existence of satisfiable path in the remaining graph; Adopt two greedy metrics to iteratively remove edges until the Relax-SAT test returns Negative result.

16 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Relax-SAT test Relax-SAT Metric For each edge e, For each path p, ϕ 1 (e) = m i=1 w e i c i λ i ϕ 1 (p) = e p ϕ(e) Similarly, define ϕ 2 (e) = m i=1 w e i c i λ i

17 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks How far from SAT Assume λ m i λ i, β(p) max i ( wp c i ); Assume p satisfies a set C s of constrains, and does not satisfy the set C s, then φ(p) = c i C s λ i and ϕ 1 (p) = w p i c i C s λ c i w j i c j C s λ c j, therefore j φ(p) ρ ϕ 1 (p) ρ (λ ρ)β(p); ϕ 1 (p) ρ φ(p) ρ; since it is hard to calculate β(p) when p is not determinate, we assert maxϕ 1 (p) = minϕ 2 (p) ρ maxφ(p) ρ; minϕ 2 (p) ρ maxφ(p) ρ; p

18 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Relax-SAT test Relax-SAT test Input: directed graph G = (V, E), constant ρ; Output: is there a satisfiable path. for every edge e E do ϕ 2 (e) m wi e i=1 λ c i ; i q shortest s-t path on metric ϕ 2 ; if ϕ 2 (q) > ρ then Return NO; if ϕ 2 (q) < ρ then Return YES;

19 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Two greedy metrics Definition (Nonlinear Mixed Edge Metric) n ( ) wi (e)π(i, e) α ξ(e) = λ i c i=1 i where w i (e) is the i th weight on edge e, π(i,e) is the length of a s t path, which has the minimum weight w.r.t the i th constraint over all s t paths containing e, and λ i is the priority parameter for the i th constraint. It is evident that the smaller ξ(e) is, the more likely it will be included in the QoS-optimal path. α [1,, + ) is tuned to increase the likelihood of this path to be selected in the optimal solution.

20 network with small amount of constraints small networks with unrestricted constraints two fast heuristic algorithms for general networks Two greedy metrics Definition (Betweenness) θ(e) i: e P i λ i where P i refer to the shortest path w.r.t the i th constraint.

21 Results To be announced.

22 Bibliography Fernado A. Kuipers and Piet F. A. Van Mieghem, Conditions that Impact the complexity of QoS Routing, IEEE transaction on Networking, 2005 Line B. Reinhardt and David Pisinger, Multi-Objective and Multi-Constrained Non-Additive Shortest Path Problems, Technique Report, DTU Management, Turgay Korkmaz and Marwan Krunz, Multi-Constrained Optimal Path Selection, Infocom, P. Kh adivi, S. Samavi and T. D. Todd, Multi-constraint QoS routing using a new single mixed metrics, Journal of Network and Computer Applications, Yuxi Li, Janelee Harms, Robert Holte, Fast Exact MultiConstraint Shortest Path Algorithms, ICC, Jack Edmonds and Richard M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, Journal of the ACM 19 (2):

Maximum Flow. Flow Networks. Flow Networks Ford-Fulkerson Method Edmonds-Karp Algorithm Push-Relabel Algorithms. Example Flow Network

Maximum Flow. Flow Networks. Flow Networks Ford-Fulkerson Method Edmonds-Karp Algorithm Push-Relabel Algorithms. Example Flow Network Flow Networks Ford-Fulkerson Method Edmonds-Karp Algorithm Push-Relabel Algorithms Maximum Flow Flow Networks A flow network is a directed graph where: Each edge (u,v) has a capacity c(u,v) 0. If (u,v)

More information

Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost

Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R

More information

Maximum flows & Maximum Matchings

Maximum flows & Maximum Matchings Chapter 9 Maximum flows & Maximum Matchings This chapter analyzes flows and matchings. We will define flows and maximum flows and present an algorithm that solves the maximum flow problem. Then matchings

More information

Solution for Homework set 3

Solution for Homework set 3 TTIC 300 and CMSC 37000 Algorithms Winter 07 Solution for Homework set 3 Question (0 points) We are given a directed graph G = (V, E), with two special vertices s and t, and non-negative integral capacities

More information

Mathematical Tools for Engineering and Management

Mathematical Tools for Engineering and Management Mathematical Tools for Engineering and Management Lecture 8 8 Dec 0 Overview Models, Data and Algorithms Linear Optimization Mathematical Background: Polyhedra, Simplex-Algorithm Sensitivity Analysis;

More information

Toward the joint design of electronic and optical layer protection

Toward the joint design of electronic and optical layer protection Toward the joint design of electronic and optical layer protection Massachusetts Institute of Technology Slide 1 Slide 2 CHALLENGES: - SEAMLESS CONNECTIVITY - MULTI-MEDIA (FIBER,SATCOM,WIRELESS) - HETEROGENEOUS

More information

LECTURES 3 and 4: Flows and Matchings

LECTURES 3 and 4: Flows and Matchings LECTURES 3 and 4: Flows and Matchings 1 Max Flow MAX FLOW (SP). Instance: Directed graph N = (V,A), two nodes s,t V, and capacities on the arcs c : A R +. A flow is a set of numbers on the arcs such that

More information

Clustering-Based Distributed Precomputation for Quality-of-Service Routing*

Clustering-Based Distributed Precomputation for Quality-of-Service Routing* Clustering-Based Distributed Precomputation for Quality-of-Service Routing* Yong Cui and Jianping Wu Department of Computer Science, Tsinghua University, Beijing, P.R.China, 100084 cy@csnet1.cs.tsinghua.edu.cn,

More information

Approximability Results for the p-center Problem

Approximability Results for the p-center Problem Approximability Results for the p-center Problem Stefan Buettcher Course Project Algorithm Design and Analysis Prof. Timothy Chan University of Waterloo, Spring 2004 The p-center

More information

Graphs and Network Flows IE411. Lecture 13. Dr. Ted Ralphs

Graphs and Network Flows IE411. Lecture 13. Dr. Ted Ralphs Graphs and Network Flows IE411 Lecture 13 Dr. Ted Ralphs IE411 Lecture 13 1 References for Today s Lecture IE411 Lecture 13 2 References for Today s Lecture Required reading Sections 21.1 21.2 References

More information

Quality of Service Routing

Quality of Service Routing Quality of Service Routing KNOM Tutorial 2004, Jeju, Korea, Nov. 4-5, 2004. November 5, 2004 Kwang-Hui Lee Communication Systems Lab, Changwon National University khlee@changwon.ac.kr Contents Introduction

More information

A Heuristic Algorithm for the Multi-constrained Multicast Tree

A Heuristic Algorithm for the Multi-constrained Multicast Tree A Heuristic Algorithm for the Multi-constrained Multicast Tree Wen-Lin Yang Department of Information Technology National Pingtung Institute of Commerce No.51, Ming-Sheng East Road, Pingtung City,Taiwan

More information

Computing a Path subject to Multiple Constraints: Advances and Challenges

Computing a Path subject to Multiple Constraints: Advances and Challenges Computing a Path subject to Multiple Constraints: Advances and Challenges Guoliang (Larry) Xue Faculty of Computer Science and Engineering School of Computing, Informatics and Decision Systems Engineering

More information

An Evolutionary Algorithm for the Multi-objective Shortest Path Problem

An Evolutionary Algorithm for the Multi-objective Shortest Path Problem An Evolutionary Algorithm for the Multi-objective Shortest Path Problem Fangguo He Huan Qi Qiong Fan Institute of Systems Engineering, Huazhong University of Science & Technology, Wuhan 430074, P. R. China

More information

Chapter 5 Graph Algorithms Algorithm Theory WS 2012/13 Fabian Kuhn

Chapter 5 Graph Algorithms Algorithm Theory WS 2012/13 Fabian Kuhn Chapter 5 Graph Algorithms Algorithm Theory WS 2012/13 Fabian Kuhn Graphs Extremely important concept in computer science Graph, : node (or vertex) set : edge set Simple graph: no self loops, no multiple

More information

1 The Traveling Salesperson Problem (TSP)

1 The Traveling Salesperson Problem (TSP) CS 598CSC: Approximation Algorithms Lecture date: January 23, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im In the previous lecture, we had a quick overview of several basic aspects of approximation

More information

Performance Evaluation of Constraint-Based Path Selection Algorithms

Performance Evaluation of Constraint-Based Path Selection Algorithms Performance Evaluation of Constraint-Based Path Selection Algorithms Fernando Kuipers, Delft University of Technology Turgay Korkmaz, University of Texas at San Antonio Marwan Krunz, University of Arizona

More information

Connected Liar s Domination in Graphs: Complexity and Algorithm 1

Connected Liar s Domination in Graphs: Complexity and Algorithm 1 Applied Mathematical Sciences, Vol. 12, 2018, no. 10, 489-494 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8344 Connected Liar s Domination in Graphs: Complexity and Algorithm 1 Chengye

More information

A Randomized Algorithm for Minimizing User Disturbance Due to Changes in Cellular Technology

A Randomized Algorithm for Minimizing User Disturbance Due to Changes in Cellular Technology A Randomized Algorithm for Minimizing User Disturbance Due to Changes in Cellular Technology Carlos A. S. OLIVEIRA CAO Lab, Dept. of ISE, University of Florida Gainesville, FL 32611, USA David PAOLINI

More information

An Introduction to Dual Ascent Heuristics

An Introduction to Dual Ascent Heuristics An Introduction to Dual Ascent Heuristics Introduction A substantial proportion of Combinatorial Optimisation Problems (COPs) are essentially pure or mixed integer linear programming. COPs are in general

More information

Cross-Virtual Concatenation for Ethernet-over-SONET/SDH Networks

Cross-Virtual Concatenation for Ethernet-over-SONET/SDH Networks Cross-Virtual Concatenation for Ethernet-over-SONET/SDH Networks Satyajeet S. Ahuja and Marwan Krunz {ahuja,krunz}@ece.arizona.edu Dept. of Electrical and Computer Engineering, The University of Arizona.

More information

2.3 Optimal paths. Optimal (shortest or longest) paths have a wide range of applications:

2.3 Optimal paths. Optimal (shortest or longest) paths have a wide range of applications: . Optimal paths Optimal (shortest or longest) paths have a wide range of applications: Google maps, GPS navigators planning and management of transportation, electrical and telecommunication networks project

More information

The Encoding Complexity of Network Coding

The Encoding Complexity of Network Coding The Encoding Complexity of Network Coding Michael Langberg Alexander Sprintson Jehoshua Bruck California Institute of Technology Email: mikel,spalex,bruck @caltech.edu Abstract In the multicast network

More information

Jessica Su (some parts copied from CLRS / last quarter s notes)

Jessica Su (some parts copied from CLRS / last quarter s notes) 1 Max flow Consider a directed graph G with positive edge weights c that define the capacity of each edge. We can identify two special nodes in G: the source node s and the sink node t. We want to find

More information

Graphs and Network Flows IE411. Lecture 21. Dr. Ted Ralphs

Graphs and Network Flows IE411. Lecture 21. Dr. Ted Ralphs Graphs and Network Flows IE411 Lecture 21 Dr. Ted Ralphs IE411 Lecture 21 1 Combinatorial Optimization and Network Flows In general, most combinatorial optimization and integer programming problems are

More information

Computer Science & Engineering 423/823 Design and Analysis of Algorithms

Computer Science & Engineering 423/823 Design and Analysis of Algorithms Computer Science & Engineering 423/823 Design and Analysis of Algorithms Lecture 07 Single-Source Shortest Paths (Chapter 24) Stephen Scott and Vinodchandran N. Variyam sscott@cse.unl.edu 1/36 Introduction

More information

Computational problems. Lecture 2: Combinatorial search and optimisation problems. Computational problems. Examples. Example

Computational problems. Lecture 2: Combinatorial search and optimisation problems. Computational problems. Examples. Example Lecture 2: Combinatorial search and optimisation problems Different types of computational problems Examples of computational problems Relationships between problems Computational properties of different

More information

Lecture 10,11: General Matching Polytope, Maximum Flow. 1 Perfect Matching and Matching Polytope on General Graphs

Lecture 10,11: General Matching Polytope, Maximum Flow. 1 Perfect Matching and Matching Polytope on General Graphs CMPUT 675: Topics in Algorithms and Combinatorial Optimization (Fall 2009) Lecture 10,11: General Matching Polytope, Maximum Flow Lecturer: Mohammad R Salavatipour Date: Oct 6 and 8, 2009 Scriber: Mohammad

More information

Reducing Directed Max Flow to Undirected Max Flow and Bipartite Matching

Reducing Directed Max Flow to Undirected Max Flow and Bipartite Matching Reducing Directed Max Flow to Undirected Max Flow and Bipartite Matching Henry Lin Division of Computer Science University of California, Berkeley Berkeley, CA 94720 Email: henrylin@eecs.berkeley.edu Abstract

More information

Cuts, Connectivity, and Flow

Cuts, Connectivity, and Flow Cuts, Connectivity, and Flow Vertex Cut and Connectivity A separating set or vertex cut of a graph G is a set S V(G) such that G S G S has more than one component A graph G is k-connected if every vertex

More information

Syllabus. simple: no multi-edges undirected edges in-/out-/degree /0: absent 1: present (strongly) connected component subgraph, induced graph

Syllabus. simple: no multi-edges undirected edges in-/out-/degree /0: absent 1: present (strongly) connected component subgraph, induced graph Introduction to Algorithms Syllabus Recap on Graphs: un/directed, weighted Shortest Paths: single-source, all-pairs Minimum Spanning Tree: Prim, Kruskal Maximum Flow: Ford-Fulkerson, Edmonds-Karp Maximum

More information

Introduction. I Given a weighted, directed graph G =(V, E) with weight function

Introduction. I Given a weighted, directed graph G =(V, E) with weight function ntroduction Computer Science & Engineering 2/82 Design and Analysis of Algorithms Lecture 05 Single-Source Shortest Paths (Chapter 2) Stephen Scott (Adapted from Vinodchandran N. Variyam) sscott@cse.unl.edu

More information

CSE 417 Network Flows (pt 3) Modeling with Min Cuts

CSE 417 Network Flows (pt 3) Modeling with Min Cuts CSE 417 Network Flows (pt 3) Modeling with Min Cuts Reminders > HW6 is due on Friday start early bug fixed on line 33 of OptimalLineup.java: > change true to false Review of last two lectures > Defined

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms Group Members: 1. Geng Xue (A0095628R) 2. Cai Jingli (A0095623B) 3. Xing Zhe (A0095644W) 4. Zhu Xiaolu (A0109657W) 5. Wang Zixiao (A0095670X) 6. Jiao Qing (A0095637R) 7. Zhang

More information

Minimum Spanning Tree (undirected graph)

Minimum Spanning Tree (undirected graph) 1 Minimum Spanning Tree (undirected graph) 2 Path tree vs. spanning tree We have constructed trees in graphs for shortest path to anywhere else (from vertex is the root) Minimum spanning trees instead

More information

Important separators and parameterized algorithms

Important separators and parameterized algorithms Important separators and parameterized algorithms Dániel Marx 1 1 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary School on Parameterized Algorithms

More information

Maximum flow problem CE 377K. March 3, 2015

Maximum flow problem CE 377K. March 3, 2015 Maximum flow problem CE 377K March 3, 2015 Informal evaluation results 2 slow, 16 OK, 2 fast Most unclear topics: max-flow/min-cut, WHAT WILL BE ON THE MIDTERM? Most helpful things: review at start of

More information

Mathematical and Algorithmic Foundations Linear Programming and Matchings

Mathematical and Algorithmic Foundations Linear Programming and Matchings Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis

More information

Bi-directional Search in QoS Routing

Bi-directional Search in QoS Routing Bi-directional Search in QoS Routing F.A. Kuipers and P. Van Mieghem Delft University of Technology Electrical Engineering, Mathematics and Computer Science P.O Box 5031, 2600 GA Delft, The Netherlands

More information

9.1 Cook-Levin Theorem

9.1 Cook-Levin Theorem CS787: Advanced Algorithms Scribe: Shijin Kong and David Malec Lecturer: Shuchi Chawla Topic: NP-Completeness, Approximation Algorithms Date: 10/1/2007 As we ve already seen in the preceding lecture, two

More information

Introduction. I Given a weighted, directed graph G =(V, E) with weight function

Introduction. I Given a weighted, directed graph G =(V, E) with weight function ntroduction Computer Science & Engineering 2/82 Design and Analysis of Algorithms Lecture 06 Single-Source Shortest Paths (Chapter 2) Stephen Scott (Adapted from Vinodchandran N. Variyam) sscott@cse.unl.edu

More information

2.3 Optimal paths. E. Amaldi Foundations of Operations Research Politecnico di Milano 1

2.3 Optimal paths. E. Amaldi Foundations of Operations Research Politecnico di Milano 1 . Optimal paths E. Amaldi Foundations of Operations Research Politecnico di Milano Optimal (minimum or maximum) paths have a wide range of applications: Google maps, GPS navigators planning and management

More information

Graph Algorithms Matching

Graph Algorithms Matching Chapter 5 Graph Algorithms Matching Algorithm Theory WS 2012/13 Fabian Kuhn Circulation: Demands and Lower Bounds Given: Directed network, with Edge capacities 0and lower bounds l for Node demands for

More information

NP-Completeness. Algorithms

NP-Completeness. Algorithms NP-Completeness Algorithms The NP-Completeness Theory Objective: Identify a class of problems that are hard to solve. Exponential time is hard. Polynomial time is easy. Why: Do not try to find efficient

More information

1 Introduction. 2 The Generic Push-Relabel Algorithm. Improvements on the Push-Relabel Method: Excess Scaling. 2.1 Definitions CMSC 29700

1 Introduction. 2 The Generic Push-Relabel Algorithm. Improvements on the Push-Relabel Method: Excess Scaling. 2.1 Definitions CMSC 29700 CMSC 9700 Supervised Reading & Research Week 6: May 17, 013 Improvements on the Push-Relabel Method: Excess Scaling Rahul Mehta 1 Introduction We now know about both the methods based on the idea of Ford

More information

Connectivity-aware Virtual Network Embedding

Connectivity-aware Virtual Network Embedding Connectivity-aware Virtual Network Embedding Nashid Shahriar, Reaz Ahmed, Shihabur R. Chowdhury, Md Mashrur Alam Khan, Raouf Boutaba Jeebak Mitra, Feng Zeng Outline Survivability in Virtual Network Embedding

More information

Retiming. Adapted from: Synthesis and Optimization of Digital Circuits, G. De Micheli Stanford. Outline. Structural optimization methods. Retiming.

Retiming. Adapted from: Synthesis and Optimization of Digital Circuits, G. De Micheli Stanford. Outline. Structural optimization methods. Retiming. Retiming Adapted from: Synthesis and Optimization of Digital Circuits, G. De Micheli Stanford Outline Structural optimization methods. Retiming. Modeling. Retiming for minimum delay. Retiming for minimum

More information

NP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions.

NP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions. CS 787: Advanced Algorithms NP-Hardness Instructor: Dieter van Melkebeek We review the concept of polynomial-time reductions, define various classes of problems including NP-complete, and show that 3-SAT

More information

Construction of Minimum-Weight Spanners Mikkel Sigurd Martin Zachariasen

Construction of Minimum-Weight Spanners Mikkel Sigurd Martin Zachariasen Construction of Minimum-Weight Spanners Mikkel Sigurd Martin Zachariasen University of Copenhagen Outline Motivation and Background Minimum-Weight Spanner Problem Greedy Spanner Algorithm Exact Algorithm:

More information

CSE 417 Network Flows (pt 4) Min Cost Flows

CSE 417 Network Flows (pt 4) Min Cost Flows CSE 417 Network Flows (pt 4) Min Cost Flows Reminders > HW6 is due Monday Review of last three lectures > Defined the maximum flow problem find the feasible flow of maximum value flow is feasible if it

More information

An Initial Study of the Multi-Constraint Routing Problem Using Genetic Algorithm

An Initial Study of the Multi-Constraint Routing Problem Using Genetic Algorithm An Initial Study of the Multi-Constraint Routing Problem Using Genetic Algorithm Zhongchao Yu Dept. of Computer Science University of Maryand College Park, MD 20742 yuzc@cs.umd.edu ABSTRACT Multi-constrained

More information

Principles of AI Planning. Principles of AI Planning. 7.1 How to obtain a heuristic. 7.2 Relaxed planning tasks. 7.1 How to obtain a heuristic

Principles of AI Planning. Principles of AI Planning. 7.1 How to obtain a heuristic. 7.2 Relaxed planning tasks. 7.1 How to obtain a heuristic Principles of AI Planning June 8th, 2010 7. Planning as search: relaxed planning tasks Principles of AI Planning 7. Planning as search: relaxed planning tasks Malte Helmert and Bernhard Nebel 7.1 How to

More information

Greedy algorithms Or Do the right thing

Greedy algorithms Or Do the right thing Greedy algorithms Or Do the right thing March 1, 2005 1 Greedy Algorithm Basic idea: When solving a problem do locally the right thing. Problem: Usually does not work. VertexCover (Optimization Version)

More information

W[1]-hardness. Dániel Marx. Recent Advances in Parameterized Complexity Tel Aviv, Israel, December 3, 2017

W[1]-hardness. Dániel Marx. Recent Advances in Parameterized Complexity Tel Aviv, Israel, December 3, 2017 1 W[1]-hardness Dániel Marx Recent Advances in Parameterized Complexity Tel Aviv, Israel, December 3, 2017 2 Lower bounds So far we have seen positive results: basic algorithmic techniques for fixed-parameter

More information

Steiner Trees and Forests

Steiner Trees and Forests Massachusetts Institute of Technology Lecturer: Adriana Lopez 18.434: Seminar in Theoretical Computer Science March 7, 2006 Steiner Trees and Forests 1 Steiner Tree Problem Given an undirected graph G

More information

Overview of Constraint-Based Path Selection Algorithms for QoS Routing

Overview of Constraint-Based Path Selection Algorithms for QoS Routing Overview of Constraint-Based Path Selection Algorithms for QoS Routing F.A. Kuipers, Delft University of Technology T. Korkmaz, University of Texas at San Antonio M. Krunz, University of Arizona P. Van

More information

LEAST COST ROUTING ALGORITHM WITH THE STATE SPACE RELAXATION IN A CENTRALIZED NETWORK

LEAST COST ROUTING ALGORITHM WITH THE STATE SPACE RELAXATION IN A CENTRALIZED NETWORK VOL., NO., JUNE 08 ISSN 896608 00608 Asian Research Publishing Network (ARPN). All rights reserved. LEAST COST ROUTING ALGORITHM WITH THE STATE SPACE RELAXATION IN A CENTRALIZED NETWORK Y. J. Lee Department

More information

3 No-Wait Job Shops with Variable Processing Times

3 No-Wait Job Shops with Variable Processing Times 3 No-Wait Job Shops with Variable Processing Times In this chapter we assume that, on top of the classical no-wait job shop setting, we are given a set of processing times for each operation. We may select

More information

Fast Exact MultiConstraint Shortest Path Algorithms

Fast Exact MultiConstraint Shortest Path Algorithms This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 27 proceedings. Fast Exact MultiConstraint Shortest Path Algorithms

More information

Robust validation of network designs under uncertain demands and failures

Robust validation of network designs under uncertain demands and failures Robust validation of network designs under uncertain demands and failures Yiyang Chang, Sanjay Rao, and Mohit Tawarmalani Purdue University USENIX NSDI 2017 Validating network design Network design today

More information

PERFECT MATCHING THE CENTRALIZED DEPLOYMENT MOBILE SENSORS THE PROBLEM SECOND PART: WIRELESS NETWORKS 2.B. SENSOR NETWORKS OF MOBILE SENSORS

PERFECT MATCHING THE CENTRALIZED DEPLOYMENT MOBILE SENSORS THE PROBLEM SECOND PART: WIRELESS NETWORKS 2.B. SENSOR NETWORKS OF MOBILE SENSORS SECOND PART: WIRELESS NETWORKS 2.B. SENSOR NETWORKS THE CENTRALIZED DEPLOYMENT OF MOBILE SENSORS I.E. THE MINIMUM WEIGHT PERFECT MATCHING 1 2 ON BIPARTITE GRAPHS Prof. Tiziana Calamoneri Network Algorithms

More information

The Size Robust Multiple Knapsack Problem

The Size Robust Multiple Knapsack Problem MASTER THESIS ICA-3251535 The Size Robust Multiple Knapsack Problem Branch and Price for the Separate and Combined Recovery Decomposition Model Author: D.D. Tönissen, Supervisors: dr. ir. J.M. van den

More information

Finding Critical Regions and Region-Disjoint Paths in a Network Term Paper

Finding Critical Regions and Region-Disjoint Paths in a Network Term Paper Finding Critical Regions and Term Paper Jean Olimb Department of Electrical & Computer Engineering Missouri University of Science and Technology SJOFX3@mst.edu, 23 February 2017 rev. 1.0 2017 solimb Outline

More information

Lecture 4: Primal Dual Matching Algorithm and Non-Bipartite Matching. 1 Primal/Dual Algorithm for weighted matchings in Bipartite Graphs

Lecture 4: Primal Dual Matching Algorithm and Non-Bipartite Matching. 1 Primal/Dual Algorithm for weighted matchings in Bipartite Graphs CMPUT 675: Topics in Algorithms and Combinatorial Optimization (Fall 009) Lecture 4: Primal Dual Matching Algorithm and Non-Bipartite Matching Lecturer: Mohammad R. Salavatipour Date: Sept 15 and 17, 009

More information

Topic: Local Search: Max-Cut, Facility Location Date: 2/13/2007

Topic: Local Search: Max-Cut, Facility Location Date: 2/13/2007 CS880: Approximations Algorithms Scribe: Chi Man Liu Lecturer: Shuchi Chawla Topic: Local Search: Max-Cut, Facility Location Date: 2/3/2007 In previous lectures we saw how dynamic programming could be

More information

Approximation Algorithms: The Primal-Dual Method. My T. Thai

Approximation Algorithms: The Primal-Dual Method. My T. Thai Approximation Algorithms: The Primal-Dual Method My T. Thai 1 Overview of the Primal-Dual Method Consider the following primal program, called P: min st n c j x j j=1 n a ij x j b i j=1 x j 0 Then the

More information

Fast and Simple Algorithms for Weighted Perfect Matching

Fast and Simple Algorithms for Weighted Perfect Matching Fast and Simple Algorithms for Weighted Perfect Matching Mirjam Wattenhofer, Roger Wattenhofer {mirjam.wattenhofer,wattenhofer}@inf.ethz.ch, Department of Computer Science, ETH Zurich, Switzerland Abstract

More information

15-854: Approximations Algorithms Lecturer: Anupam Gupta Topic: Direct Rounding of LP Relaxations Date: 10/31/2005 Scribe: Varun Gupta

15-854: Approximations Algorithms Lecturer: Anupam Gupta Topic: Direct Rounding of LP Relaxations Date: 10/31/2005 Scribe: Varun Gupta 15-854: Approximations Algorithms Lecturer: Anupam Gupta Topic: Direct Rounding of LP Relaxations Date: 10/31/2005 Scribe: Varun Gupta 15.1 Introduction In the last lecture we saw how to formulate optimization

More information

/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18

/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18 601.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18 22.1 Introduction We spent the last two lectures proving that for certain problems, we can

More information

Lecture 7: Asymmetric K-Center

Lecture 7: Asymmetric K-Center Advanced Approximation Algorithms (CMU 18-854B, Spring 008) Lecture 7: Asymmetric K-Center February 5, 007 Lecturer: Anupam Gupta Scribe: Jeremiah Blocki In this lecture, we will consider the K-center

More information

6 Randomized rounding of semidefinite programs

6 Randomized rounding of semidefinite programs 6 Randomized rounding of semidefinite programs We now turn to a new tool which gives substantially improved performance guarantees for some problems We now show how nonlinear programming relaxations can

More information

Taking Stock. IE170: Algorithms in Systems Engineering: Lecture 20. Example. Shortest Paths Definitions

Taking Stock. IE170: Algorithms in Systems Engineering: Lecture 20. Example. Shortest Paths Definitions Taking Stock IE170: Algorithms in Systems Engineering: Lecture 20 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University March 19, 2007 Last Time Minimum Spanning Trees Strongly

More information

The Ordered Covering Problem

The Ordered Covering Problem The Ordered Covering Problem Uriel Feige Yael Hitron November 8, 2016 Abstract We introduce the Ordered Covering (OC) problem. The input is a finite set of n elements X, a color function c : X {0, 1} and

More information

PERFECT MATCHING THE CENTRALIZED DEPLOYMENT MOBILE SENSORS THE PROBLEM SECOND PART: WIRELESS NETWORKS 2.B. SENSOR NETWORKS OF MOBILE SENSORS

PERFECT MATCHING THE CENTRALIZED DEPLOYMENT MOBILE SENSORS THE PROBLEM SECOND PART: WIRELESS NETWORKS 2.B. SENSOR NETWORKS OF MOBILE SENSORS SECOND PART: WIRELESS NETWORKS.B. SENSOR NETWORKS THE CENTRALIZED DEPLOYMENT OF MOBILE SENSORS I.E. THE MINIMUM WEIGHT PERFECT MATCHING ON BIPARTITE GRAPHS Prof. Tiziana Calamoneri Network Algorithms A.y.

More information

Repetition: Primal Dual for Set Cover

Repetition: Primal Dual for Set Cover Repetition: Primal Dual for Set Cover Primal Relaxation: k min i=1 w ix i s.t. u U i:u S i x i 1 i {1,..., k} x i 0 Dual Formulation: max u U y u s.t. i {1,..., k} u:u S i y u w i y u 0 Harald Räcke 428

More information

α Coverage to Extend Network Lifetime on Wireless Sensor Networks

α Coverage to Extend Network Lifetime on Wireless Sensor Networks Noname manuscript No. (will be inserted by the editor) α Coverage to Extend Network Lifetime on Wireless Sensor Networks Monica Gentili Andrea Raiconi Received: date / Accepted: date Abstract An important

More information

Simple Quality-of-Service Path First Protocol and Modeling Analysis*

Simple Quality-of-Service Path First Protocol and Modeling Analysis* Simple Quality-of-Service Path First Protocol and Modeling Analysis* Lin Shen, Mingwei Xu, Ke Xu, Yong Cui, Youjian Zhao Department of Computer Science, Tsinghua University, Beijing, P.R.China, 100084

More information

Optimal Topology Design for Overlay Networks

Optimal Topology Design for Overlay Networks Optimal Topology Design for Overlay Networks Mina Kamel 1, Caterina Scoglio 1, and Todd Easton 2 1 Electrical and computer Engineering Department 2 Industrial and Manufacturing Systems Engineering Department

More information

Introduction to Approximation Algorithms

Introduction to Approximation Algorithms Introduction to Approximation Algorithms Dr. Gautam K. Das Departmet of Mathematics Indian Institute of Technology Guwahati, India gkd@iitg.ernet.in February 19, 2016 Outline of the lecture Background

More information

An O(log n/ log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem

An O(log n/ log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem An O(log n/ log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem and more recent developments CATS @ UMD April 22, 2016 The Asymmetric Traveling Salesman Problem (ATSP) Problem

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design and Analysis LECTURE 29 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/7/2016 Approximation

More information

CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 14: Combinatorial Problems as Linear Programs I. Instructor: Shaddin Dughmi

CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 14: Combinatorial Problems as Linear Programs I. Instructor: Shaddin Dughmi CS599: Convex and Combinatorial Optimization Fall 2013 Lecture 14: Combinatorial Problems as Linear Programs I Instructor: Shaddin Dughmi Announcements Posted solutions to HW1 Today: Combinatorial problems

More information

Communication Networks I December 4, 2001 Agenda Graph theory notation Trees Shortest path algorithms Distributed, asynchronous algorithms Page 1

Communication Networks I December 4, 2001 Agenda Graph theory notation Trees Shortest path algorithms Distributed, asynchronous algorithms Page 1 Communication Networks I December, Agenda Graph theory notation Trees Shortest path algorithms Distributed, asynchronous algorithms Page Communication Networks I December, Notation G = (V,E) denotes a

More information

OPTICAL NETWORKS. Virtual Topology Design. A. Gençata İTÜ, Dept. Computer Engineering 2005

OPTICAL NETWORKS. Virtual Topology Design. A. Gençata İTÜ, Dept. Computer Engineering 2005 OPTICAL NETWORKS Virtual Topology Design A. Gençata İTÜ, Dept. Computer Engineering 2005 Virtual Topology A lightpath provides single-hop communication between any two nodes, which could be far apart in

More information

Solutions for the Exam 6 January 2014

Solutions for the Exam 6 January 2014 Mastermath and LNMB Course: Discrete Optimization Solutions for the Exam 6 January 2014 Utrecht University, Educatorium, 13:30 16:30 The examination lasts 3 hours. Grading will be done before January 20,

More information

Basic Approximation algorithms

Basic Approximation algorithms Approximation slides Basic Approximation algorithms Guy Kortsarz Approximation slides 2 A ρ approximation algorithm for problems that we can not solve exactly Given an NP-hard question finding the optimum

More information

On the Robustness of Distributed Computing Networks

On the Robustness of Distributed Computing Networks 1 On the Robustness of Distributed Computing Networks Jianan Zhang, Hyang-Won Lee, and Eytan Modiano Lab for Information and Decision Systems, Massachusetts Institute of Technology, USA Dept. of Software,

More information

Randomized rounding of semidefinite programs and primal-dual method for integer linear programming. Reza Moosavi Dr. Saeedeh Parsaeefard Dec.

Randomized rounding of semidefinite programs and primal-dual method for integer linear programming. Reza Moosavi Dr. Saeedeh Parsaeefard Dec. Randomized rounding of semidefinite programs and primal-dual method for integer linear programming Dr. Saeedeh Parsaeefard 1 2 3 4 Semidefinite Programming () 1 Integer Programming integer programming

More information

W[1]-hardness. Dániel Marx 1. Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary

W[1]-hardness. Dániel Marx 1. Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary W[1]-hardness Dániel Marx 1 1 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary School on Parameterized Algorithms and Complexity Będlewo, Poland

More information

Designing robust network topologies for wireless sensor networks in adversarial environments

Designing robust network topologies for wireless sensor networks in adversarial environments Designing robust network topologies for wireless sensor networks in adversarial environments Aron Laszka a, Levente Buttyán a, Dávid Szeszlér b a Department of Telecommunications, Budapest University of

More information

On the Robustness of Distributed Computing Networks

On the Robustness of Distributed Computing Networks 1 On the Robustness of Distributed Computing Networks Jianan Zhang, Hyang-Won Lee, and Eytan Modiano Lab for Information and Decision Systems, Massachusetts Institute of Technology, USA Dept. of Software,

More information

On the Min-Max 2-Cluster Editing Problem

On the Min-Max 2-Cluster Editing Problem JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 29, 1109-1120 (2013) On the Min-Max 2-Cluster Editing Problem LI-HSUAN CHEN 1, MAW-SHANG CHANG 2, CHUN-CHIEH WANG 1 AND BANG YE WU 1,* 1 Department of Computer

More information

Lecture 6: Linear Programming for Sparsest Cut

Lecture 6: Linear Programming for Sparsest Cut Lecture 6: Linear Programming for Sparsest Cut Sparsest Cut and SOS The SOS hierarchy captures the algorithms for sparsest cut, but they were discovered directly without thinking about SOS (and this is

More information

2386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 6, JUNE 2006

2386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 6, JUNE 2006 2386 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 6, JUNE 2006 The Encoding Complexity of Network Coding Michael Langberg, Member, IEEE, Alexander Sprintson, Member, IEEE, and Jehoshua Bruck,

More information

Outline: Finish uncapacitated simplex method Negative cost cycle algorithm The max-flow problem Max-flow min-cut theorem

Outline: Finish uncapacitated simplex method Negative cost cycle algorithm The max-flow problem Max-flow min-cut theorem Outline: Finish uncapacitated simplex method Negative cost cycle algorithm The max-flow problem Max-flow min-cut theorem Uncapacitated Networks: Basic primal and dual solutions Flow conservation constraints

More information

UNIVERSITY OF TORONTO Department of Computer Science April 2014 Final Exam CSC373H1S Robert Robere Duration - 3 hours No Aids Allowed.

UNIVERSITY OF TORONTO Department of Computer Science April 2014 Final Exam CSC373H1S Robert Robere Duration - 3 hours No Aids Allowed. UNIVERSITY OF TORONTO Department of Computer Science April 2014 Final Exam CSC373H1S Robert Robere Duration - 3 hours No Aids Allowed. PLEASE COMPLETE THE SECTION BELOW AND THE SECTION BEHIND THIS PAGE:

More information

Unicast and Multicast QoS Routing with Multiple Constraints

Unicast and Multicast QoS Routing with Multiple Constraints 481 Unicast and Multicast QoS Routing with Multiple Constraints Dan Wang 1, Funda Ergun 1,andZhanXu 2 1 School of Computer Science, Simon Fraser University, Burnaby BC V5A 1S6, Canada {danw,funda}@cs.sfu.ca

More information

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 3, MARCH 2016 1907 Joint Flow Routing and DoF Allocation in Multihop MIMO Networks Xiaoqi Qin, Student Member, IEEE, Xu Yuan, Student Member,

More information

11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

More information

Algorithms for Integer Programming

Algorithms for Integer Programming Algorithms for Integer Programming Laura Galli November 9, 2016 Unlike linear programming problems, integer programming problems are very difficult to solve. In fact, no efficient general algorithm is

More information