ScalingUp Urban Network Security Games with Graph Contraction


 Myles Ferguson
 2 years ago
 Views:
Transcription
1 ScalingUp Urban Network Security Games with Graph Contraction Hiroaki Iwashita Kotaro Ohori Hirokazu Anai Fujitsu Laboratories Ltd. Kawasaki , Japan Abstract Urban network security games, which model attackers and defenders on road networks, provide a promising framework for planning effective strategies of patrolling and inspection. This paper focuses on improving scalability of the algorithm to solve them, which is one of the important challenges in security game problems. Stateoftheart solvers have been scaled up to handle realworld networks with tens of thousands of edges; however, it can take days or more to solve them when we vary problem settings of sources, targets, and security resources. This paper provides a novel method MiCANS to improve the scalability drastically, which includes two features: (1) the strategy spaces of both players are reduced dramatically through edge contraction by identifying the edges that are not essential for defender s optimal strategy, based on the technique that we call cut arrangement ; and (2) the reduced problem is solved efficiently with a mixedinteger quadratic programg oracle. Although theoretical analysis have not been made yet enough, experimental results show that it finds optimal solutions in most cases and achieves several orders of magnitude speedup compared to the existing algorithms. 1 Introduction Effective safety policies are required to prevent terrorism and drug dealing with the deteriorating security situation around the world. However, all possible security checkpoints cannot be covered at every moment because we have limited security resources. Over recent years, gametheoretic approaches have been applied to plan effective resource allocation strategies for patrolling and inspection. The security games modeled based on a Stackelberg game [Von Stackelberg, 1934] have provided various applications in realworld domains, for example, LA International Airport [Pita et al., 2008], US Federal Air Marshals Service [Tsai et al., 2009], and US Coast Guard [Shieh et al., 2012; Fang et al., 2013]. As one of the important game types, there is a security game for urban networks, which models an attacker and a defender who take a decision on the network consisting of nodes and edges. The defender s strategies represent the allocations of their limited resources to edges on the network, while the attacker has paths from any source node to any target one as its strategies. This paper focuses on improving scalability of the algorithms for solving urban network security problems, which is one of important challenges in security games [Tambe et al., 2014]. The strategy spaces for both the defender and the attacker increase exponentially with the number of security resources and the size of the network, respectively. There have been remarkable studies to scale up to realistic problem sizes in the urban network security games. [Tsai et al., 2010] provided RANGER which obtains an approximate solution for the defender by operating on the marginal probabilities of edges rather than solving for a distribution over allocations. [Jain et al., 2011] presented RUGGED based on a doubleoracle approach [McMahan et al., 2003] which need not to enumerate the all pure strategies for either of the attacker and the defender. Although the algorithm can obtain a global equilibrium strategy by computing the best responses iteratively, it is restricted to scale up to large scale networks because the large number of iterations is required to converge the strategy. To improve the scalability, [Jain et al., 2013] introduced SNARES that can obtain optimal strategies for Mumbai road network comprising 9,503 nodes and 20,416 edges. However, the algorithm in the previous research cannot be always applied in the real world urban road network without any condition. The runtime required by an algorithm in security games varies significantly depending on the number of targets and resources [Jain et al., 2012]. In our experiment with the SNARES algorithm, a larger the number of targets and resources resulted in an extraordinarily high computing time to gain an optimal solution. Moreover, experiments when varying only the places of sources and targets showed widely different runtime for every run. This paper provides a novel method MiCANS (MinCut Arrangement for Network Security) to improve the scalability drastically, the method of which yields relatively robust results even if varying the placements of resources and targets. MiCANS has the following two features: (1) the strategy spaces of both players are reduced dramatically through edge contraction by identifying the edges that are not essential for
2 defender s optimal strategy, based on the technique that we call cut arrangement; and (2) the reduced problem is solved efficiently with a mixedinteger quadratic programg oracle. The solutions gained using the method generate approximately same values to optimal ones with SNARES. The method also show orders of magnitude speedups compared to SNARES, and so can be applied to various types of realistic problems. 2 Urban Network Security Problem 2.1 Problem Definition Urban network security game [Tsai et al., 2010] is a zerosum game with two players: a defender and an attacker. The network security game is modeled using a graph G = (V, E) describing a road network, a set of source nodes S V a set of target nodes T V a payoff U(t) > 0 for each target t T, and the number of defender resources k. The defender allocates up to the k defender resources to any k on the E edges. The attacker moves along a path on the edges of G starting at any one of the source node s S to any one of the targets s S. The attacker fails if a defender resource is on its path to the target t, otherwise the attacker gains U(t) for a successful attack on target t. Conversely, the defender gets U(t), which we call damage, if the attacker successfully attacks t. The objective of the security game is to find an optimal mixed strategy for the defender, which means the most effective allocation plan of the defender resources, in terms of imizing the maximum expected damage. Since this is a zerosum game, its optimal solution is a Nash equilibrium as well as a strong Stackelberg equilibrium [Yin et al., 2010]. We note that an attacker is able to start from any source, so there is no reasonable efficacy for the attacker to choose a path via some sources. Therefore, in the case of network security game with multiple sources, the contraction of the associated graph G through merging multiple sources to a single source does not change its optimal solution. In the same way, the targets with the same payoff can be contracted. Let D be a set of all defender allocations and A be a set of all attacker paths. We denote a defender s mixed strategy by x : D [0,1] and expectation of the damage by z. The attacker s path a A gets U(a) if the attack succeeds. The probability that the attacker s path a becomes successful against defender allocation d is denoted by P(d, a) {0, 1}. Then the network security game can be formulated as a imax linear programg problem described by the following formulas: x,z z (1) z U(a) (P(d,a) x(d)), a A (2) d D x(d) = 1 (3) d D 0 x(d) 1, d D. (4) By solving formulas (1) (4), we obtain an optimal defender s mixed strategy and expectation of the damage at that time. Moreover, the solution of the dual problem provides an optimal attacker s mixed strategy y : A [0, 1]. This basic formulation, however, cannot be applied directly to realworld problems since D and A increase exponentially with the problem size. 2.2 MinCut Method It is obvious that a defender can defend completely by allocating resources to all cut edges if the size of cuts separating sources and targets (i.e., total number of cut edges) is not larger than the number k of resources. It is shown in [Washburn and Wood, 1995] that a defender s optimal strategy is to allocate resources to k cut edges randomly chosen among a cut if there is only one target (or all targets have same values) even in the case that the size of cuts is larger than k. This method enables to reduce the problem with exponential complexity in the size of a given graph G into polynomial time complexity. However, it is not always true that a unique cut which separate sources from targets provides an optimal solution for the cases targets values are not uniform. 2.3 DoubleOracle Method As stateoftheart algorithms to solve general urban network security games, RUGGED [Jain et al., 2011] and SNARES [Jain et al., 2013] based on doubleoracle methods have been proposed. RUGGED, for solving the imax problem given in formulas (1) (4), is a doubleoracle method [McMahan et al., 2003], which corresponds to constraint and column generation of linear programg problems, in order to prevent the enumeration of all defender allocations D and all attacker paths A. The doubleoracle method reaches an optimal solution asymptotically by adding the best response strategy against an equilibrium strategy to the game instead of finding an optimal mixed strategy after enumerating all pure strategies. The outline of doubleoracle method in RUGGED is shown in Algorithm 1. Computation of best responses of both play Algorithm 1: Doubleoracle method 1 Initialize D by arbitrary defender allocations 2 Initialize A by arbitrary attacker paths 3 repeat 4 Solve the imax LP problem of equations (1) (4), and let x and y be mixed strategies of the defender and the attacker over D and A respectively 5 Find defender s best response to y and add it to D 6 Find attacker s best response to x and add it to A 7 until convergence 8 return x ers in lines 5 and 6 is achieved by solving mixedinteger linear programg problems. We note that the modules require long computation time in the whole procedure in RUGGED. In SNARES, Algorithm 1 has been improved by adding the following two features. (1) On computing optimal responses for both players, SNARES first tries to find better responses
3 Figure 1: Overview of MiCANS quickly with greedy heuristics; if it fails, then uses a mixedinteger linear prograg solver. (2) A method called cutfanout is used to improve initial values of D and A. Mincutfanout first constructs a cut which separates the target with the highest value from the sources; and then initializes D by defender allocations such that each allocation cover k edges of the cut, and A by attacker s best responses to them. 3 MiCANS Algorithm In this section, we present MiCANS, a novel method to deal with massive network security games. Figure 1 illustrates an overview of MiCANS. It includes an algorithm, which we call cut arrangement, to generate a reduced game incrementally from the original massive game. MiCANS also provides an oraclebased efficient solution to the reduced game. 3.1 MinCut Arrangement When we investigate optimal strategies of the defender, it is often found that many of the graph edges are never used for defender allocations. Supposing that those edges can be detered in advance, we have great potential for reducing the problem. Leaving discussion on how to reduce the problem until the next subsection, here we introduce an algorithm to pick out good candidate edges for defender allocations. The cut method for onetarget problems [Washburn and Wood, 1995] and cutfanout method in SNARES [Jain et al., 2013] implies that cuts between sources and targets are keys to find them. A cut separating multiple targets from the sources gives the best set of edges for uniformly enhancing security of those targets. Mincut arrangement combines a plurality of such cuts to construct a set of candidate edges. The outline of MiCANS is shown in Algorithm 2, where E is a current set of candidate edges and x is a current mixed strategy of the defender. The initial values of E and x are empty, meaning that defender resources are not allocated on any edge. A critical target is the one that will suffer the highest expectation of the damage (the product of its value and success probability of attacker s path to it) under x. In each iteration, candidate edges are added in such a way that protection of all critical targets can be reinforced uniformly. It is repeated until no new candidate edges are added. Although it is not proven that the algorithm always finds sufficient candidate edges for an optimal solution, we have experimentally confirmed that they would be enough for practical use. Algorithm 2: MiCANS 1 Let S be the set of source nodes 2 Initialize E and x to be empty 3 repeat 4 Let T be a set of all critical targets under x 5 Add cut edges between S and T to E 6 Update x by solving the security game under the constraint that only the edges in E can be used 7 until convergence 8 return x 3.2 Security Games under Edge Restriction Supposing that the defender has selected a subset E E of edges in the original graph G = (V,E) as a set of candidate edges, we can reduce strategy space of the attacker as well as the defender. Since the defender allocates resources only on candidate edges, the attacker can move freely through all other edges without any risk; both players do not need to take attacker s individual paths on those free edges into account for optimizing their strategies. Therefore, edge contraction of every e E \ E, which makes the graph composed only of candidate edges, reduces the problem safely. Contraction of many edges can lead to many occurrences of multiple edges, which are two or more edges that are incident to the same two vertices. An optimal strategy of the defender should use those edges uniformly since they are topologically equivalent. We do not need to distinguish each member of multiple edges and they can be simplified as a single edge with the capacity that represents the number of original edges. For each pair of nodes connected by n edges, we merge them into a single edge e with capacity C e = n. After all, the original graph G = (V,E) is reduced to a smaller weighted graph Ĝ = ( ˆV,Ê), in which each edge e Ê has capacity C e of a positive integer. Now let d e {0,...,C e } denote the number of resources on edge e in defender allocation d. When the defender and the attacker play d and a respectively on Ĝ, the attack succeeds at probability e a (1 d e /C e ). Therefore, this game is formulated as follows: x,z z (5) ( ( z U(a) 1 d ) ) e x(d), a A (6) d D e a C e x(d) = 1 (7) d D 0 x(d) 1, d D; (8) where D and A are all defender allocations and all attacker paths on Ĝ respectively, x : D [0,1] is the defender s mixed strategy, z is an expectation of the damage, and U(a) is the amount of damage when attacker path a A succeeds. 3.3 A Small Example We demonstrate an execution example of MiCANS algorithm using a small graph illustrated in Figure 2(a), where s is a
4 (a) Original graph (b) Step 1 (c) Step 2 (d) Step 3 (e) Step 3 Figure 2: An execution example source node, t 1 is a target node with value of 1, and t 2 is another target node with value of 2. Let us suppose that two resources can be used by the defender. The critical target at the first step is t 2 and a cut between s and t 2, namely, c 1 is computed. Now, candidate edges are the two edges crossing c 1. A weighted graph of the first step is shown in Figure 2(b), which leads the first solution that completely guards t 2 by placing the two resources on the single edge. At the second step, the critical target changes to t 1 and the weighted graph is updated as Figure 2(c). As the optimal mixed strategy for the defender balances expected damage of attacks on t 1 and t 2 to 2/3, both targets become critical at the third step. There are multiple options of cuts such as c 3, c 3 and c 3 in this step; taking c 3 makes the weighted graph as Figure 2(d) and others as Figure 2(e). We can get an optimal solution of which expected damage is 1/2, from either graph even though they are topologically different. 3.4 Further Reduction of the Strategy Space Let D e = {0,...,C e } and D e = D e \ {0,C e }. We say that edge e is partially blocked by defender allocation d when d e D e. The following theorem allows us to restrict the defender s strategy space to only the defender allocations that do not partially block two or more edges. Theorem 1. At least one optimal mixed strategy for the defender includes no defender allocation by which two or more edges are partially blocked. Proof. Let us assume that defender allocation d partially blocks edges i and j. Let d and d be defender allocations such that d i i if e = i, d i + j if e = i, d e = d j + i if e = j, d e = d j j if e = j, otherwise; otherwise; d e where i and j are set to the largest acceptable value: i = (d i,c j d j ) and j = (d j,c i d i ). Note that at most one of edges i and j are partially blocked by d and by d. We replace d at probability p in any mixed strategy with d and d at probabilities j i + j p and i i + j p respectively. It can be shown by comparing probability of successful attacks that this replacement never makes the strategy worse. By repeating it, we can safely eliate all the defender allocations that partially block two or more edges from any optimal mixed strategy. d e On the other hand, attacker s strategy space can be reduced by excluding paths passing over some target that is more valuable than their destinations. 3.5 Efficient Solution with MIQP Oracle The attacker s strategy space becomes small enough on the reduced weighted graphs, while the defender s strategy space tends to remain rather large even when the restriction in Section 3.4 is applied. Algorithm 3 summarizes the procedure to solve the security game in MiCANS, where we use a single oracle of defender s best response. Algorithm 3: Singleoracle method in MiCANS 1 Initialize D by arbitrary defender allocation 2 Initialize A by all attacker paths 3 repeat 4 Solve the imax LP problem of equations (5) (8), and let x and y be mixed strategies of the defender and the attacker over D and A respectively 5 Find defender s best response to y and add it to D 6 until convergence 7 return x The oracle finds one of the best defender allocation d under attacker s mixed strategy y. It is formulated as a mixedinteger nonlinear programg (MINLP) problem: d a A ( U(a)y(a) e a ( 1 d e C e ) ) (9) d e k (10) e Ê d e D e, e Ê. (11) Unfortunately, the above MINLP can not be accepted directly by many generic optimization solvers. We transform it into a mixedinteger quadratic programg (MIQP) problem using the restriction on d introduced in Section 3.4. Let λ(e,r) be a binary variable representing if the number of resources allocated on edge e is r, or d e = r. From the restriction, no two r D e satisfies λ(e,r) = 1 for all e Ê. It allows us to rewrite equations (9) (11) as follows: λ,ω a A e Ê U(a) y(a) 1 e a r D e r λ(e,r) (1 ω(a)) C e (12) r D e r λ(e,r) k (13) e Ê r D e λ(e,r) 1 (14) r D e λ(e,r) 1, e Ê (15) ω(a) λ(e,c e ), a A (16) e a λ(e,r) {0,1}, e Ê, r D e (17) ω(a) [0,1], a A. (18)
5 Equation (13) says that the total number of allocated resources does not exceed k. Equations (14) (15) corresponds to the restriction introduced in Section 3.4. Equation (16) defines the relation between ω and λ; ω(a) = 1 if attacker path a is completely blocked by the defender at some edge e a, otherwise ω(a) = 0. 4 Experimental Results We have implemented MiCANS and SNARES algorithms in Python with NetworkX graph library [Schult and Swart, 2008] and Gurobi mathematical programg solver. All experiments were performed on a machine with 2.9GHz CPU and 10GB memory. We used three types of network models: 1. Random road network models called GRE (Grid model with Random Edges) [Peng et al., 2014], which is a planar connected graph made of l w square grid of nodes, horizontal/vertical edges between neighbors are controlled by probability p, and diagonal ones by q. It is reported that the value of p spread in [0.3,0.9] and q in [0.1, 0.7] in models that best matches with road networks of Europe. We fixed sources to w nodes at the bottom end, while the target nodes are selected randomly. 2. Random geometric graphs, which also have been shown to mimic some properties of real road networks [Eppstein and Goodrich, 2008] and was used to evaluate algorithms in [Jain et al., 2013]. We have distributed n nodes uniformly at random in a unit square and connected all two nodes between which distance is not more than given threshold d. Source and target nodes are selected randomly. 3. A real road network data of Mumbai, made of 21,132 nodes and 33,603 edges. We have extracted it from a rectangle area of latitude to and longitude to of the Open Street Map [Haklay and Weber, 2008]. Source and target nodes are selected randomly. 4.1 Effectiveness of Graph Reduction We generated random 400 instances of the GRE model, of which horizontal and vertical sizes are chosen randomly from {10,...,300}, p from [0.3,0.9], and q from [0.1,0.7]. Figure 3(a) presents the relation between the number of nodes and edges of the final reduced graph made by cut arrangement. It is confirmed that every graph has small edge/node ratio, which prevents explosion of the number of attacker paths since most nodes have degree of 1 or 2. The actual number of attacker paths enumerated on every reduced graph was less than 200 in these examples. The distribution of original graph size versus reduced one in Figure 3(b) shows that size of the graphs have been reduced by several orders of magnitude. It should be further emphasized that no direct correlation was found between them. Figure 3(c) and Figure 3(d) shows that the size of reduced graph is strongly related to the number of targets but not to sources. It is obvious when sufficient resources are available with respect to targets. (a) Vs. nodes after reduction (c) Vs. number of sources (b) Vs. edges before reduction (d) Vs. number of targets Figure 3: Number of edges after reduction (a) Varying number of nodes (c) Varying number of targets (b) Varying edge probabilities (d) Varying number of resources Figure 4: Runtime for GRE models 4.2 Comparison with SNARES We compared performance of MiCANS with SNARES using the GRE models and random geometric graphs. For the GRE models, we have experimented around the following central parameter values: nodes, (p, q) = (0.6, 0.4), 3 targets, and 3 resources. Figure 4 presents the results when varying (a) the number of nodes, (b) edge probabilities, (c) the number of targets, and (d) the number of resources, from the central setting. Each experiment is repeated 10 times with different random seeds to exae uncertainty of the runtime. The marks on the top margin indicates the number of runs of SNARES that have not finished within 1,000 seconds. All these results show that execution time of SNARES varies extremely, while MiCANS is very robust and all runs are finished within a second. Figure 5 presents results for random geometric graphs, where we selected central parameter values as 100 nodes,
6 (a) Varying number of nodes (b) Varying distance threshold (a) Varying number of nodes (b) Varying edge parameters (c) Varying number of targets (d) Varying number of resources (c) Varying number of targets (d) Varying number of resources Figure 5: Runtime for random geometric graphs Figure 7: Runtime for large GRE models Figure 6: Total number of inner iterations for GRE models (a) Varying numver of targets (b) Varying number of resources d = 0.2, 3 sources, 3 targets, and 3 resources. We can find almost the same trends as the GRE models. Comparison of the numbers of inner iterations (the number of runs of imax optimizations) in the algorithms is plotted in Figure 6. It also shows significant reduction by MiCANS and no clear correlation of difficulties between the original problem and the reduced one. We have compared expected payoff values for all 308 problems solved by both SNARES and MiCANS and confirmed that they are always the same; it follows that MiCANS have successfully found optimal solutions for those problems. 4.3 Scalability We evaluated scalability of MiCANS using more difficult problems, which our implementation of SNARES could not solve. Experiments are performed using 10 random instances for each parameter setting. Figure 7 presents results for larger GRE models, where we selected central parameter values as nodes, (p,q) = (0.6,0.4), 20 targets, and 20 resources. Note that the runtime is not logscaled, showing that MiCANS is still stable for large problems. When the original graph size is very large (e.g. >100k nodes), runtime was doated by cut computation. Results were similar for large random geometric graphs. Figure 8: Runtime for Mumbai road network Figure 8(a) presents the results for the Mumbai road network when varying the number of targets from 10 to 40 and fixing both sources and resources to 20. Figure 8(b) presents the results when varying the number of resources from 10 to 60 and fixing both sources and targets to 20. Those problem settings are much harder than the ones in [Jain et al., 2013]. All results confirm that MiCANS is robust enough and shows gradual growth of the computational cost with respect to size of the problems. 5 Conclusions We have proposed MiCANS, the method that drastically improves the scalability of solvers for urban network security games. Mincut arrangement effectively reduces the original graph to a very small one by abstracting out the edges that are not essential for defender s optimal strategy. We have also presented a MIQP formulation of a defender oracle for efficient solution of the reduced game. Although MiCANS can actually find optimal solutions for realistic examples, it is still an unresolved issue for us to discover a strict method to check if the set of candidate edges are sufficient for constructing an optimal solution.
7 References [Eppstein and Goodrich, 2008] David Eppstein and Michael T Goodrich. Studying (nonplanar) road networks through an algorithmic lens. In Proceedings of the 16th ACM SIGSPATIAL international conference on Advances in geographic information systems, page 16. ACM, [Fang et al., 2013] Fei Fang, Albert Xin Jiang, and Milind Tambe. Optimal patrol strategy for protecting moving targets with multiple mobile resources. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages , [Haklay and Weber, 2008] Mordechai Haklay and Patrick Weber. Openstreetmap: Usergenerated street maps. Pervasive Computing, IEEE, 7(4):12 18, [Jain et al., 2011] Manish Jain, Dmytro Korzhyk, Ondřej Vaněk, Vincent Conitzer, Michal Pěchouček, and Milind Tambe. A double oracle algorithm for zerosum security games on graphs. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages , [Jain et al., 2012] Manish Jain, Kevin LeytonBrown, and Milind Tambe. The deploymenttosaturation ratio in security games. In Conference on Artificial Intelligence (AAAI), pages , [Jain et al., 2013] Manish Jain, Vincent Conitzer, and Milind Tambe. Security scheduling for realworld networks. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages , [McMahan et al., 2003] H Brendan McMahan, Geoffrey J Gordon, and Avrim Blum. Planning in the presence of cost functions controlled by an adversary. In International Conference on Machine Learning, pages , [Peng et al., 2014] Wei Peng, Guohua Dong, Kun Yang, and Jinshu Su. A random road network model and its effects on topological characteristics of mobile delaytolerant networks. IEEE Transactions on Mobile Computing, 13(12): , [Pita et al., 2008] James Pita, Manish Jain, Janusz Marecki, Fernando Ordóñez, Christopher Portway, Milind Tambe, Craig Western, Praveen Paruchuri, and Sarit Kraus. Deployed armor protection: the application of a game theoretic model for security at the los angeles international airport. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages , [Schult and Swart, 2008] Daniel A Schult and PJ Swart. Exploring network structure, dynamics, and function using networkx. In Proceedings of the 7th Python in Science Conferences (SciPy 2008), volume 2008, pages 11 16, [Shieh et al., 2012] Eric Shieh, Bo An, Rong Yang, Milind Tambe, Craig Baldwin, Joseph DiRenzo, Ben Maule, and Garrett Meyer. Protect: A deployed game theoretic system to protect the ports of the united states. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 13 20, [Tambe et al., 2014] Milind Tambe, Albert Xin Jiang, Bo An, and Manish Jain. Computational game theory for security: Progress and challenges. In AAAI Spring Symposium on Applied Computational Game Theory, [Tsai et al., 2009] Jason Tsai, Shyamsunder Rathi, Christopher Kiekintveld, Fernando Ordonez, and Milind Tambe. Irisa tool for strategic security allocation in transportation networks. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), [Tsai et al., 2010] Jason Tsai, Zhengyu Yin, Junyoung Kwak, David Kempe, Christopher Kiekintveld, and Milind Tambe. Urban security: Gametheoretic resource allocation in networked physical domains. In Conference on Artificial Intelligence (AAAI), pages , [Von Stackelberg, 1934] Heinrich Von Stackelberg. Marktform und Gleichgewicht. J. Springer, [Washburn and Wood, 1995] Alan Washburn and Kevin Wood. Twoperson zerosum games for network interdiction. Operations Research, 43(2): , [Yin et al., 2010] Zhengyu Yin, Dmytro Korzhyk, Christopher Kiekintveld, Vincent Conitzer, and Milind Tambe. Stackelberg vs. nash in security games: Interchangeability, equivalence, and uniqueness. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages , 2010.
Game Theory for Security: A RealWorld Challenge Problem for Multiagent Systems and Beyond
AAAI Technical Report SS1203 Game Theory for Security, Sustainability and Health Game Theory for Security: A RealWorld Challenge Problem for Multiagent Systems and Beyond Milind Tambe, Bo An Computer
More informationSecurity Games with Protection Externalities
Jiarui Gan The Key Laboratory of Intelligent Information Processing, ICT, CAS University of Chinese Academy of Sciences Beijing 100190, China ganjr@ics.ict.ac.cn Security Games with Protection Externalities
More informationManish Jain. Phone: +1 (213)
Manish Jain Phone: +1 (213) 400 0882 http://teamcore.usc.edu/manish Email: manish.jain@usc.edu Research Interests: Computational Gametheory, Largescale Optimization, Realworld Security Applications,
More informationGame Theory for Security: Key Algorithmic Principles, Deployed Systems, Lessons Learned
Game Theory for Security: Key Algorithmic Principles, Deployed Systems, Lessons Learned Milind Tambe*, Manish Jain*, James Adam Pita*, Albert Xin Jiang* Abstract Security is a critical concern around the
More informationStackelberg Security Games for Security. Fernando Ordóñez Universidad de Chile
Stackelberg Security Games for Security Fernando Ordóñez Universidad de Chile Stackelberg Games for Security Fernando Ordóñez Universidad de Chile Stackelberg Games for Security Fernando Ordóñez Milind
More informationComputing GameTheoretic Solutions and Applications to Security
Computing GameTheoretic Solutions and Applications to Security Vincent Conitzer Departments of Computer Science and Economics Duke University Durham, NC 27708, USA conitzer@cs.duke.edu Abstract The multiagent
More informationStackelberg Security Games (SSG) Basics and Application Overview
Stackelberg Security Games (SSG) Basics and Application Overview Bo An School of Computer Engineering Nanyang Technological University Singapore, 639798 boan@ntu.edu.sg Milind Tambe, Arunesh Sinha Computer
More informationComplexity of Computing Optimal Stackelberg Strategies in Security Resource Allocation Games
Complexity of Computing Optimal Stackelberg Strategies in Security Resource Allocation Games Dmytro Korzhyk, Vincent Conitzer, Ronald Parr Duke University, Durham, NC 27708, USA {dima, conitzer, parr}@cs.duke.edu
More informationSecurity Games Applied to RealWorld: Research Contributions and Challenges
Security Games Applied to RealWorld: Research Contributions and Challenges Manish Jain and Bo An and Milind Tambe Abstract The goal of this chapter is to introduce a challenging realworld problem for
More informationBo An. Research Appointments. Chinese Academy of Sciences. Institute of Computing Technology
Bo An Institute of Computing Technology Phone: (010) 62601343 Chinese Academy of Sciences Fax: (010) 62601343 No.6 Kexueyuan South Road, Rm 1343 Email: boan@ict.ac.cn Haidian District Beijing, China 100190
More informationAnalyzing the Effectiveness of Adversary Modeling in Security Games
Proceedings of the TwentySeventh AAAI Conference on Artificial Intelligence Analyzing the Effectiveness of Adversary Modeling in Security Games Thanh H. Nguyen 1, Rong Yang 1, Amos Azaria 2, Sarit Kraus
More informationSoftware Assistants for Randomized Patrol Planning for The LAX Airport Police and The Federal Air Marshals Service
INTERFACES Vol. 00, No. 0, Xxxxx 0000, pp. 000 000 ISSN 00922102 EISSN 1526551X 00 0000 0001 INFORMS DOI 10.1287/xxxx.0000.0000 c 0000 INFORMS Software Assistants for Randomized Patrol Planning for The
More informationCompact Representations and Approximations for Compuation in Games
Compact Representations and Approximations for Compuation in Games Kevin Swersky April 23, 2008 Abstract Compact representations have recently been developed as a way of both encoding the strategic interactions
More informationPreventing Illegal Logging: Simultaneous Optimization of Resource Teams and Tactics for Security
Preventing Illegal Logging: Simultaneous Optimization of Resource Teams and Tactics for Security Sara Mc Carthy, Milind Tambe University of Southern California Christopher Kiekintveld University of Texas
More informationChapter 13: Binary and MixedInteger Programming
Chapter 3: Binary and MixedInteger Programming The general branch and bound approach described in the previous chapter can be customized for special situations. This chapter addresses two special situations:
More informationGame Theory Meets Information Security Management
Game Theory Meets Information Security Management Andrew Fielder 1, Emmanouil Panaousis 2, Pasquale Malacaria 2, Chris Hankin 1, and Fabrizio Smeraldi 2 1 Imperial College London {andrew.fielder,c.hankin}@imperial.ac.uk
More informationHuman Adversaries in Opportunistic Crime Security Games: How Past success (or failure) affects future behavior
Abstract There are a growing number of automated decision aids based on gametheoretic algorithms in daily use by security agencies to assist in allocating or scheduling their limited security resources.
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationMultilayer Structure of Data Center Based on Steiner Triple System
Journal of Computational Information Systems 9: 11 (2013) 4371 4378 Available at http://www.jofcis.com Multilayer Structure of Data Center Based on Steiner Triple System Jianfei ZHANG 1, Zhiyi FANG 1,
More informationAn Overview of Recent Application Trends at the AAMAS conference: Security, Sustainability and Safety
An Overview of Recent Application Trends at the AAMAS conference: Security, Sustainability and Safety Manish Jain, Bo An, Milind Tambe {manish.jain, boa, tambe}@usc.edu University of Southern California,
More informationBSS BSS. Ondřej Vaněk, Štěpán Kopřiva, Ondřej Hrstka, Jiří Zelinka, Michal Pěchouček. Blindspot Solutions
Modeling Maritime Contraband Trafficking Activities with the Agentbased Approach Ondřej Vaněk, Štěpán Kopřiva, Ondřej Hrstka, Jiří Zelinka, Michal Pěchouček Ondřej Vaněk CEO ondrej.vanek@blindspotsolutions.com
More informationNetwork Security Validation Using Game Theory
Network Security Validation Using Game Theory Vicky Papadopoulou and Andreas Gregoriades Computer Science and Engineering Dep., European University Cyprus, Cyprus {v.papadopoulou,a.gregoriades}@euc.ac.cy
More informationAlbert X. Jiang. Postdoctoral Research Associate Powell Hall of Engineering 208
Albert X. Jiang Postdoctoral Research Associate Powell Hall of Engineering 208 TEAMCORE Research Group 3737 Watt Way Computer Science Department Los Angeles, CA 90089 University of Southern California
More informationAn Empirical Study of Two MIS Algorithms
An Empirical Study of Two MIS Algorithms Email: Tushar Bisht and Kishore Kothapalli International Institute of Information Technology, Hyderabad Hyderabad, Andhra Pradesh, India 32. tushar.bisht@research.iiit.ac.in,
More informationThe multiinteger set cover and the facility terminal cover problem
The multiinteger set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.
More informationPlaying Games for Security: An Efficient Exact Algorithm for Solving Bayesian Stackelberg Games
Playing Games for Security: An Efficient Exact Algorithm for Solving Bayesian Stackelberg Games Praveen Paruchuri Intelligent Automation Inc., Rockville, MD pparuchuri@iai.com Jonathan P. Pearce, Janusz
More informationFei Fang. Ph.D. Candidate Salvatori Computer Science Center 300
Fei Fang Ph.D. Candidate Salvatori Computer Science Center 300 TEAMCORE Research Group 941 Bloom Walk Computer Science Department Los Angeles, CA 900890781 University of Southern California Phone: +1
More informationFrom the Lab to the Classroom and Beyond: Extending a GameBased Research Platform for Teaching AI to Diverse Audiences
From the Lab to the Classroom and Beyond: Extending a GameBased Research Platform for Teaching AI to Diverse Audiences Nicole Sintov 1, Debarun Kar 1, Thanh Nguyen 1, Fei Fang 1, Kevin Hoffman 2, Arnaud
More informationOutline. NPcompleteness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NPcompleteness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2pairs sum vs. general Subset Sum Reducing one problem to another Clique
More informationA Load Balancing Algorithm based on the Variation Trend of Entropy in Homogeneous Cluster
, pp.1120 http://dx.doi.org/10.14257/ ijgdc.2014.7.2.02 A Load Balancing Algorithm based on the Variation Trend of Entropy in Homogeneous Cluster Kehe Wu 1, Long Chen 2, Shichao Ye 2 and Yi Li 2 1 Beijing
More informationExistence of pure Nash equilibria (NE): Complexity of computing pure NE: Approximating the social optimum: Empirical results:
Existence Theorems and Approximation Algorithms for Generalized Network Security Games V.S. Anil Kumar, Rajmohan Rajaraman, Zhifeng Sun, Ravi Sundaram, College of Computer & Information Science, Northeastern
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More information(Refer Slide Time: 01:35)
Artificial Intelligence Prof. Sudeshna Sarkar Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture  6 Informed Search 2 Today we start our 6 th lecture. This
More informationMath 119 Pretest Review Answers
Math 9 Pretest Review Answers Linear Equations Find the slope of the line passing through the given points:. (, 5); (0, ) 7/. (, 4); (, 0) Undefined. (, ); (, 4) 4. Find the equation of the line with
More informationMINIMIZING STORAGE COST IN CLOUD COMPUTING ENVIRONMENT
MINIMIZING STORAGE COST IN CLOUD COMPUTING ENVIRONMENT 1 SARIKA K B, 2 S SUBASREE 1 Department of Computer Science, Nehru College of Engineering and Research Centre, Thrissur, Kerala 2 Professor and Head,
More informationThe Role of Size Normalization on the Recognition Rate of Handwritten Numerals
The Role of Size Normalization on the Recognition Rate of Handwritten Numerals Chun Lei He, Ping Zhang, Jianxiong Dong, Ching Y. Suen, Tien D. Bui Centre for Pattern Recognition and Machine Intelligence,
More informationParallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation
Parallel Computing for Option Pricing Based on the Backward Stochastic Differential Equation Ying Peng, Bin Gong, Hui Liu, and Yanxin Zhang School of Computer Science and Technology, Shandong University,
More informationThe tipping point is often referred to in popular literature. First introduced in Schelling 78 and Granovetter 78
Paulo Shakarian and Damon Paulo Dept. Electrical Engineering and Computer Science and Network Science Center U.S. Military Academy West Point, NY ASONAM 2012 The tipping point is often referred to in popular
More informationA Game Theoretic Model to Handle Network Intrusions over Multiple Packets
A Game Theoretic Model to Handle Network Intrusions over Multiple Packets Mona Mehrandish, Chadi M. Assi, and Mourad Debbabi Concordia Institute for Information Systems Engineering Concordia University,
More informationVoronoi Treemaps in D3
Voronoi Treemaps in D3 Peter Henry University of Washington phenry@gmail.com Paul Vines University of Washington paul.l.vines@gmail.com ABSTRACT Voronoi treemaps are an alternative to traditional rectangular
More informationAn Undergraduate Course in the Intersection of Computer Science and Economics
An Undergraduate Course in the Intersection of Computer Science and Economics Vincent Conitzer Duke Unversity Durham, NC 27708, USA conitzer@cs.duke.edu Abstract In recent years, major research advances
More informationA Game Theoretical Framework on Intrusion Detection in Heterogeneous Networks Lin Chen, Member, IEEE, and Jean Leneutre
IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL 4, NO 2, JUNE 2009 165 A Game Theoretical Framework on Intrusion Detection in Heterogeneous Networks Lin Chen, Member, IEEE, and Jean Leneutre
More informationMedial Axis Construction and Applications in 3D Wireless Sensor Networks
Medial Axis Construction and Applications in 3D Wireless Sensor Networks Su Xia, Ning Ding, Miao Jin, Hongyi Wu, and Yang Yang Presenter: Hongyi Wu University of Louisiana at Lafayette Outline Introduction
More informationMapReduce and Distributed Data Analysis. Sergei Vassilvitskii Google Research
MapReduce and Distributed Data Analysis Google Research 1 Dealing With Massive Data 2 2 Dealing With Massive Data Polynomial Memory Sublinear RAM Sketches External Memory Property Testing 3 3 Dealing With
More informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More informationInstituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra
Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra João Clímaco and Marta Pascoal A new method to detere unsupported nondoated solutions
More informationEFFICIENT KNOWLEDGE BASE MANAGEMENT IN DCSP
EFFICIENT KNOWLEDGE BASE MANAGEMENT IN DCSP Hong Jiang Mathematics & Computer Science Department, Benedict College, USA jiangh@benedict.edu ABSTRACT DCSP (Distributed Constraint Satisfaction Problem) has
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationRandom vs. StructureBased Testing of AnswerSet Programs: An Experimental Comparison
Random vs. StructureBased Testing of AnswerSet Programs: An Experimental Comparison Tomi Janhunen 1, Ilkka Niemelä 1, Johannes Oetsch 2, Jörg Pührer 2, and Hans Tompits 2 1 Aalto University, Department
More informationDECENTRALIZED LOAD BALANCING IN HETEROGENEOUS SYSTEMS USING DIFFUSION APPROACH
DECENTRALIZED LOAD BALANCING IN HETEROGENEOUS SYSTEMS USING DIFFUSION APPROACH P.Neelakantan Department of Computer Science & Engineering, SVCET, Chittoor pneelakantan@rediffmail.com ABSTRACT The grid
More informationECON 40050 Game Theory Exam 1  Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.
1 ECON 40050 Game Theory Exam 1  Answer Key Instructions: 1) You may use a pen or pencil, a handheld nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,
More informationOn the Hardness of Topology Inference
On the Hardness of Topology Inference H. B. Acharya 1 and M. G. Gouda 2 1 The University of Texas at Austin, USA acharya@cs.utexas.edu 2 The National Science Foundation, USA mgouda@nsf.gov Abstract. Many
More informationFPGA area allocation for parallel C applications
1 FPGA area allocation for parallel C applications VladMihai Sima, Elena Moscu Panainte, Koen Bertels Computer Engineering Faculty of Electrical Engineering, Mathematics and Computer Science Delft University
More informationBig Data Analytics of MultiRelationship Online Social Network Based on MultiSubnet Composited Complex Network
, pp.273284 http://dx.doi.org/10.14257/ijdta.2015.8.5.24 Big Data Analytics of MultiRelationship Online Social Network Based on MultiSubnet Composited Complex Network Gengxin Sun 1, Sheng Bin 2 and
More informationFleet Assignment Using Collective Intelligence
Fleet Assignment Using Collective Intelligence Nicolas E Antoine, Stefan R Bieniawski, and Ilan M Kroo Stanford University, Stanford, CA 94305 David H Wolpert NASA Ames Research Center, Moffett Field,
More informationAnalysis of an Artificial Hormone System (Extended abstract)
c 2013. This is the author s version of the work. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purpose or for creating
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationA Mathematical Programming Solution to the Mars Express Memory Dumping Problem
A Mathematical Programming Solution to the Mars Express Memory Dumping Problem Giovanni Righini and Emanuele Tresoldi Dipartimento di Tecnologie dell Informazione Università degli Studi di Milano Via Bramante
More informationEquilibrium computation: Part 1
Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationIntegrating Benders decomposition within Constraint Programming
Integrating Benders decomposition within Constraint Programming Hadrien Cambazard, Narendra Jussien email: {hcambaza,jussien}@emn.fr École des Mines de Nantes, LINA CNRS FRE 2729 4 rue Alfred Kastler BP
More informationOn the Interaction and Competition among Internet Service Providers
On the Interaction and Competition among Internet Service Providers Sam C.M. Lee John C.S. Lui + Abstract The current Internet architecture comprises of different privately owned Internet service providers
More informationContinuous Fastest Path Planning in Road Networks by Mining RealTime Traffic Event Information
Continuous Fastest Path Planning in Road Networks by Mining RealTime Traffic Event Information Eric HsuehChan Lu ChiWei Huang Vincent S. Tseng Institute of Computer Science and Information Engineering
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationWhat is Data Mining, and How is it Useful for Power Plant Optimization? (and How is it Different from DOE, CFD, Statistical Modeling)
data analysis data mining quality control webbased analytics What is Data Mining, and How is it Useful for Power Plant Optimization? (and How is it Different from DOE, CFD, Statistical Modeling) StatSoft
More informationSolving Singledigit Sudoku Subproblems
Solving Singledigit Sudoku Subproblems David Eppstein Int. Conf. Fun with Algorithms, June 2012 Sudoku An ab ab array of cells, partitioned into a b blocks, partially filled in with numbers from 1 to
More informationCompetitive Analysis of On line Randomized Call Control in Cellular Networks
Competitive Analysis of On line Randomized Call Control in Cellular Networks Ioannis Caragiannis Christos Kaklamanis Evi Papaioannou Abstract In this paper we address an important communication issue arising
More informationA Comparison of General Approaches to Multiprocessor Scheduling
A Comparison of General Approaches to Multiprocessor Scheduling JingChiou Liou AT&T Laboratories Middletown, NJ 0778, USA jing@jolt.mt.att.com Michael A. Palis Department of Computer Science Rutgers University
More informationA Sequential Game Perspective and Optimization of the Smart Grid with Distributed Data Centers
A Sequential Game Perspective and Optimization of the Smart Grid with Distributed Data Centers Yanzhi Wang, Xue Lin, and Massoud Pedram Department of Electrical Engineering University of Southern California
More informationA Direct Numerical Method for Observability Analysis
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method
More informationAn Evaluation of Network Survivability When Defense Levels Are Discounted by the Accumulated Experience of Attackers
An Evaluation of Network Survivability When Defense Levels Are Discounted by the Accumulated Experience of Attackers Frank YeongSung Lin National Tatiwan University, Taiwan yslin@im.ntu.edu.tw PeiYu
More informationA Network Flow Approach in Cloud Computing
1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges
More informationNetwork (Tree) Topology Inference Based on Prüfer Sequence
Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,
More informationEntropy based Graph Clustering: Application to Biological and Social Networks
Entropy based Graph Clustering: Application to Biological and Social Networks Edward C Kenley YoungRae Cho Department of Computer Science Baylor University Complex Systems Definition Dynamically evolving
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationDistributed Dynamic Load Balancing for IterativeStencil Applications
Distributed Dynamic Load Balancing for IterativeStencil Applications G. Dethier 1, P. Marchot 2 and P.A. de Marneffe 1 1 EECS Department, University of Liege, Belgium 2 Chemical Engineering Department,
More informationCOLORED GRAPHS AND THEIR PROPERTIES
COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring
More informationChapter 1 Quadratic Equations in One Unknown (I)
Tin Ka Ping Secondary School 015016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real
More informationLecture Notes on Spanning Trees
Lecture Notes on Spanning Trees 15122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model
More informationFrans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages
Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages Article (Published version) (Refereed) Original citation: de Ruiter, Frans and Biggs, Norman (2015) Applications
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationHighperformance local search for planning maintenance of EDF nuclear park
Highperformance local search for planning maintenance of EDF nuclear park Frédéric Gardi Karim Nouioua Bouygues elab, Paris fgardi@bouygues.com Laboratoire d'informatique Fondamentale  CNRS UMR 6166,
More informationALGEBRA 1/ALGEBRA 1 HONORS
ALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More information(Refer Slide Time: 01:52)
Software Engineering Prof. N. L. Sarda Computer Science & Engineering Indian Institute of Technology, Bombay Lecture  2 Introduction to Software Engineering Challenges, Process Models etc (Part 2) This
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationStrategic planning in LTL logistics increasing the capacity utilization of trucks
Strategic planning in LTL logistics increasing the capacity utilization of trucks J. Fabian Meier 1,2 Institute of Transport Logistics TU Dortmund, Germany Uwe Clausen 3 Fraunhofer Institute for Material
More informationDecentralized Utilitybased Sensor Network Design
Decentralized Utilitybased Sensor Network Design Narayanan Sadagopan and Bhaskar Krishnamachari University of Southern California, Los Angeles, CA 900890781, USA narayans@cs.usc.edu, bkrishna@usc.edu
More informationA Hierarchical Structure based Coverage Repair in Wireless Sensor Networks
A Hierarchical Structure based Coverage Repair in Wireless Sensor Networks Jie Wu Computer Science & Engineering Department Florida Atlantic University Boca Raton, FL 3343, USA Email: jie@cse.fau.edu
More informationPolicy Distribution Methods for Function Parallel Firewalls
Policy Distribution Methods for Function Parallel Firewalls Michael R. Horvath GreatWall Systems WinstonSalem, NC 27101, USA Errin W. Fulp Department of Computer Science Wake Forest University WinstonSalem,
More informationBargaining Solutions in a Social Network
Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,
More informationHigh School Algebra 1 Common Core Standards & Learning Targets
High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities NQ.1. Use units as a way to understand problems
More informationFeature Selection with MonteCarlo Tree Search
Feature Selection with MonteCarlo Tree Search Robert Pinsler 20.01.2015 20.01.2015 Fachbereich Informatik DKE: Seminar zu maschinellem Lernen Robert Pinsler 1 Agenda 1 Feature Selection 2 Feature Selection
More informationMoving Target Search. 204 Automated Reasoning
Moving Target Search Toru Ishida NTT Communications and Information Processing Laboratories 12356, Take, Yokosuka, 23803, JAPAN ishida%nttkb.ntt.jp@relay.cs.net Richard E. Korf Computer Science Department
More informationTwo General Methods to Reduce Delay and Change of Enumeration Algorithms
ISSN 13465597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII2003004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration
More informationAmajor benefit of MonteCarlo schedule analysis is to
2005 AACE International Transactions RISK.10 The Benefits of Monte Carlo Schedule Analysis Mr. Jason Verschoor, P.Eng. Amajor benefit of MonteCarlo schedule analysis is to expose underlying risks to
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More information