Optimization with several prize categories
6.2 Integer linear program formulation
We start by modifying the optimization problem in (3.12):
maximize
xtf
minn P
v∈V p(i)v min{1, sv(xtf)} : i∈ Io subject to Atfxtf=btf
xtfj∈ {0,1}, for allj= 1, . . . ,|Etf|
(6.1)
where isIis the set of categories andp(i)v denotes the prize from categoryi∈ Ithat nodevholds. The linear mapsv(·)was defined in (3.11).
By introducing an epigraph variable, we can reformulate (6.1) into maximize
xtf,y y subject to y≤P
v∈V p(i)v min{1, sv(xtf)}, fori∈ I Atfxtf=btf
xtfj ∈ {0,1}, for allj= 1, . . . ,|Etf|
(6.2)
whereyis the (scalar) epigraph variable.
Finally, we can get rid of theminoperators via an additional set of variablest= (tv)v∈V: maximize
xtf,y,t y subject to y≤P
v∈V p(i)v tv, fori∈ I tv≤1, forv∈V
tv≤sv(xtf), forv∈V Atfxtf=btf
xtfj∈ {0,1}, for allj= 1, . . . ,|Etf|.
(6.3)
Formulation (6.3) is an integer linear optimization problem.
6.3 Recursive greedy search
It is straightforward to extend the recursive greedy search (RGS) methods from chapter 5 to the new multi-category prize setting. We just need to modify the oracle functionfX(P), i.e., the auxiliary method that computes the merit of a given sequence of nodes P, given that we have already visited a set of nodesX.
The output of this oracle function will now be the minimum value returned by the original oracle functions in each of the measurement categories:
fXI(P, X,ΨI) = min
i∈I{fX(P;p(i))} (6.4)
whereiindexes the setI of measurement categories,ΨI ={p(1), . . . , p(i), . . . , p(|I|)}denotes the set of all prize distributions for all categories of measurement, andp(i)refers to thei-th prize distribution.
Chapter 7
Conclusions
We explored several approaches for addressing the prize accumulation problem. We found that the two best approaches are:
• our integer linear program (ILP) formulation—from chapter 3—which opens the door for solving the prize accumulation problem via any ILP solver. Computer simulations have shown that an ILP solver can outperform a brute-force search;
• our fortified recursive greedy search method(FRGS)—from chapter 5—which boosts the perfor-mance of the algorithm in Chekuri and Pal [2005] by wrapping it in a fortified rollout framework Bert-sekas et al. [1997]. Computer simulations that the FRGS performs close to the ILP solver; yet, it has a predictable running time and is much simpler to implement.
Future work The following topics are provided as suggestions of possible future work:
• Study the Minimum Weight Connected Graph problem, so that we are not constrained to paths in sensor networks.
• Evaluate the performance of the studied algorithms in terms of prize distribution variance among the network.
• Evaluate how the performance of the studied algorithms is affected in terms of the structure of the network; that would be accomplished by running those algorithms in batches of several random networks, where each batch would be characterized by a constant number of nodes and maximum node degree.
Bibliography
Boost C++ libraries. URLhttp://www.boost.org.
T. Achterberg. SCIP: Solving constraint integer programs. Mathematical Programming Computation, 1 (1):1–41, July 2009. http://mpc.zib.de/index.php/MPC/article/view/4.
R. K. Ahuja, T. L. Magnanti, and J. B. Orlin. Network Flows: Theory, Algorithms, and Applications, chapter 3, pages 79–83. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1993. ISBN 0-13-617549-X.
D. Bertsekas. Rollout algorithms for discrete optimization: a survey. In P. M. Pardalos, D.-Z. Du, and R. L. Graham, editors, Handbook of Combinatorial Optimization, pages 2989–3013. Springer New York, 2013. ISBN 978-1-4419-7996-4. doi: 10.1007/978-1-4419-7997-1 8. URL http://dx.doi.
org/10.1007/978-1-4419-7997-1_8.
D. P. Bertsekas and D. A. Castanon. Rollout algorithms for stochastic scheduling problems. Journal of Heuristics, 5(1):89–108, Apr. 1999. ISSN 1381-1231. doi: 10.1023/A:1009634810396. URLhttp:
//dx.doi.org/10.1023/A:1009634810396.
D. P. Bertsekas, J. N. Tsitsiklis, and C. Wu. Rollout algorithms for combinatorial optimization. Journal of Heuristics, 3(3):245–262, Nov. 1997. ISSN 1381-1231. doi: 10.1023/A:1009635226865. URL http://dx.doi.org/10.1023/A:1009635226865.
F. Bian, D. Kempe, and R. Govindan. Utility-based sensor selection. In Information Processing in Sensor Networks, 2006. IPSN 2006. The Fifth International Conference on, pages 11–18, 2006. doi:
10.1109/IPSN.2006.244032.
C. Chekuri and M. Pal. A recursive greedy algorithm for walks in directed graphs. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’05, pages 245–253, Washington, DC, USA, 2005. IEEE Computer Society. ISBN 0-7695-2468-0. doi: 10.1109/SFCS.
2005.9. URLhttp://dx.doi.org/10.1109/SFCS.2005.9.
D. Gleich. MatlabBGL, a Matlab graph library. URLhttp://dgleich.github.io/matlab-bgl.
Y.-W. Huang, N. Jing, and E. A. Rundensteiner. Optimizing path query performance: graph clus-tering strategies. Transportation Research Part C: Emerging Technologies, 8(1–6):381 – 408,
2000. ISSN 0968-090X. doi: http://dx.doi.org/10.1016/S0968-090X(00)00049-8. URLhttp://www.
sciencedirect.com/science/article/pii/S0968090X00000498.
B. Johansson, M. Rabi, and M. Johansson. A randomized incremental subgradient method for distributed optimization in networked systems. SIAM Journal on Optimization, 20(3):1157–1170, 2010. doi:
10.1137/08073038X. URLhttp://dx.doi.org/10.1137/08073038X.
S. Joshi and S. Boyd. Sensor selection via convex optimization. Signal Processing, IEEE Transactions on, 57(2):451–462, Feb 2009. ISSN 1053-587X. doi: 10.1109/TSP.2008.2007095.
R. Kala, A. Shukla, and R. Tiwari. Optimized graph search using multi-level graph clustering.
In S. Ranka, S. Aluru, R. Buyya, Y.-C. Chung, S. Dua, A. Grama, S. Gupta, R. Kumar, and V. Phoha, editors, Contemporary Computing, volume 40 of Communications in Computer and In-formation Science, pages 103–114. Springer Berlin Heidelberg, 2009. ISBN 978-3-642-03546-3. doi:
10.1007/978-3-642-03547-0 11. URLhttp://dx.doi.org/10.1007/978-3-642-03547-0_11.
S. Kar and J. Moura. Gossip and distributed kalman filtering: Weak consensus under weak detectability.
Signal Processing, IEEE Transactions on, 59(4):1766–1784, April 2011. ISSN 1053-587X. doi: 10.
1109/TSP.2010.2100385.
H. F. Lee and D. R. Dooly. Algorithms for the constrained maximum-weight connected graph problem. Naval Research Logistics (NRL), 43(7):985–1008, 1996. ISSN 1520-6750. doi:
10.1002/(SICI)1520-6750(199610)43:7h985::AID-NAV4i3.0.CO;2-9. URL http://dx.doi.org/10.
1002/(SICI)1520-6750(199610)43:7<985::AID-NAV4>3.0.CO;2-9.
H. F. Lee and D. R. Dooly. Decomposition algorithms for the maximum-weight connected graph prob-lem. Naval Research Logistics (NRL), 45(8):817–837, 1998. ISSN 1520-6750. doi: 10.1002/(SICI) 1520-6750(199812)45:8h817::AID-NAV4i3.0.CO;2-1. URL http://dx.doi.org/10.1002/(SICI) 1520-6750(199812)45:8<817::AID-NAV4>3.0.CO;2-1.
H. Liu, L. J. Latecki, and S. Yan. Revealing cluster structure of graph by path following replicator dynamic.
Computing Research Repository, abs/1303.2643, 2013. URLhttp://arxiv.org/abs/1303.2643.
S. Nowozin and C. Lampert. Global connectivity potentials for random field models. InComputer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 818–825, June 2009. doi:
10.1109/CVPR.2009.5206567.
S. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. Prentice Hall Press, Upper Saddle River, NJ, USA, 3rd edition, 2009. ISBN 0136042597, 9780136042594.
C. Sanderson. Armadillo: An open source C++ linear algebra library for fast prototyping and com-putationally intensive experiments. Technical report, NICTA, September 2010. URLhttp://arma.
sourceforge.net.
T. Yamamoto, H. Bannai, M. Nagasaki, and S. Miyano. Better decomposition heuristics for the maximum-weight connected graph problem using betweenness centrality. In J. Gama, V. Costa,
A. Jorge, and P. Brazdil, editors, Discovery Science, volume 5808 of Lecture Notes in Com-puter Science, pages 465–472. Springer Berlin Heidelberg, 2009. ISBN 978-3-642-04746-6. doi:
10.1007/978-3-642-04747-3 40. URLhttp://dx.doi.org/10.1007/978-3-642-04747-3_40.