Efficiently computing succinct trade-off curves

Trade-off (aka Pareto) curves are typically used to represent the trade-off among different objectives in multiobjective optimization problems. Although trade-off curves are exponentially large for typical combinatorial optimization problems (and infinite for continuous problems), it was observed in Papadimitriou and Yannakakis [On the approximability of trade-offs and optimal access of web sources, in: Proc. 41st IEEE Symp. on Foundations of Computer Science, 2000] that there exist polynomial size e approximations for any e > 0, and that under certain general conditions, such approximate e-Pareto curves can be constructed in polynomial time. In this paper we seek general-purpose algorithms for the efficient approximation of trade-off curves using as few points as possible. In the case of two objectives, we present a general algorithm that efficiently computes an e-Pareto curve that uses at most 3 times the number of points of the smallest such curve; we show that no algorithm can be better than 3-competitive in this setting. If we relax e to any e' > e, then we can efficiently construct an e'-curve that uses no more points than the smallest e-curve. With three objectives we show that no algorithm can be c-competitive for any constant c unless it is allowed to use a larger e value. We present an algorithm that is 4-competitive for any e' > (1 + e)2 - 1. We explore the problem in high dimensions and give hardness proofs showing that (unless P=NP) no constant approximation factor can be achieved efficiently even if we relax e by an arbitrary constant.

[1]  Mihalis Yannakakis,et al.  Multiobjective query optimization , 2001, PODS '01.

[2]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[3]  Mihalis Yannakakis.  On a class of totally unimodular matrices , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[4]  Mihalis Yannakakis,et al.  On the Complexity of Database Queries , 1999, J. Comput. Syst. Sci..

[5]  Mihalis Yannakakis,et al.  On the approximation of maximum satisfiability , 1992, SODA '92.

[6]  Catriel Beeri,et al.  Properties of acyclic database schemes , 1981, STOC '81.

[7]  Mihalis Yannakakis,et al.  A Note on Succinct Representations of Graphs , 1986, Inf. Control..

[8]  Alfred V. Aho,et al.  Bounds on the size and transmission rate of communications protocols , 1982 .

[9]  Rajeev Alur,et al.  Distinguishing tests for nondeterministic and probabilistic machines , 1995, STOC '95.

[10]  Ashish Goel,et al.  Efficient computation of delay-sensitive routes from one source to all destinations , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[11]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[12]  AlurRajeev,et al.  Model checking of hierarchical state machines , 1998 .

[13]  Santosh S. Vempala,et al.  A convex relaxation for the asymmetric TSP , 1999, SODA '99.

[14]  Mihalis Yannakakis,et al.  Fundamental discrepancies between average-case analyses under discrete and continuous distributions: a bin packing case study , 1991, STOC '91.

[15]  Mihalis Yannakakis,et al.  Permuting Elements Within Columns of a Matrix in Order to Minimize Maximum Row Sum , 1984, Math. Oper. Res..

[16]  Rajeev Alur,et al.  Model Checking of Message Sequence Charts , 1999, CONCUR.

[17]  Rina Panigrahy,et al.  An O(log*n) approximation algorithm for the asymmetric p-center problem , 1996, SODA '96.

[18]  Mihalis Yannakakis,et al.  Independent database schemas , 1982, PODS '82.

[19]  Kousha Etessami,et al.  Checking LTL properties of recursive Markov chains , 2005, Second International Conference on the Quantitative Evaluation of Systems (QEST'05).

[20]  Mihalis Yannakakis,et al.  Graph-theoretic methods in database theory , 1990, PODS.

[21]  Refael Hassin,et al.  Approximation Schemes for the Restricted Shortest Path Problem , 1992, Math. Oper. Res..

[22]  Kousha Etessami,et al.  Efficient Qualitative Analysis of Classes of Recursive Markov Decision Processes and Simple Stochastic Games , 2006, STACS.

[23]  K. K. Nambiar,et al.  Foundations of Computer Science , 2001, Lecture Notes in Computer Science.

[24]  Thanasis Hadzilacos,et al.  Deleting completed transactions , 1985, PODS '86.

[25]  Venkatesan Guruswami,et al.  Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems , 1999, STOC '99.

[26]  Swarat Chaudhuri,et al.  Compression of Partially Ordered Strings , 2003, CONCUR.

[27]  Christos H. Papadimitriou,et al.  Approximately dominating representatives , 2005, Theor. Comput. Sci..

[28]  Sergei Vassilvitskii,et al.  Efficiently computing succinct trade-off curves , 2005, Theoretical Computer Science.

[29]  Rajeev Alur,et al.  Inference of message sequence charts , 2000, Proceedings of the 2000 International Conference on Software Engineering. ICSE 2000 the New Millennium.

[30]  David Lee,et al.  An Efficient Algorithm for Minimizing Real-Time Transition Systems , 1997, Formal Methods Syst. Des..

[31]  Mihalis Yannakakis.  Serializability by Locking , 1984, JACM.

[32]  Catriel Beeri,et al.  On the Desirability of Acyclic Database Schemes , 1983, JACM.

[33]  Mihalis Yannakakis,et al.  Black Box Checking , 1999, FORTE.

[34]  Mihalis Yannakakis,et al.  The complexity of facets (and some facets of complexity) , 1982, STOC '82.

[35]  Mihalis Yannakakis,et al.  A polynomial algorithm for the MIN CUT linear arrangement of trees , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[36]  Mihalis Yannakakis,et al.  Linear and Book Embeddings of Graphs , 1986, Aegean Workshop on Computing.

[37]  Sudipto Guha,et al.  Asymmetric k-center is log* n-hard to approximate , 2005, JACM.

[38]  Mihalis Yannakakis,et al.  The Analysis of Local Search Problems and Their Heuristics , 1990, STACS.

[39]  Mihalis Yannakakis.  Testing and Checking of Finite State Systems , 2002, LATIN.

[40]  Mihalis Yannakakis,et al.  Searching a Fixed Graph , 1996, ICALP.

[41]  Aaron Archer,et al.  Two O (log* k)-Approximation Algorithms for the Asymmetric k-Center Problem , 2001, IPCO.

[42]  Dimitrios Gunopulos,et al.  Efficient Approximation Of Optimization Queries Under Parametric Aggregation Constraints , 2003, VLDB.

[43]  Mihalis Yannakakis,et al.  On nested depth first search , 1996, The Spin Verification System.

[44]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[45]  Mihalis Yannakakis,et al.  Worst-case ration for planar graphs and the method of induction on faces , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[46]  Mihalis Yannakakis.  Querying weak instances , 1984, PODS '84.

[47]  Mihalis Yannakakis,et al.  Suboptimal Cuts: Their Enumeration, Weight and Number (Extended Abstract) , 1992, ICALP.

[48]  Tao Jiang,et al.  Linear approximation of shortest superstrings , 1991, STOC '91.

[49]  David Lee,et al.  Testing for Finite State Systems , 1998, CSL.

[50]  Mihalis Yannakakis,et al.  On the value of information in distributed decision-making (extended abstract) , 1991, PODC '91.

[51]  H IbarraOscar,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975 .

[52]  Mihalis Yannakakis,et al.  The complexity of reliable concurrency control , 1985, PODS '85.

[53]  Mihalis Yannakakis,et al.  Approximation of Multiobjective Optimization Problems , 2001, WADS.

[54]  Rajeev Alur,et al.  Model checking of hierarchical state machines , 1998, TOPL.

[55]  Mihalis Yannakakis,et al.  High-probability parallel transitive closure algorithms , 1990, SPAA '90.

[56]  David S. Johnson,et al.  Scheduling Opposing Forests , 1983 .

[57]  Mihalis Yannakakis,et al.  Deadlock-Freedom (and Safety) of Transactions in a Distributed Database , 1986, J. Comput. Syst. Sci..

[58]  Mihalis Yannakakis,et al.  Verifying temporal properties of finite-state probabilistic programs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[59]  Rajeev Alur,et al.  Realizability and verification of MSC graphs , 2005, Theor. Comput. Sci..

[60]  Mihalis Yannakakis,et al.  Existence of Reduction Hierarchies , 1997, CSL.

[61]  Mihalis Yannakakis,et al.  Tie-Breaking Semantics and Structural Totality , 1997, J. Comput. Syst. Sci..

[62]  Mihalis Yannakakis,et al.  Towards an architecture-independent analysis of parallel algorithms , 1990, STOC '88.

[63]  Mihalis Yannakakis,et al.  Testing hierarchical systems , 2005, SODA '05.

[64]  Mihalis Yannakakis,et al.  Shortest Paths Without a Map , 1989, Theor. Comput. Sci..

[65]  Rajeev Alur,et al.  Analysis of recursive state machines , 2001, TOPL.

[67]  Kousha Etessami,et al.  From Rule-based to Automata-based Testing , 2000, FORTE.

[68]  Mihalis Yannakakis,et al.  Recent Developments on the Approximability of Combinatorial Problems , 1993, ISAAC.

[69]  Mihalis Yannakakis,et al.  The Maximum k-Colorable Subgraph Problem for Chordal Graphs , 1987, Inf. Process. Lett..

[70]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[71]  Sudipto Guha,et al.  Asymmetric k-center is log* n-hard to approximate , 2004, STOC '04.

[72]  Mihalis Yannakakis,et al.  Tools for Template Dependencies , 1983, SIAM J. Comput..

[73]  Mihalis Yannakakis,et al.  Multiway Cuts in Directed and Node Weighted Graphs , 1994, ICALP.

[74]  E. Polak,et al.  On Multicriteria Optimization , 1976 .

[75]  Mihalis Yannakakis,et al.  Minimum and maximum delay problems in real-time systems , 1991, Formal Methods Syst. Des..

[76]  Mihalis Yannakakis,et al.  On the complexity of local search , 1990, STOC '90.

[77]  Mihalis Yannakakis,et al.  Testing, optimization, and games , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[78]  Mihalis Yannakakis,et al.  On limited nondeterminism and the complexity of the V-C dimension , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[79]  R. Ravi,et al.  Many birds with one stone: multi-objective approximation algorithms , 1993, STOC '93.

[80]  Mihalis Yannakakis,et al.  Optimization problems from feature testing of communication protocols , 1996, Proceedings of 1996 International Conference on Network Protocols (ICNP-96).

[81]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[82]  Oliver Vornberger,et al.  The Complexity of Testing Whether a Graph is a Superconcentrator , 1981, Inf. Process. Lett..

[83]  Mihalis Yannakakis,et al.  Linear programming without the matrix , 1993, STOC.

[84]  Kousha Etessami,et al.  Algorithmic Verification of Recursive Probabilistic State Machines , 2005, TACAS.

[85]  Mihalis Yannakakis,et al.  Algorithms for Acyclic Database Schemes , 1981, VLDB.

[86]  Mihalis Yannakakis,et al.  On the Complexity of Protein Folding , 1998, J. Comput. Biol..

[87]  T. C. Edwin Cheng,et al.  Bicriterion Single Machine Scheduling with Resource Dependent Processing Times , 1998, SIAM J. Optim..

[88]  Mihalis Yannakakis,et al.  Pfaffian Orientations, 0/1 Permanents, and Even Cycles in Directed Graphs , 1988, ICALP.

[89]  Mihalis Yannakakis,et al.  On the approximability of trade-offs and optimal access of Web sources , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[90]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[91]  Mihalis Yannakakis,et al.  On Datalog vs. Polynomial Time , 1995, J. Comput. Syst. Sci..

[92]  Mihalis Yannakakis,et al.  On monotone formulae with restricted depth , 1984, STOC '84.

[93]  Alex Groce,et al.  Adaptive Model Checking , 2002, Log. J. IGPL.

[94]  Mihalis Yannakakis,et al.  Some Open Problems in Approximation , 1994, CIAC.

[95]  Mihalis Yannakakis,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[96]  Sampath Kannan,et al.  Communicating Hierarchical State Machines , 1999, ICALP.

[97]  Arthur Warburton,et al.  Approximation of Pareto Optima in Multiple-Objective, Shortest-Path Problems , 1987, Oper. Res..

[98]  Mihalis Yannakakis,et al.  Four pages are necessary and sufficient for planar graphs , 1986, Symposium on the Theory of Computing.

[99]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..

[100]  Alex Groce,et al.  AMC: An Adaptive Model Checker , 2002, CAV.

[101]  Mihalis Yannakakis,et al.  The input/output complexity of transitive closure , 2005, Annals of Mathematics and Artificial Intelligence.

[102]  S French,et al.  Multicriteria Analysis , 1998, J. Oper. Res. Soc..

[103]  Benny K. Nielsen,et al.  On the location of Steiner points in uniformly-oriented Steiner trees , 2002, Inf. Process. Lett..

[104]  Mihalis Yannakakis.  Issues of correctness in database concurrency control by locking , 1981, STOC '81.

[105]  Carsten Lund,et al.  The Approximation of Maximum Subgraph Problems , 1993, ICALP.

[106]  Mihalis Yannakakis,et al.  Hierarchical State Machines , 2000, IFIP TCS.

[107]  Alfred V. Aho,et al.  On notions of information transfer in VLSI circuits , 1983, STOC.

[108]  Danny Raz,et al.  A simple efficient approximation scheme for the restricted shortest path problem , 2001, Oper. Res. Lett..

[109]  Pierre Hansen,et al.  Bicriterion Path Problems , 1980 .

[110]  Funda Ergün,et al.  An improved FPTAS for Restricted Shortest Path , 2002, Inf. Process. Lett..

[111]  David Lee,et al.  Protocol System Integration, Interface and Interoperability , 2004, OPODIS.

[112]  Alon Itai,et al.  Timing Verification by Successive Approximation , 1992, CAV.

[113]  Mihalis Yannakakis.  Perspectives on database theory , 1996, SIGA.

[114]  Mihalis Yannakakis,et al.  Expressing combinatorial optimization problems by linear programs , 1991, STOC '88.

[115]  David Lee,et al.  Testing finite state machines , 1991, STOC '91.

[116]  Pierre Wolper,et al.  Memory-efficient algorithms for the verification of temporal properties , 1990, Formal Methods Syst. Des..