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Hardness-Aware Restart Policies. Yongshao Ruan, Eric Horvitz, & Henry Kautz. IJCAI 2003 Workshop on Stochastic Search. Randomized Restart Strategies for Backtrack Search. Simple idea: randomize branching heuristic, restart with new seed if solution is not found in a reasonable amount of time

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Hardness aware restart policies

Hardness-Aware Restart Policies

Yongshao Ruan, Eric Horvitz, & Henry Kautz

IJCAI 2003 Workshop on Stochastic Search

Randomized restart strategies for backtrack search
Randomized Restart Strategies for Backtrack Search

  • Simple idea: randomize branching heuristic, restart with new seed if solution is not found in a reasonable amount of time

  • Effective on a wide range of structured problems (Luby 1993, Gomes et al 1997)

  • Issue: when to restart?

No knowledge of rtd
No Knowledge of RTD

  • Can we do better?

    • Information about progress of the current run (looking good?)

    • Partial knowledge of RTD


Observation horizon



Median run time


  • (UAI 2001) – Can predict a particular run’s time to solution (very roughly) based on features of the solver’s trace during an initial window

  • (AAAI 2002) – Can improve time to solution by immediately pruning runs that are predicted to be long

    • Scenario: You know RTD of a problem ensemble. Each run is from a different randomly-selected problem. Goal is solve some problem as soon as possible (i.e., you can skip ones that look hard).

    • In general: optimal policy is to set cutoff conditionally on value of observed features.

Answers continued
Answers (continued)

  • (CP 2002) – Given partial knowledge about an ensemble RTD, the optimal strategy uses the information gained from each failed run to update its beliefs about the shape of the RTD.

    • Scenario: There is a set of k different problem ensembles, and you know the ensemble RTD of each. Nature chooses one of the ensembles at random, but does not tell you which one. Each run is from a different randomly-chosen problem from that ensemble. Your goal is to solve some problem as soon as possible.

    • In general: cutoffs change for each run.

Answers final
Answers (final!)

  • (IJCAI 2003 Workshop) – The unknown RTD of a particular problem instance can be approximated by the RTD of a sub-ensemble

    • Scenario: You are allowed to sample a problem distribution and consider various ways of clustering instances that have similar instance RTD’s. Then you are given a new random instance and must solve it as quickly as possible (i.e., you cannot skip over problems!)

    • Most realistic?

Partitioning ensemble rtd by instance median run times
Partitioning ensemble RTD by instance median run-times

Instance median > ensemble median

Ensemble RTD

Instance median < ensemble median

Computing the restart strategy
Computing the restart strategy

  • Assume that the (unknown) RTD of the given instance is well-approximated by the RTD of one of the clusters

  • Strategy depends upon your state of belief about which cluster that is

  • Formalize as an POMDP:

    • State = state of belief

    • Actions = use a particular cutoff K

    • Effect = { solved, not solved }


  • Bellman equation:

  • Solve by dynamic programming (ouch!)

Optimal expected time to solution from belief state s

Probability that running with cutoff t in state s fails (resulting in state s’)

Simple example
Simple Example

  • Suppose RTD of each instance is a scaled Pareto controlled by a parameter b  Uniform[11, 200]

  • Median run time = 2b, so medians are uniformly distributedin [22, 200]

  • Cluster into two sub-distributions

    • Median  110

    • Median > 110

  • Dynamic programming solution:

    201 ,222 ,234 ,239 ,242 ,244 …


  • Last piece in basic mathematics for optimal restarts with partial information

  • See paper for details of incorporating observations

  • RTD alone gives log speedup over Luby universal (still can be significant)

  • Unlimited potential for speedup with more accurate run-time predictors!