scheduler.py

__author__ = "Johannes Köster"
__copyright__ = "Copyright 2022, Johannes Köster"
__email__ = "johannes.koester@uni-due.de"
__license__ = "MIT"

import os, signal, sys
import datetime
import threading
import operator
import time
import math
import asyncio

from functools import partial
from collections import defaultdict
from itertools import chain, accumulate, product
from contextlib import ContextDecorator


from snakemake.executors import DryrunExecutor, TouchExecutor, CPUExecutor
from snakemake.executors import (
    GenericClusterExecutor,
    SynchronousClusterExecutor,
    DRMAAExecutor,
    KubernetesExecutor,
    TibannaExecutor,
)
from snakemake.executors.google_lifesciences import GoogleLifeSciencesExecutor
from snakemake.executors.ga4gh_tes import TaskExecutionServiceExecutor
from snakemake.exceptions import RuleException, WorkflowError, print_exception
from snakemake.shell import shell
from snakemake.common import async_run
from snakemake.logging import logger

from fractions import Fraction


def cumsum(iterable, zero=[0]):
    return list(chain(zero, accumulate(iterable)))


_ERROR_MSG_FINAL = (
    "Exiting because a job execution failed. " "Look above for error message"
)

_ERROR_MSG_ISSUE_823 = (
    "BUG: Out of jobs ready to be started, but not all files built yet."
    " Please check https://github.com/snakemake/snakemake/issues/823 for more information."
)


class DummyRateLimiter(ContextDecorator):
    def __enter__(self):
        return self

    def __exit__(self, *args):
        return False


class JobScheduler:
    def __init__(
        self,
        workflow,
        dag,
        local_cores=1,
        dryrun=False,
        touch=False,
        cluster=None,
        cluster_status=None,
        cluster_config=None,
        cluster_sync=None,
        cluster_cancel=None,
        cluster_cancel_nargs=None,
        cluster_sidecar=None,
        drmaa=None,
        drmaa_log_dir=None,
        kubernetes=None,
        container_image=None,
        tibanna=None,
        tibanna_sfn=None,
        google_lifesciences=None,
        google_lifesciences_regions=None,
        google_lifesciences_location=None,
        google_lifesciences_cache=False,
        tes=None,
        precommand="",
        preemption_default=None,
        preemptible_rules=None,
        tibanna_config=False,
        jobname=None,
        quiet=False,
        printreason=False,
        printshellcmds=False,
        keepgoing=False,
        max_jobs_per_second=None,
        max_status_checks_per_second=100,
        latency_wait=3,
        greediness=1.0,
        force_use_threads=False,
        assume_shared_fs=True,
        keepincomplete=False,
        keepmetadata=True,
        scheduler_type=None,
        scheduler_ilp_solver=None,
    ):
        """Create a new instance of KnapsackJobScheduler."""
        from ratelimiter import RateLimiter

        cores = workflow.global_resources["_cores"]

        self.cluster = cluster
        self.cluster_config = cluster_config
        self.cluster_sync = cluster_sync
        self.dag = dag
        self.workflow = workflow
        self.dryrun = dryrun
        self.touch = touch
        self.quiet = quiet
        self.keepgoing = keepgoing
        self.running = set()
        self.failed = set()
        self.finished_jobs = 0
        self.greediness = 1
        self.max_jobs_per_second = max_jobs_per_second
        self.keepincomplete = keepincomplete
        self.keepmetadata = keepmetadata
        self.scheduler_type = scheduler_type
        self.scheduler_ilp_solver = scheduler_ilp_solver
        self._tofinish = []
        self._toerror = []
        self.handle_job_success = True
        self.update_resources = True
        self.print_progress = not self.quiet and not self.dryrun
        self.update_dynamic = not self.dryrun

        self.global_resources = {
            name: (sys.maxsize if res is None else res)
            for name, res in workflow.global_resources.items()
        }

        if workflow.global_resources["_nodes"] is not None:
            # Do not restrict cores locally if nodes are used (i.e. in case of cluster/cloud submission).
            self.global_resources["_cores"] = sys.maxsize
        self.resources = dict(self.global_resources)

        use_threads = (
            force_use_threads
            or (os.name != "posix")
            or cluster
            or cluster_sync
            or drmaa
        )
        self._open_jobs = threading.Semaphore(0)
        self._lock = threading.Lock()

        self._errors = False
        self._executor_error = None
        self._finished = False
        self._job_queue = None
        self._last_job_selection_empty = False
        self._submit_callback = self._noop
        self._finish_callback = self._proceed

        self._local_executor = None
        if dryrun:
            self._executor = DryrunExecutor(
                workflow,
                dag,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
            )
        elif touch:
            self._executor = TouchExecutor(
                workflow,
                dag,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
            )
        elif cluster or cluster_sync or (drmaa is not None):
            if not workflow.immediate_submit:
                # No local jobs when using immediate submit!
                # Otherwise, they will fail due to missing input
                self._local_executor = CPUExecutor(
                    workflow,
                    dag,
                    local_cores,
                    printreason=printreason,
                    quiet=quiet,
                    printshellcmds=printshellcmds,
                    latency_wait=latency_wait,
                    cores=local_cores,
                    keepincomplete=keepincomplete,
                    keepmetadata=keepmetadata,
                )
            if cluster or cluster_sync:
                if cluster_sync:
                    constructor = SynchronousClusterExecutor
                else:
                    constructor = partial(
                        GenericClusterExecutor,
                        statuscmd=cluster_status,
                        cancelcmd=cluster_cancel,
                        cancelnargs=cluster_cancel_nargs,
                        sidecarcmd=cluster_sidecar,
                        max_status_checks_per_second=max_status_checks_per_second,
                    )

                self._executor = constructor(
                    workflow,
                    dag,
                    None,
                    submitcmd=(cluster or cluster_sync),
                    cluster_config=cluster_config,
                    jobname=jobname,
                    printreason=printreason,
                    quiet=quiet,
                    printshellcmds=printshellcmds,
                    latency_wait=latency_wait,
                    assume_shared_fs=assume_shared_fs,
                    keepincomplete=keepincomplete,
                    keepmetadata=keepmetadata,
                )
                if workflow.immediate_submit:
                    self.update_dynamic = False
                    self.print_progress = False
                    self.update_resources = False
                    self.handle_job_success = False
            else:
                self._executor = DRMAAExecutor(
                    workflow,
                    dag,
                    None,
                    drmaa_args=drmaa,
                    drmaa_log_dir=drmaa_log_dir,
                    jobname=jobname,
                    printreason=printreason,
                    quiet=quiet,
                    printshellcmds=printshellcmds,
                    latency_wait=latency_wait,
                    cluster_config=cluster_config,
                    assume_shared_fs=assume_shared_fs,
                    max_status_checks_per_second=max_status_checks_per_second,
                    keepincomplete=keepincomplete,
                    keepmetadata=keepmetadata,
                )
        elif kubernetes:
            self._local_executor = CPUExecutor(
                workflow,
                dag,
                local_cores,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                cores=local_cores,
                keepincomplete=keepincomplete,
                keepmetadata=keepmetadata,
            )

            self._executor = KubernetesExecutor(
                workflow,
                dag,
                kubernetes,
                container_image=container_image,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                cluster_config=cluster_config,
                keepincomplete=keepincomplete,
                keepmetadata=keepmetadata,
            )
        elif tibanna:
            self._local_executor = CPUExecutor(
                workflow,
                dag,
                local_cores,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                use_threads=use_threads,
                latency_wait=latency_wait,
                cores=local_cores,
                keepincomplete=keepincomplete,
                keepmetadata=keepmetadata,
            )

            self._executor = TibannaExecutor(
                workflow,
                dag,
                cores,
                tibanna_sfn,
                precommand=precommand,
                tibanna_config=tibanna_config,
                container_image=container_image,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                keepincomplete=keepincomplete,
                keepmetadata=keepmetadata,
            )
        elif google_lifesciences:
            self._local_executor = CPUExecutor(
                workflow,
                dag,
                local_cores,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                cores=local_cores,
            )

            self._executor = GoogleLifeSciencesExecutor(
                workflow,
                dag,
                cores,
                container_image=container_image,
                regions=google_lifesciences_regions,
                location=google_lifesciences_location,
                cache=google_lifesciences_cache,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                preemption_default=preemption_default,
                preemptible_rules=preemptible_rules,
            )
        elif tes:
            self._local_executor = CPUExecutor(
                workflow,
                dag,
                local_cores,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                cores=local_cores,
                keepincomplete=keepincomplete,
            )

            self._executor = TaskExecutionServiceExecutor(
                workflow,
                dag,
                cores=local_cores,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                latency_wait=latency_wait,
                tes_url=tes,
                container_image=container_image,
            )

        else:
            self._executor = CPUExecutor(
                workflow,
                dag,
                cores,
                printreason=printreason,
                quiet=quiet,
                printshellcmds=printshellcmds,
                use_threads=use_threads,
                latency_wait=latency_wait,
                cores=cores,
                keepincomplete=keepincomplete,
                keepmetadata=keepmetadata,
            )
        if self.max_jobs_per_second and not self.dryrun:
            max_jobs_frac = Fraction(self.max_jobs_per_second).limit_denominator()
            self.rate_limiter = RateLimiter(
                max_calls=max_jobs_frac.numerator, period=max_jobs_frac.denominator
            )

        else:
            # essentially no rate limit
            self.rate_limiter = DummyRateLimiter()

        # Choose job selector (greedy or ILP)
        self.job_selector = self.job_selector_greedy
        if scheduler_type == "ilp":
            import pulp

            if pulp.apis.LpSolverDefault is None:
                logger.warning(
                    "Falling back to greedy scheduler because no default "
                    "solver is found for pulp (you have to install either "
                    "coincbc or glpk)."
                )
            else:
                self.job_selector = self.job_selector_ilp

        self._user_kill = None
        try:
            signal.signal(signal.SIGTERM, self.exit_gracefully)
        except ValueError:
            # If this fails, it is due to scheduler not being invoked in the main thread.
            # This can only happen with --gui, in which case it is fine for now.
            pass
        self._open_jobs.release()

    def executor_error_callback(self, exception):
        with self._lock:
            self._executor_error = exception
            # next scheduling round to catch and raise error
            self._open_jobs.release()

    @property
    def stats(self):
        try:
            return self._executor.stats
        except AttributeError:
            raise TypeError("Executor does not support stats")

    @property
    def open_jobs(self):
        """Return open jobs."""
        jobs = self.dag.ready_jobs

        if not self.dryrun:
            jobs = [
                job
                for job in jobs
                if not job.dynamic_input and not self.dag.dynamic(job)
            ]
        return jobs

    @property
    def remaining_jobs(self):
        """Return jobs to be scheduled including not yet ready ones."""
        return [
            job
            for job in self.dag.needrun_jobs
            if job not in self.running and not self.dag.finished(job)
        ]

    def schedule(self):
        """Schedule jobs that are ready, maximizing cpu usage."""
        try:
            while True:
                # work around so that the wait does not prevent keyboard interrupts
                # while not self._open_jobs.acquire(False):
                #    time.sleep(1)
                self._open_jobs.acquire()

                # obtain needrun and running jobs in a thread-safe way
                with self._lock:
                    self._finish_jobs()
                    self._error_jobs()
                    needrun = set(self.open_jobs)
                    running = list(self.running)
                    errors = self._errors
                    executor_error = self._executor_error
                    user_kill = self._user_kill

                # handle errors
                if user_kill or (not self.keepgoing and errors) or executor_error:
                    if user_kill == "graceful":
                        logger.info(
                            "Will exit after finishing currently running jobs (scheduler)."
                        )

                    if executor_error:
                        print_exception(executor_error, self.workflow.linemaps)

                    if executor_error or not running:
                        logger.info("Shutting down, this might take some time.")
                        self._executor.shutdown()
                        if not user_kill:
                            logger.error(_ERROR_MSG_FINAL)
                        return False
                    continue

                # all runnable jobs have finished, normal shutdown
                if not needrun and (not running or self.workflow.immediate_submit):
                    self._executor.shutdown()
                    if errors:
                        logger.error(_ERROR_MSG_FINAL)
                    # we still have unfinished jobs. this is not good. direct
                    # user to github issue
                    if self.remaining_jobs:
                        logger.error(_ERROR_MSG_ISSUE_823)
                        logger.error(
                            "Remaining jobs:\n"
                            + "\n".join(
                                " - " + str(job) + ": " + ", ".join(job.output)
                                for job in self.remaining_jobs
                            )
                        )
                        return False
                    return not errors

                # continue if no new job needs to be executed
                if not needrun:
                    continue

                # select jobs by solving knapsack problem (omit with dryrun)
                if self.dryrun:
                    run = needrun
                else:
                    logger.debug(
                        "Resources before job selection: {}".format(self.resources)
                    )
                    logger.debug(
                        "Ready jobs ({}):\n\t".format(len(needrun))
                        + "\n\t".join(map(str, needrun))
                    )

                    if not self._last_job_selection_empty:
                        logger.info("Select jobs to execute...")
                    run = self.job_selector(needrun)
                    self._last_job_selection_empty = not run

                    logger.debug(
                        "Selected jobs ({}):\n\t".format(len(run))
                        + "\n\t".join(map(str, run))
                    )
                    logger.debug(
                        "Resources after job selection: {}".format(self.resources)
                    )
                # update running jobs
                with self._lock:
                    self.running.update(run)
                    # remove from ready_jobs
                    self.dag.register_running(run)

                # reset params and resources because they might contain TBDs
                if not self.dryrun:
                    for job in run:
                        job.reset_params_and_resources()

                # actually run jobs
                local_runjobs = [job for job in run if job.is_local]
                runjobs = [job for job in run if not job.is_local]
                self.run(local_runjobs, executor=self._local_executor or self._executor)
                self.run(runjobs)
        except (KeyboardInterrupt, SystemExit):
            logger.info(
                "Terminating processes on user request, this might take some time."
            )
            self._executor.cancel()
            return False

    def _finish_jobs(self):
        # must be called from within lock
        # clear the global tofinish such that parallel calls do not interfere
        for job in self._tofinish:
            if self.handle_job_success:
                try:
                    self.get_executor(job).handle_job_success(job)
                except (RuleException, WorkflowError) as e:
                    # if an error occurs while processing job output,
                    # we do the same as in case of errors during execution
                    print_exception(e, self.workflow.linemaps)
                    self._handle_error(job)
                    continue

            if self.update_resources:
                # normal jobs have len=1, group jobs have len>1
                self.finished_jobs += len(job)
                self.running.remove(job)
                self._free_resources(job)

            if self.print_progress:
                if job.is_group():
                    for j in job:
                        logger.job_finished(jobid=j.jobid)
                else:
                    logger.job_finished(jobid=job.jobid)
                self.progress()

            self.dag.finish(job, update_dynamic=self.update_dynamic)
        self._tofinish.clear()

    def _error_jobs(self):
        # must be called from within lock
        for job in self._toerror:
            self._handle_error(job)
        self._toerror.clear()

    def run(self, jobs, executor=None):
        if executor is None:
            executor = self._executor
        executor.run_jobs(
            jobs,
            callback=self._finish_callback,
            submit_callback=self._submit_callback,
            error_callback=self._error,
        )

    def get_executor(self, job):
        if job.is_local and self._local_executor is not None:
            return self._local_executor
        return self._executor

    def _noop(self, job):
        pass

    def _free_resources(self, job):
        for name, value in job.resources.items():
            if name in self.resources:
                value = self.calc_resource(name, value)
                self.resources[name] += value

    def _proceed(self, job):
        """Do stuff after job is finished."""
        with self._lock:
            self._tofinish.append(job)

            if self.dryrun:
                if len(self.running) - len(self._tofinish) - len(self._toerror) <= 0:
                    # During dryrun, only release when all running jobs are done.
                    # This saves a lot of time, as self.open_jobs has to be
                    # evaluated less frequently.
                    self._open_jobs.release()
            else:
                # go on scheduling if there is any free core
                self._open_jobs.release()

    def _error(self, job):
        with self._lock:
            self._toerror.append(job)
            self._open_jobs.release()

    def _handle_error(self, job):
        """Clear jobs and stop the workflow.

        If Snakemake is configured to restart jobs then the job might have
        "restart_times" left and we just decrement and let the scheduler
        try to run the job again.
        """
        self.get_executor(job).handle_job_error(job)
        self.running.remove(job)
        self._free_resources(job)
        # attempt starts counting from 1, but the first attempt is not
        # a restart, hence we subtract 1.
        if job.restart_times > job.attempt - 1:
            logger.info("Trying to restart job {}.".format(self.dag.jobid(job)))
            job.attempt += 1
            # add job to those being ready again
            self.dag._ready_jobs.add(job)
        else:
            self._errors = True
            self.failed.add(job)
            if self.keepgoing:
                logger.info("Job failed, going on with independent jobs.")

    def exit_gracefully(self, *args):
        with self._lock:
            self._user_kill = "graceful"
        self._open_jobs.release()

    def job_selector_ilp(self, jobs):
        """
        Job scheduling by optimization of resource usage by solving ILP using pulp
        """
        import pulp
        from pulp import lpSum
        from stopit import ThreadingTimeout as Timeout, TimeoutException

        if len(jobs) == 1:
            logger.debug(
                "Using greedy selector because only single job has to be scheduled."
            )
            return self.job_selector_greedy(jobs)

        with self._lock:
            if not self.resources["_cores"]:
                return set()

            # assert self.resources["_cores"] > 0
            scheduled_jobs = {
                job: pulp.LpVariable(
                    "job_{}".format(idx), lowBound=0, upBound=1, cat=pulp.LpInteger
                )
                for idx, job in enumerate(jobs)
            }

            def size_gb(f):
                if self.touch:
                    # In case of touch mode, there is no need to prioritize based on size.
                    # We cannot access it anyway, because the files might be temporary and
                    # not present.
                    return 0
                else:
                    return f.size / 1e9

            temp_files = {
                temp_file for job in jobs for temp_file in self.dag.temp_input(job)
            }

            temp_job_improvement = {
                temp_file: pulp.LpVariable(
                    "temp_file_{}".format(idx), lowBound=0, upBound=1, cat="Continuous"
                )
                for idx, temp_file in enumerate(temp_files)
            }

            temp_file_deletable = {
                temp_file: pulp.LpVariable(
                    "deletable_{}".format(idx),
                    lowBound=0,
                    upBound=1,
                    cat=pulp.LpInteger,
                )
                for idx, temp_file in enumerate(temp_files)
            }
            prob = pulp.LpProblem("JobScheduler", pulp.LpMaximize)

            total_temp_size = max(
                sum([size_gb(temp_file) for temp_file in temp_files]), 1
            )
            total_core_requirement = sum(
                [max(job.resources.get("_cores", 1), 1) for job in jobs]
            )
            # Objective function
            # Job priority > Core load
            # Core load > temp file removal
            # Instant removal > temp size
            prob += (
                2
                * total_core_requirement
                * 2
                * total_temp_size
                * lpSum([job.priority * scheduled_jobs[job] for job in jobs])
                + 2
                * total_temp_size
                * lpSum(
                    [
                        max(job.resources.get("_cores", 1), 1) * scheduled_jobs[job]
                        for job in jobs
                    ]
                )
                + total_temp_size
                * lpSum(
                    [
                        temp_file_deletable[temp_file] * size_gb(temp_file)
                        for temp_file in temp_files
                    ]
                )
                + lpSum(
                    [
                        temp_job_improvement[temp_file] * size_gb(temp_file)
                        for temp_file in temp_files
                    ]
                )
            )

            # Constraints:
            for name in self.workflow.global_resources:
                prob += (
                    lpSum(
                        [
                            scheduled_jobs[job] * job.resources.get(name, 0)
                            for job in jobs
                        ]
                    )
                    <= self.resources[name]
                )

            # Choose jobs that lead to "fastest" (minimum steps) removal of existing temp file
            remaining_jobs = self.remaining_jobs
            for temp_file in temp_files:
                prob += temp_job_improvement[temp_file] <= lpSum(
                    [
                        scheduled_jobs[job] * self.required_by_job(temp_file, job)
                        for job in jobs
                    ]
                ) / lpSum(
                    [self.required_by_job(temp_file, job) for job in remaining_jobs]
                )

                prob += (
                    temp_file_deletable[temp_file] <= temp_job_improvement[temp_file]
                )

        try:
            with Timeout(10, swallow_exc=False):
                self._solve_ilp(prob)
        except TimeoutException as e:
            logger.warning(
                "Failed to solve scheduling problem with ILP solver in time (10s). "
                "Falling back to greedy solver."
            )
            return self.job_selector_greedy(jobs)
        except pulp.apis.core.PulpSolverError as e:
            logger.warning(
                "Failed to solve scheduling problem with ILP solver. Falling back to greedy solver. "
                "Run Snakemake with --verbose to see the full solver output for debugging the problem."
            )
            return self.job_selector_greedy(jobs)

        selected_jobs = set(
            job for job, variable in scheduled_jobs.items() if variable.value() == 1.0
        )

        if not selected_jobs:
            # No selected jobs. This could be due to insufficient resources or a failure in the ILP solver
            # Hence, we silently fall back to the greedy solver to make sure that we don't miss anything.
            return self.job_selector_greedy(jobs)

        for name in self.workflow.global_resources:
            self.resources[name] -= sum(
                [job.resources.get(name, 0) for job in selected_jobs]
            )
        return selected_jobs

    def _solve_ilp(self, prob):
        import pulp

        old_path = os.environ["PATH"]
        if self.workflow.scheduler_solver_path is None:
            # Temporarily prepend the given snakemake env to the path, such that the solver can be found in any case.
            # This is needed for cluster envs, where the cluster job might have a different environment but
            # still needs access to the solver binary.
            os.environ["PATH"] = "{}:{}".format(
                self.workflow.scheduler_solver_path, os.environ["PATH"]
            )
        try:
            solver = (
                pulp.get_solver(self.scheduler_ilp_solver)
                if self.scheduler_ilp_solver
                else pulp.apis.LpSolverDefault
            )
        finally:
            os.environ["PATH"] = old_path
        solver.msg = self.workflow.verbose
        prob.solve(solver)

    def required_by_job(self, temp_file, job):
        return 1 if temp_file in self.dag.temp_input(job) else 0

    def job_selector_greedy(self, jobs):
        """
        Using the greedy heuristic from
        "A Greedy Algorithm for the General Multidimensional Knapsack
        Problem", Akcay, Li, Xu, Annals of Operations Research, 2012

        Args:
            jobs (list):    list of jobs
        """
        with self._lock:
            if not self.resources["_cores"]:
                return set()
            # each job is an item with one copy (0-1 MDKP)
            n = len(jobs)
            x = [0] * n  # selected jobs
            E = set(range(n))  # jobs still free to select
            u = [1] * n
            a = list(map(self.job_weight, jobs))  # resource usage of jobs
            c = list(map(self.job_reward, jobs))  # job rewards

            def calc_reward():
                return [c_j * y_j for c_j, y_j in zip(c, y)]

            b = [
                self.resources[name] for name in self.global_resources
            ]  # resource capacities

            while True:
                # Step 2: compute effective capacities
                y = [
                    (
                        min(
                            (min(u[j], b_i // a_j_i) if a_j_i > 0 else u[j])
                            for b_i, a_j_i in zip(b, a[j])
                            if a_j_i
                        )
                        if j in E
                        else 0
                    )
                    for j in range(n)
                ]
                if not any(y):
                    break
                y = [
                    (max(1, int(self.greediness * y_j)) if y_j > 0 else 0) for y_j in y
                ]

                # Step 3: compute rewards on cumulative sums
                reward = calc_reward()
                j_sel = max(E, key=reward.__getitem__)  # argmax

                # Step 4: batch increment
                y_sel = y[j_sel]

                # Step 5: update information
                x[j_sel] += y_sel
                b = [b_i - (a_j_i * y_sel) for b_i, a_j_i in zip(b, a[j_sel])]
                u[j_sel] -= y_sel
                if not u[j_sel] or self.greediness == 1:
                    E.remove(j_sel)
                if not E:
                    break

            solution = set(job for job, sel in zip(jobs, x) if sel)
            # update resources
            for name, b_i in zip(self.global_resources, b):
                self.resources[name] = b_i
            return solution

    def calc_resource(self, name, value):
        gres = self.global_resources[name]
        if value > gres:
            if name == "_cores":
                name = "threads"
            raise WorkflowError(
                "Job needs {name}={res} but only {name}={gres} "
                "are available. This is likely because two "
                "jobs are connected via a pipe or a service output and have to run "
                "simultaneously. Consider providing more "
                "resources (e.g. via --cores).".format(name=name, res=value, gres=gres)
            )
        return value

    def rule_weight(self, rule):
        res = rule.resources
        return [
            self.calc_resource(name, res.get(name, 0)) for name in self.global_resources
        ]

    def job_weight(self, job):
        res = job.resources
        return [
            self.calc_resource(name, res.get(name, 0)) for name in self.global_resources
        ]

    def job_reward(self, job):
        if self.touch or self.dryrun or self.workflow.immediate_submit:
            temp_size = 0
            input_size = 0
        else:
            try:
                temp_size = self.dag.temp_size(job)
                input_size = job.inputsize
            except FileNotFoundError:
                # If the file is not yet present, this shall not affect the
                # job selection.
                temp_size = 0
                input_size = 0

        # Usually, this should guide the scheduler to first schedule all jobs
        # that remove the largest temp file, then the second largest and so on.
        # Since the weight is summed up, it can in theory be that it sometimes
        # prefers a set of many jobs that all depend on smaller temp files though.
        # A real solution to the problem is therefore to use dummy jobs that
        # ensure selection of groups of jobs that together delete the same temp
        # file.

        return (job.priority, temp_size, input_size)

    def progress(self):
        """Display the progress."""
        logger.progress(done=self.finished_jobs, total=len(self.dag))