Oops I posted the wrong version of the code:

sub runFile {
  local($fileName, $timelimit) = @_;
  my $child_pid;
  local $SIG{ALRM} = sub {
    $didTimeout=1;
    print OUTPUT "timeout";
    print OUTPUT "-", $child_pid;
    kill $child_pid, 9;
  };
  if ($child_pid = fork ) { # create child
    waitpid($child_pid, 0); # Wait for child to terminate
  } elsif (defined $child_pid) {
      alarm $timelimit;
      $cmd = "time $solver1 $fileName >> output.txt";
      exec ($cmd);
      alarm 0;
      exit;
  } else { # Something terrible has happened, as fork failed
    die "Cannot fork!\n";
  }
}
[download]

This code times out the function but allows it to still run in the background.