<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def fix_variables(bqm, sampling_mode=True):
 """Determine assignments for some variables of a binary quadratic model. Roof duality finds a lower bound for the minimum of a quadratic polynomial. It can also find minimizing assignments for some of the polynomial's variables; these fixed variables take the same values in all optimal solutions [BHT]_ [BH]_. A quadratic pseudo-Boolean function can be represented as a network to find the lower bound through network-flow computations. `fix_variables` uses maximum flow in the implication network to correctly fix variables. Consequently, you can find an assignment for the remaining variables that attains the optimal value. Args: bqm (:obj:`.BinaryQuadraticModel`) A binary quadratic model. sampling_mode (bool, optional, default=True):
 In sampling mode, only roof-duality is used. When `sampling_mode` is false, strongly connected components are used to fix more variables, but in some optimal solutions these variables may take different values. Returns: dict: Variable assignments for some variables of the specified binary quadratic model. Examples: This example creates a binary quadratic model with a single ground state and fixes the model's single variable to the minimizing assignment. {'a': -1} This example has two ground states, :math:`a=b=-1` and :math:`a=b=1`, with no variable having a single value for all ground states, so neither variable is fixed. {} This example turns sampling model off, so variables are fixed to an assignment that attains the ground state. {'a': 1, 'b': 1} .. [BHT] Boros, E., P.L. Hammer, G. Tavares. Preprocessing of Unconstraint Quadratic Binary Optimization. Rutcor Research Report 10-2006, April, 2006. .. [BH] Boros, E., P.L. Hammer. Pseudo-Boolean optimization. Discrete Applied Mathematics 123, (2002), pp. 155-225 """

    try:
        from dimod.roof_duality._fix_variables import fix_variables_wrapper
    except ImportError:
        raise ImportError("c++ extension roof_duality is not built")

    if sampling_mode:
        method = 2  # roof-duality only
    else:
        method = 1  # roof-duality and strongly connected components

    linear = bqm.linear
    if all(v in linear for v in range(len(bqm))):
        # we can work with the binary form of the bqm directly
        fixed = fix_variables_wrapper(bqm.binary, method)
    else:
        try:
            inverse_mapping = dict(enumerate(sorted(linear)))
        except TypeError:
            # in python3 unlike types cannot be sorted
            inverse_mapping = dict(enumerate(linear))
        mapping = {v: i for i, v in inverse_mapping.items()}

        fixed = fix_variables_wrapper(bqm.relabel_variables(mapping, inplace=False).binary, method)
        fixed = {inverse_mapping[v]: val for v, val in fixed.items()}

    if bqm.vartype is Vartype.SPIN:
        return {v: 2*val - 1 for v, val in fixed.items()}
    else:
        return fixed