MINimum Relevant Variables in Linear System (Min-RVLS) is a problem in mathematical optimization. Given a linear program, it is required to find a feasible solution in which the number of non-zero variables is as small as possible.
 The problem is known to be NP-hard and even hard to approximate.
 A Min-RVLS problem is defined by:
 The linear system is given by: A x R b. It is assumed to be feasible (i.e., satisfied by at least one x). Depending on R, there are four different variants of this system: A x = b, A x ≥ b, A x > b, A x ≠ b.
 The goal is to find an n-by-1 vector x that satisfies the system A x R b, and subject to that, contains as few as possible nonzero elements.
 The problem Min-RVLS[=] was presented by Garey and Johnson, who called it "minimum weight solution to linear equations". They proved it was NP-hard, but did not consider approximations.
 The Min-RVLS problem is important in machine learning and linear discriminant analysis. Given a set of positive and negative examples, it is required to minimize the number of features that are required to correctly classify them. The problem is known as the minimum feature set problem. An algorithm that approximates Min-RVLS within a factor of 
  
    
      
        O
        (
        log
        ⁡
        (
        m
        )
        )
      
    
    {\displaystyle O(\log(m))}
  
 could substantially reduce the number of training samples required to attain a given accuracy level.
 The shortest codeword problem in coding theory is the same problem as Min-RVLS[=] when the coefficients are in GF(2).
 In MINimum Unsatisfied Linear Relations (Min-ULR), we are given a binary relation R and a linear system A x R b, which is now assumed to be  infeasible.  The goal is to find a vector x that violates as few relations as possible, while satisfying all the others.
 Min-ULR[≠] is trivially solvable, since any system with real variables and a finite number of inequality constraints is feasible. As for the other three variants:
 In the complementary problem MAXimum Feasible Linear Subsystem (Max-FLS), the goal is to find a maximum subset of the constraints that can be satisfied simultaneously.  
 All four variants of Min-RVLS are hard to approximate. In particular all four variants cannot be approximated within a factor of 
  
    
      
        
          2
          
            
              log
              
                1
                −
                ε
              
            
            ⁡
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    {\displaystyle 2^{\log ^{1-\varepsilon }n}}
  
, for any 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
, unless NP is contained in DTIME(
  
    
      
        
          n
          
            polylog
            ⁡
            (
            n
            )
          
        
      
    
    {\displaystyle n^{\operatorname {polylog} (n)}}
  
).: 247–250  The hardness is proved by reductions:
 On the other hand, there is a reduction from Min-RVLS[=] to Min-ULR[=]. It also applies to Min-ULR[≥] and Min-ULR[>], since each equation can be replaced by two complementary inequalities.
 Therefore, when R is in {=,>,≥}, Min-ULR and Min-RVLS are equivalent in terms of approximation hardness.