Minimum relevant variables in linear system

A Min-RVLS problem is defined by:[1]

• A binary relation R, which is one of {=, ≥, >, ≠};
• An m-by-n matrix A (where m is the number of constraints and n the number of variables);
• An m-by-1 vector b.

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,[2] 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.[3] The problem is known as the minimum feature set problem. An algorithm that approximates Min-RVLS within a factor of ${\displaystyle O(\log(m))}$ could substantially reduce the number of training samples required to attain a given accuracy level.[4]

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:

• Min-ULR[=,>,≥] are NP-hard even with homogeneous systems and bipolar coefficients (coefficients in {1,-1}). [5]
• The NP-complete problem Minimum feedback arc set reduces to Min-ULR[≥], with exactly one 1 and one -1 in each constraint, and all right-hand sides equal to 1.[6]
• Min-ULR[=,>,≥] are polynomial if the number of variables n is constant: they can be solved polynomially using an algorithm of Greer[7] in time ${\displaystyle O(n\cdot m^{n}/2^{n-1})}$.
• Min-ULR[=,>,≥] are linear if the number of constraints m is constant, since all subsystems can be checked in time O(n).
• Min-ULR[≥] is polynomial in some special case.[6]
• Min-ULR[=,>,≥] can be approximated within n + 1 in polynomial time.[1]
• Min-ULR[>,≥] are minimum-dominating-set-hard, even with homogeneous systems and ternary coefficients (in {−1,0,1}).
• Min-ULR[=] cannot be approximated within a factor of ${\displaystyle 2^{\log ^{1-\varepsilon }n}}$, for any ${\displaystyle \varepsilon >0}$, unless NP is contained in DTIME(${\displaystyle n^{\operatorname {polylog} (n)}}$).[8]

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.[5]

• Max-FLS[≠] can be solved in polynomial time.
• Max-FLS[=] is NP-hard even with homogeneous systems and bipolar coefficients.

• . With integer coefficients, it is hard to approximate within ${\displaystyle m^{\varepsilon }}$. With coefficients over GF[q], it is q-approximable.
• Max-FLS[>] and Max-FLS[≥] are NP-hard even with homogeneous systems and bipolar coefficients. They are 2-approximable, but they cannot be approximated within any smaller constant factor.

All four variants of Min-RVLS are hard to approximate. In particular all four variants cannot be approximated within a factor of ${\displaystyle 2^{\log ^{1-\varepsilon }n}}$, for any ${\displaystyle \varepsilon >0}$, unless NP is contained in DTIME(${\displaystyle n^{\operatorname {polylog} (n)}}$).[1]^{: 247–250} The hardness is proved by reductions:

• There is a reduction from Min-ULR[=] to Min-RVLS[=]. It also applies to Min-RVLS[≥] and Min-RVLS[>], since each equation can be replaced by two complementary inequalities.
• There is a reduction from minimum-dominating-set to Min-RVLS[≠].

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.