Minimum relevant variables in linear system

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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.

Definition

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.

Special case

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.

Applications

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 [math]\displaystyle{ O(\log(m)) }[/math] 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).

Related problems

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 [math]\displaystyle{ O(n\cdot m^n / 2^{n-1}) }[/math].
  • 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 [math]\displaystyle{ 2^{\log^{1-\varepsilon}n} }[/math], for any [math]\displaystyle{ \varepsilon\gt 0 }[/math], unless NP is contained in DTIME([math]\displaystyle{ n^{\operatorname{polylog}(n)} }[/math]).[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 [math]\displaystyle{ m^{\varepsilon} }[/math]. 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.

Hardness of approximation

All four variants of Min-RVLS are hard to approximate. In particular all four variants cannot be approximated within a factor of [math]\displaystyle{ 2^{\log^{1-\varepsilon}n} }[/math], for any [math]\displaystyle{ \varepsilon\gt 0 }[/math], unless NP is contained in DTIME([math]\displaystyle{ n^{\operatorname{polylog}(n)} }[/math]).[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.

References

  1. ↑ 1.0 1.1 1.2 Amaldi, Edoardo; Kann, Viggo (December 1998). "On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems". Theoretical Computer Science 209 (1–2): 237–260. doi:10.1016/S0304-3975(97)00115-1. 
  2. ↑ Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman. ISBN 978-0-7167-1044-8. 
  3. ↑ Koehler, Gary J. (November 1991). "Linear Discriminant Functions Determined by Genetic Search". ORSA Journal on Computing 3 (4): 345–357. doi:10.1287/ijoc.3.4.345. 
  4. ↑ Van Horn, Kevin S.; Martinez, Tony R. (January 1994). "The minimum feature set problem". Neural Networks 7 (3): 491–494. doi:10.1016/0893-6080(94)90082-5. 
  5. ↑ 5.0 5.1 Amaldi, Edoardo; Kann, Viggo (August 1995). "The complexity and approximability of finding maximum feasible subsystems of linear relations". Theoretical Computer Science 147 (1–2): 181–210. doi:10.1016/0304-3975(94)00254-G. 
  6. ↑ 6.0 6.1 Sankaran, Jayaram K (February 1993). "A note on resolving infeasibility in linear programs by constraint relaxation". Operations Research Letters 13 (1): 19–20. doi:10.1016/0167-6377(93)90079-V. 
  7. ↑ Greer, R. (2011). Trees and Hills: Methodology for Maximizing Functions of Systems of Linear Relations. Elsevier. ISBN 978-0-08-087207-0. [page needed]
  8. ↑ Arora, Sanjeev; Babai, László; Stern, Jacques; Sweedyk, Z (April 1997). "The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations". Journal of Computer and System Sciences 54 (2): 317–331. doi:10.1006/jcss.1997.1472. 



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