A strongly polynomial algorithm for general matrices with fixed m

We end this section by giving a brief description of how the solution method for systems satisfying Property OneFP could be adapted to find integer solutions to any TSS, but that in doing so we may lose efficiency. Since we can convert any general two-sided system into a system with separated variables we discuss systems with separated variables only. Let A ∈ Rm×n, B ∈ Rm×k and suppose (A, B) satisfies Property One+_{FP. In this case}

for each row i of Ax = By we will have a number of pairs (air(i,s), bir0_(i,s)), some integer

Ax = By we can identify a single pair of active entries for each row of the equation, and hence the pair (x, y) is also an integer solution to the system Ax = B−y where (A, B−) satisfies Property OneFP and B− is obtained from B by slightly decreasing each inactive entry in B with respect to y.

In general Ax = By if and only if there exists an m-tuple (k1, ..., km), ki ∈ {1, ..., k},

a real number 0 < δ  1 and a matrix B− = (b−_{ij) ∈ R}m×k with

b−_ij =        bij, if j = ki; bij ⊗ δ−1, otherwise such that Ax = B−y.

Hence given a pair (A, B) satisfying Property One+FP we can generate a number of pairs (A, B(t)_{), t ∈ N such that x, y is an integer solution to Ax = By if and only is there} exists t such that Ax = B(t)_{y. Note that each B}(t) _{is obtained from B by decreasing}

the value of all but one element, bir0_(i,s), per row and the pairs (A, B(t)) satisfy Property

OneFP. We can therefore determine whether an integer solution to Ax = B(t)_{y exists in}

strongly polynomial time.

Unfortunately in the worst case there could be as many as nk pairs per row and thus (nk)m matrices to check, so the complexity of this method is O(m3nmkm). However we can say that, for fixed m, a strongly polynomial method for finding integer solutions to TSS exists.

5.3

Conclusion

We began by constructing Algorithms 5.1 (SEP-INT-TSS) and 5.4 (GEN-INT-TSS) which, for finite input matrices, can determine whether an integer solution to Ax = By and Ax = Bx respectively, exist. Further we proved that these algorithms run in pseudopoly-

nomial time for finite input matrices.

We defined a class of TSSs for which the entire set of integer solutions could be described in strongly polynomial time, a key result being Theorem 5.14. This was any TSS for which the pair of input matrices satisfied Property OneFP. We used this class of matrices to show that, for fixed m, it is possible to find integer solutions to TSSs in strongly polynomial time.

Further research could be done to find more strongly polynomially solvable cases, and to give an efficient full description of all integer solutions to a TSS. It is also known, see [27], that the set of integer solutions to a TSS can be written as an intersection of the solution sets of one sided systems. While the set of integer solutions to each of these simpler systems can be described in strongly polynomial time, the number of systems involved in the description is too large to determine whether an integer solution to the original TSS existed efficiently.

At the time of writing, for two-sided systems which do not satisfy the generic prop- erty, it is unknown whether an integer solution, or indeed any solution, can be found in polynomial time. If we remove the integrality requirement, then it is known that finding a solution to a max-algebraic two-sided system is equivalent to finding a solution to a mean payoff game [10]. Mean payoff games are a well known class of problems in NP ∩ co-NP, it is expected that a polynomial solution method will be found in the future.

6. Integer max-linear programs

In this chapter we investigate integer solutions to max-linear programming problems. We briefly consider the case when the constraints are in the form of a one-sided equality, showing that methods for finding real solutions can be adapted to find integer solutions, and that the optimal objective value, and an optimal solution, can be found in strongly polynomial time. The main focus of this chapter is problems with two-sided constraints. Using Algorithm 5.4 (GEN-INT-TSS) for solving TSSs, we describe a bisection method to find an optimal solution to an integer max-linear program in pseudopolynomial time, this is Algorithms INT-MAXLINMIN and INT-MAXLINMAX. Finally, we consider the IMLP where the input matrices satisfy Property OneFP. Key results are Theorems 6.33 and 6.34, which describe the optimal objective value and find an optimal solution to finite input systems in strongly polynomial time. These results are then used to prove that, for any input matrix, the problem is strongly polynomially solvable if the input matrices satisfy Property OneFP.

Recall that, in max-algebra, we have to consider the maximisation problem and the minimisation problem independently, as there is no easy way to switch between them.

6.1

One-sided constraints

Let A ∈ Rm×n b ∈ Rm, c ∈ Rn. We will assume throughout this section that A is doubly R-astic. The one-sided max linear program (OMLP) is stated below.

cT ⊗ x → min or max (OM LP )

s.t A ⊗ x = b x ∈ Rn

When we additionally require that x ∈ Zn _{we have the one-sided integer max-linear}

program (OIMLP). We will show that the OIMLP is strongly polynomially solvable. We use OMLPmax and OMLPmin to denote the problems maximising and minimising the objective function respectively. Similarly for OIMLP.