Integer Decomposition for Polyhedra Defined by Nearly Totally Unimodular Matrices

Abstract

We call a matrix A nearly totally unimodular if it can be obtained from a totally unimodular matrixà by adding to each row ofà an integer multiple of some fixed row a T ofÃ. For an integer vector b and a nearly totally unimodular matrix A, we denote by P A,b the integer hull of the set {x ∈ R n | Ax ≤ b}. We show that P A,b has the integer decomposition property and that we can find a decomposition of a given integer vector x ∈ kP A,b in polynomial time.An interesting special case that plays a role in many cyclic scheduling problems, is when A is a circular-ones matrix. In this case, we show that given a nonnegative integer k and an integer vector x, testing if x ∈ kP A,b and finding a decomposition of x into k integer vectors in P A,b can be done time O(n(n + m) + size(x)), where A is an m × n matrix. We show that the method unifies some known results on colouring circular arc graphs and edge colouring nearly bipartite graphs. It also gives an efficient algorithm for a packet scheduling problem for smart antennas posed by E. Amaldi et al. in [1,2].

The Representation Polyhedron of a Semiorder

“…Although parametric versions of some combinatorial optimization problems have been thoroughly studied (see, e.g., Young, Tarjan and Orlin [27] or Agarwal, Eppstein, Guibas and Henzinger [1]), it does not seem to be the case for potential problems. Some particular parametric potential problems have been studied before, for instance in connection with problems whose constraint matrix has the circular ones property (see, e.g., Bartholdi, Orlin and Ratliff [6], Eisenbrand, Oriolo, Stauffer and Ventura [11] or Gijswijt [15]). To the best of our knowledge, we lack a characterization of the parametric potential problems with a totally dual integral system of constraints.…”

Coloring Fuzzy Circular Interval Graphs

“…An application of a result by Gijswijt [4] shows that so called integer decompositions, see e.g. [8], in the stable set polytope of circular interval graphs can be computed efficiently.…”

“…For given k ∈ N and a fuzzy circular interval graph of order n with weights w, it decides the existence of a weighted k coloring in time O (n 3 ) and computes a weighted k coloring in time O (n 4 + size(w)), where size(w) denotes the binary encoding length of w. The weighted coloring number alone can be computed in time O (n 2 size(w)). The algorithm is based on a reduction to circular interval graphs using an algorithm for maximum b-matching and an algorithm of Gijswijt [4] to solve the weighted coloring problem on circular interval graphs.…”

Testing additive integrality gaps

“…There are many examples of tuples (A, c) stemming from combinatorial optimization problems that have the integer rounding property [1,5,18,21]. In this case the integral problem can also be solved exactly by linear programming techniques, see e.g.…”