Shinji Mizuno We study the weighted induced bipartite subgraph problem (WIBSP). The goal of WIBSP is, given a graph and nonnegative weights for the nodes, to find a set W of nodes with the maximum total weight such that a subgraph induced by W is bipartite. WIBSP is also referred as to the graph bipartization problem or … Read more
We study the covering-type k-violation linear program where at most $k$ of the constraints can be violated. This problem is formulated as a mixed integer program and known to be strongly NP-hard. In this paper, we present a simple (k+1)-approximation algorithm using a natural LP relaxation. We also show that the integrality gap of the … Read more
We present an improved approximation algorithm for the covering 0-1 integer program (CIP), a well-known problem as a natural generalization of the set cover problem. Our algorithm uses a primal-dual algorithm for CIP by Fujito (2004) as a subroutine and achieves an approximation ratio of (f- (f-1)/m) when m is greater than or equal to … Read more
The partial covering 0-1 integer program (PCIP) is a relaxed problem of the covering 0-1 integer program (CIP) such that some fixed number of constraints may not be satisfied. This type of relaxation is also discussed in the partial set multi-cover problem (PSMCP) and the partial set cover problem (PSCP). In this paper, we propose … Read more
Carnes and Shmoys (2015) presented a 2-approximation algorithm for the minimum knapsack problem. We extend their algorithm to the minimum knapsack problem with a forcing graph (MKPFG), which has a forcing constraint for each edge in the graph. The forcing constraint means that at least one item (vertex) of the edge must be packed in … Read more
In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin’s primal-based modification and Mizuno’s use of the simplex method, we introduce a modification of Tardos’ algorithm considering only the primal problem and using simplex method … Read more
Kitahara and Mizuno get new bounds for the number of distinct solutions generated by the simplex method for linear programming (LP). In this paper, we combine results of Kitahara and Mizuno and Tardos’s strongly polynomial algorithm, and propose an algorithm for solving a standard form LP problem. The algorithm solves polynomial number of artificial LP … Read more
We will derive two main results related to the primal simplex method for an LP on a 0-1 polytope. One of the results is that, for any 0-1 polytope and any two vertices of it, there exists an LP instance for which the simplex method finds a path between them, whose length is at most … Read more
Kitahara and Mizuno (2011a) obtained an upper bound for the number of different solutions generated by the primal simplex method with Dantzig’s (the most negative) pivoting rule. In this paper, we obtain an upper bound with any pivoting rule which chooses an entering variable whose reduced cost is negative at each iteration. The bound is … Read more
Naddef shows that the Hirsch conjecture is true for (0,1)-polytopes by proving that the diameter of any $(0,1)$-polytope in $d$-dimensional Euclidean space is at most $d$. In this short paper, we give a simple proof for the diameter. The proof is based on the number of solutions generated by the simplex method for a linear … Read more