Abstract In 1992, Kalai and Kleitman proved that the diameter of a d -dimensional polyhedron with n facets is at most n 2 + log 2 d . In 2014, Todd improved the Kalai–Kleitman bound to ( n − d ) log 2 d . We improve the Todd bound to ( n − d ) − 1 + log 2 d for n ≥ d ≥ 7 , ( n − d ) − 2 + log 2 d for n ≥ d ≥ 37 , and ( n − d ) − 3 + log 2 d + O 1 ∕ d for n ≥ d ≥ 1 .
Another well-studied parameter would be @math which is defined as the largest absolute value of a subdeterminant of the constraint matrix @math associated to a polyhedron. @cite_6 strengthened and extended the Dyer and Frieze upper bound @cite_22 holding for totally unimodular case; i.e., when @math . Complexity analyses based on @math for the shadow vertex algorithm and the primal-simplex based Tardos' algorithm were proposed by Dadush and H " a hnle @cite_12 , and Mizuno, Sukegawa, and Deza @cite_21 @cite_7 , respectively.
@cite_22 @cite_7 @cite_21 @cite_6 @cite_12
2005598280 2132914290 1758756115 2089729462
We derive a new upper bound on the diameter of a polyhedron @math P = x ? R n : A x ≤ b , where @math A ? Z m × n . The bound is polynomial in @math n and the largest absolute value of a sub-determinant of @math A , denoted by @math Δ . More precisely, we show that the diameter of @math P is bounded by @math O ( Δ 2 n 4 log n Δ ) . If @math P is bounded, then we show that the diameter of @math P is at most @math O ( Δ 2 n 3.5 log n Δ ) . For the special case in which @math A is a totally unimodular matrix, the bounds are @math O ( n 4 log n ) and @math O ( n 3.5 log n ) respectively. This improves over the previous best bound of @math O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1---16, 1994). We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope @math along the edges of P, where A∈ℝm ×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1 δ that is a measure for the flatness of the vertices of P. For integer matrices A∈ℤm ×n we show a connection between δ and the largest absolute value Δ of any sub-determinant of A, yielding a bound of O(Δ4mn4) for the length of the computed path. This bound is expressed in the same parameter Δ as the recent non-constructive bound of O(Δ2n4 log(n Δ)) by [1]. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(mn4), which significantly improves the previously best known constructive bound of O(m16n3 log3 (mn)) by Dyer and Frieze [7]. 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 in the dimension of the input. 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 the simplex method to solve the auxiliary problems. The proposed algorithm is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate. A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1 e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1 2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1 2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1 4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1 6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k (k-1), such that any algorithm achieving a better than (1-1 nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1 2 (1-1 nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1 2 (for submodular maximization subject to a matroid constraint) and 1-1 nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal.