Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted $$l_\infty $$ l ∞ Norm

Abstract

The inverse max + sum spanning tree (IMSST) problem is studied, which is the first inverse problem on optimization problems with combined minmax---minsum objective functions. Given an edge-weighted undirected network $$G(V,E,c,w)$$ G ( V , E , c , w ) , the MSST problem is to find a spanning tree $$T$$ T which minimizes the combined weight $$\max _{e\in T}w(e)+\sum _{e\in T}c(e)$$ max e ? T w ( e ) + ? e ? T c ( e ) , which can be solved in $$O(m\log n)$$ O ( m log n ) time, where $$m:=|E|$$ m : = | E | and $$n:=|V|$$ n : = | V | . Whereas, in an IMSST problem, a spanning tree $$T_0$$ T 0 of $$G$$ G is given, which is not an optimal MSST. A new sum-cost vector $$\bar{c}$$ c ¯ is to be identified so that $$T_0$$ T 0 becomes an optimal MSST of the network $$G(V,E,\bar{c},w)$$ G ( V , E , c ¯ , w ) , where $$0\le c-l\le \bar{c} \le c+u$$ 0 ≤ c - l ≤ c ¯ ≤ c + u and $$l,u\ge 0$$ l , u ? 0 . The objective is to minimize the cost $$\max _{e\in E}q(e)|\bar{c}(e)-c(e)|$$ max e ? E q ( e ) | c ¯ ( e ) - c ( e ) | incurred by modifying the sum-cost vector $$c$$ c under weighted $$l_\infty $$ l ? norm, where $$q(e)\ge 1$$ q ( e ) ? 1 . We show that the unbounded IMSST problem is a linear fractional combinatorial optimization (LFCO) problem and develop a discrete type Newton method to solve it. Furthermore, we prove an $$O(m)$$ O ( m ) bound on the number of iterations, although most LFCO problems can be solved in $$O(m^2 \log m)$$ O ( m 2 log m ) iterations. Therefore, both the unbounded and bounded IMSST problems can be solved by solving $$O(m)$$ O ( m ) MSST problems. Computational results show that the algorithms can efficiently solve the problems.