We consider the minimum spanning tree problem with conflict constraints (MSTC). It is observed that computing an $\epsilon$-optimal solution to MSTC is NP-hard for any $\epsilon >0$. For a general conflict graph, computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded … Read more
We address optimization of nonlinear functions of the form $f(Wx)$~, where $f:\R^d\rightarrow \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $\calF$ of integer points in $\R^n$~. Generally, such problems are intractable, so we obtain positive algorithmic results by looking at broad natural classes … Read more
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is … Read more