Abstract.To determine if a graph has a spanning tree with few leaves is NP-hard as HAMIL-TONIAN PATH is a special case. In this paper we study the parametric dual of this problem, k-INTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time. We also give a 2-approximation algorithm for the problem.However, the main contribution of this paper is that we show the following remarkable structural bindings between k-INTERNAL SPANNING TREE and k-VERTEX COVER:• NO for k-VERTEX COVER implies YES for k-INTERNAL SPANNING TREE.• YES for k-VERTEX COVER implies NO for (2k + 1)-INTERNAL SPANNING TREE.We give a polynomial-time algorithm that produces either a vertex cover of size k or a spanning tree with at least k internal vertices. We show how to use this inherent vertex cover structure to design algorithms for FPT problems, here illustrated mainly by k-INTERNAL SPANNING TREE. We also briefly discuss the application of this vertex cover methodology to the parametric dual of the DOMINATING SET problem. This design technique seems to apply to many other FPT problems.

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Therapies consisting of a combination of agents are an attractive proposition, especially in the context of diseases such as cancer, which can manifest with a variety of tumor types in a single case. However uncovering usable drug combinations is expensive both financially and temporally. By employing computational methods to identify candidate combinations with a greater likelihood of success we can avoid these problems, even when the amount of data is prohibitively large. Hitting Set is a combinatorial problem that has useful application across many fields, however as it is NP-complete it is traditionally considered hard to solve exactly. We introduce a more general version of the problem (α,β,d)-Hitting Set, which allows more precise control over how and what the hitting set targets. Employing the framework of Parameterized Complexity we show that despite being NP-complete, the (α,β,d)-Hitting Set problem is fixed-parameter tractable with a kernel of size O(αdkd) when we parameterize by the size k of the hitting set and the maximum number α of the minimum number of hits, and taking the maximum degree d of the target sets as a constant. We demonstrate the application of this problem to multiple drug selection for cancer therapy, showing the flexibility of the problem in tailoring such drug sets. The fixed-parameter tractability result indicates that for low values of the parameters the problem can be solved quickly using exact methods. We also demonstrate that the problem is indeed practical, with computation times on the order of 5 seconds, as compared to previous Hitting Set applications using the same dataset which exhibited times on the order of 1 day, even with relatively relaxed notions for what constitutes a low value for the parameters. Furthermore the existence of a kernelization for (α,β,d)-Hitting Set indicates that the problem is readily scalable to large datasets.