We prove that every digraph of circumference l has DAG-width at most l and this is best possible. As a consequence of our result we deduce that the k-linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in [2]. We also prove that the weak k-linkage problem (where we ask for arc-disjoint paths) is polynomial...

Unknown relation between P and NP [3] complexity classes remains to be one of significant non solved problems in complexity theory. P complexity class consists of problems solvable by Deterministic Turing Machine (DTM) in polynomially bounded time, while NP complexity class consists of problem solvable by Non Deterministic Turing Machine (NDTM) in polynomially bounded time. This means that DTM ...

The genus g(G) of a graph G is the smallest number g such that G can be embedded on the orientable surface of genus g. Given a graph G and a natural number k one may ask: Is g(G) I k? This problem, called the graph genus problem, is one of the remaining basic open problems, listed by Garey and Johnson 121, for which there is neither a polynomially bounded algorithm nor a proof that the problem ...

Proof (Continued from the previous class) We have already shown in the previous class that the algorithm halts after polynomially many executions of the while-loop as the index i can decrease for at most O(n logA) times in step 6. In order to complete the proof, we also need to show that the size of the numerator and denominator of any rational number involved in the computation is polynomially...

We prove that for any definable subset $X\subset \mathbb {R}^{n}$ in a polynomially bounded o-minimal structure, with ${\rm dim}(X) \lt n$, there is finite set of regular projections (in the sense Mostowski). also give weak version this theor

In this paper we give an elementary proof of the unique, global-in-time solvability of the coagulation-(multiple) fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate. The proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument.

We prove that polynomial size discrete Hoppeld networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly, i.e...

A logic program with function symbols is called finitely ground if there is a finite propositional logic program whose stable models are exactly the same as the stable models of this program. Finite groundability is an important property for logic programs with function symbols because it makes feasible to compute such program’s stable models using traditional ASP solvers. In this paper, we int...

As the title suggests, this brief note is a follow-up to [5] (my first published paper), which the reader is assumed to have at hand. I make more readily available some results from my thesis [6, Chapter IV] that generalize some of the main results from [5], the latter being written just before the technology became available for proving more general results. Though I think these extensions are...