Klaus Jansen, Marek Karpinski, Andrzej Lingas

The Max-Bisection and Min-Bisection are the problems of finding

partitions of the vertices of a given graph into two equal size subsets so as

to maximize or minimize, respectively, the number of edges with exactly one

endpoint in each subset.

In this paper we design the first ...
more >>>

Gerhard J. Woeginger

We derive results of the following flavor:

If a combinatorial optimization problem can be formulated via a dynamic

program of a certain structure and if the involved cost and transition

functions satisfy certain arithmetical and structural conditions, then

the optimization problem automatically possesses a fully polynomial time

approximation scheme (FPTAS).

Stasys Jukna

We consider so-called ``incremental'' dynamic programming (DP) algorithms, and are interested in the number of subproblems produced by them. The standard DP algorithm for the n-dimensional Knapsack problem is incremental, and produces nK subproblems, where K is the capacity of the knapsack. We show that any incremental algorithm for this ... more >>>

Stasys Jukna

Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In this paper we present some lower ... more >>>

Stasys Jukna, Georg Schnitger

We prove a general lower bound on the size of branching programs over any semiring of zero characteristic, including the (min,+) semiring. Using it, we show that the classical dynamic programming algorithm of Bellman, Ford and Moore for the shortest s-t path problem is optimal, if only Min and Sum ... more >>>

Stasys Jukna

Many dynamic programming algorithms for discrete optimization problems are "pure" in that they only use min/max and addition operations in their recursions. Some of them, in particular those for various shortest path problems, are even "incremental" in that one of the inputs to the addition operations is a variable. We ... more >>>

Stasys Jukna

We consider probabilistic circuits working over the real numbers, and using arbitrary semialgebraic functions of bounded description complexity as gates. We show that such circuits can be simulated by deterministic circuits with an only polynomial blowup in size. An algorithmic consequence is that randomization cannot substantially speed up dynamic programming. ... more >>>

Stasys Jukna, Hannes Seiwert

We develop general lower bound arguments for approximating tropical

(min,+) and (max,+) circuits, and use them to prove the

first non-trivial, even super-polynomial, lower bounds on the size

of such circuits approximating some explicit optimization

problems. In particular, these bounds show that the approximation

powers of pure dynamic programming algorithms ...
more >>>

Stasys Jukna, Hannes Seiwert

Many discrete minimization problems, including various versions of the shortest path problem, can be efficiently solved by dynamic programming (DP) algorithms that are ``pure'' in that they only perform basic operations, as $\min$, $\max$, $+$, but no conditional branchings via if-then-else in their recursion equations. It is known that any ... more >>>

Stasys Jukna, Hannes Seiwert, Igor Sergeev

It is known that the size of monotone arithmetic $(+,\ast)$ circuits can be exponentially decreased by allowing just one division "at the very end," at the output gate. A natural question is: can the size of $(+,\ast)$ circuits be substantially reduced if we allow divisions "at the very beginning," that ... more >>>