Closing the Gap for Pseudo-Polynomial Strip Packing

Two-dimensional packing problems are a fundamental class of optimization problems and Strip Packing is one of the most natural and famous among them. Indeed it can be defined in just one sentence: Given a set of rectangular axis parallel items and a strip with bounded width and infinite height, the objective is to find a packing of the items into the strip minimizing the packing height. We speak of pseudo-polynomial Strip Packing if we consider algorithms with pseudo-polynomial running time with respect to the width of the strip. It is known that there is no pseudo-polynomial time algorithm for Strip Packing with a ratio better than 5/4 unless P = NP. The best algorithm so far has a ratio of 4/3 + epsilon. In this paper, we close the gap between inapproximability result and currently known algorithms by presenting an algorithm with approximation ratio 5/4 + epsilon. The algorithm relies on a new structural result which is the main accomplishment of this paper. It states that each optimal solution can be transformed with bounded loss in the objective such that it has one of a polynomial number of different forms thus making the problem tractable by standard techniques, i.e., dynamic programming. To show the conceptual strength of the approach, we extend our result to other problems as well, e.g., Strip Packing with 90 degree rotations and Contiguous Moldable Task Scheduling, and present algorithms with approximation ratio 5/4 + epsilon for these problems as well

On Integer Programming, Discrepancy, and Convolution

Integer programs with a constant number of constraints are solvable in pseudo-polynomial time. We give a new algorithm with a better pseudo-polynomial running time than previous results. Moreover, we establish a strong connection to the problem (min, +)-convolution. (min, +)-convolution has a trivial quadratic time algorithm and it has been conjectured that this cannot be improved significantly. We show that further improvements to our pseudo-polynomial algorithm for any fixed number of constraints are equivalent to improvements for (min, +)-convolution. This is a strong evidence that our algorithm's running time is the best possible. We also present a faster specialized algorithm for testing feasibility of an integer program with few constraints and for this we also give a tight lower bound, which is based on the SETH.Comment: A preliminary version appeared in the proceedings of ITCS 201

The interconnection problem

AbstractIn this paper the problem of interconnecting circuit modules in microprocessor and digital system design is studied. The data transfers are expressed by m sets Ti of directed edges between modules. An interconnecting schema, which is given by an assignment of the data transfers to buses, consists of the links between modules and buses. At first we show, that the problem of finding an assignment with minimum number of links is NP-complete. After that we prove that the problem of using a given interconnection schema is NP-complete, too

A (3/2+ɛ) approximation algorithm for scheduling malleable and non-malleable parallel tasks

In this paper we study a scheduling problem with malleable and non-malleable parallel tasks on $m$ processors. A non-malleable parallel task is one that runs in parallel on a specific given number of processors. The goal is to find a non-preemptive schedule on the $m$ processors which minimizes the makespan, or the latest task completion time. The previous best result is the list scheduling algorithm with an absolute approximation ratio of $2$. On the other hand, there does not exist an approximation algorithm for scheduling non-malleable parallel tasks with ratio smaller than $1.5$, unless $P=NP$. In this paper we show that a schedule with length $(1.5 +\epsilon) OPT$ can be computed for the scheduling problem in time $O(n \log n) + f(1/\epsilon)$. Furthermore we present an $(1.5 + \epsilon)$ approximation algorithm for scheduling malleable parallel tasks. Finally, we show how to extend our algorithms to the variant with additional release dates

A fast approximation scheme for the multiple knapsack problem

In this paper we propose an improved efficient approximation scheme for the multiple knapsack problem (MKP). Given a set $A$ of $n$ items and set $B$ of $m$ bins with possibly different capacities, the goal is to find a subset $A'/ \subseteq A$ of maximum total profit that can be packed into $B$ without exceeding the capacities of the bins. Kellerer gave a PTAS for MKP with identical capacities and Chekuri and Khanna presented a PTAS for MKP with arbitrary capacities with running time $n^{O(1/ \epsilon^8 \log(1/\epsilon))}$. Recently we found an EPTAS for MKP with running time $2^{O(1/\epsilon^5 \log(1/\epsilon))} poly(n)$. Here we present an improved EPTAS with running time $2^{O(1/\epsilon \log(1/\epsilon)^4)} poly(n)$. If the modified round-up property for bin packing with different sizes is true, the running time can be improved to $2^{O(1/\epsilon \log(1/\epsilon)^2)} poly(n)$