We show for a broad class of counting problems, correlation decay (strong spatial mixing) implies FPTAS on planar graphs. The framework for the counting problems considered by us is the Holant problems with arbitrary constant-size domain and symmetric constraint functions. We define a notion of regularity on the constraint functions, which covers a wide range of natural and important counting p...

Through an industrial application, we were confronted with the planning of experiments where human intervention of a chemists is required to handle the starting and termination. This gives rise to a new type of scheduling problems, namely problems of finding schedules with time periods when the tasks can neither start nor finish. We consider in this paper the natural case of small periods where...

Discrete gate sizing is a critical optimization in VLSI circuit design. Given a set of available gate sizes, discrete gate sizing problem asks to assign a size to each gate such that the delay of a combinational circuit is minimized while the cost constraint is satisfied. It is one of the most studied problems in VLSI computer-aided design. Despite this, all of the existing techniques are heuri...

In this paper, we consider the problem (denoted as EMDRT) of minimizing the earth mover’s distance between two sets of weighted points A and B in a fixed dimensional Rd space under rigid transformation. EMDRT is an important problem in both theory and applications and has received considerable attentions in recent years. Previous research on this problem has resulted in only constant approximat...

We study two related problems in non-preemptive scheduling and packing of malleable tasks with precedence constraints to minimize the makespan. We distinguish the scheduling variant, in which we allow the free choice of processors, and the packing variant, in which a task must be assigned to a contiguous subset of processors. For precedence constraints of bounded width, we completely resolve th...

We prove that computing a geometric minimum-dilation graph on a given set of points in the plane, using not more than a given number of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. We also show that the problem remains NP-hard even when a minimum-dilation tour or path is sought; not even an FPTAS exists in this case.

In the previous lecture we covered polynomial time reductions and approximation algorithms for vertex cover and set cover problems. By reductions we showed that SAT, 3SAT, Independent Set, Vertex Cover, Integer Programming, and Clique problems are NP-Hard. In this lecture we will continue to cover approximation algorithms for maximum coverage and metric TSP problems. We will also cover Strong N...

We present PTASes for the disk cover problem: given geometric objects and a finite set of disk centers, minimize the total cost for covering those objects with disks under a polynomial cost function on the disks’ radii. We describe the first FPTAS for covering a line segment when the disk centers form a discrete set, and a PTAS for when a set of geometric objects, described by polynomial algebr...

Recently, Jain, Mahdian and Saberi [5] had given a FPTAS for the problem of computing a market equilibrium in the Arrow-Debreu setting, when the utilities are linear functions. Their running time depended on the size of the numbers representing the utilities and endowments of the buyers. In this paper, we give a strongly polynomial time approximation scheme for this problem. Our algorithm build...

We present a polynomial time approximation scheme for the real-time scheduling problem with fixed priorities when resource augmentation is allowed. For a fixed ε > 0, our algorithm computes an assignment using at most (1+ε)·OPT+1 processors in polynomial time, which is feasible if the processors have speed 1 + ε. We also show that, unless P = NP , there does not exist an asymptotic FPTAS for th...