Multiobjective Versions of Polynomially Solvable Problems

2005 ◽  
pp. 221-270
Keyword(s):  
Polynomially Solvable ◽  
Solvable Problems

Privacy in Multidimensional Databases

2003 ◽  
pp. 310-360 ◽  
Author(s):  
Francesco M. Malvestuto ◽  
Marina Moscarini
Keyword(s):  
Sensitive Data ◽  
Multidimensional Database ◽  
Graphical Data ◽  
Multidimensional Databases ◽  
Confidential Data ◽  
Hard Problems ◽  
Polynomially Solvable ◽  
Query System ◽  
Future Work ◽  
Solvable Problems

When answering queries that ask for summary statistics, the query-system of a multidimensional database should guard confidential data, that is, it should avoid revealing (directly or indirectly) individual data, which could be exactly calculated or accurately estimated from the values of answered queries. In order to prevent the disclosure of confidential data, the query-system should be provided with an auditing procedure which, each time a new query is processed, checks that its answer does not allow a (knowledgeable) user to disclose any sensitive data. A promising approach consists in keeping track of (or auditing) answered queries by means a dynamic graphical data structure, here called the answer map, whose size increases with the number of answered queries and with the number of dimensions of the database, so that the problem of the existence of an efficient auditing procedure naturally arises. This chapter reviews recent results on this problem for “additive” queries (such as COUNT and SUM queries) by listing some polynomially solvable problems as well as some hard problems, and suggests directions for future work.

Polynomially Solvable Problems

1994 ◽  
pp. 69-185
Author(s):  
V. S. Tanaev ◽  
V. S. Gordon ◽  
Y. M. Shafransky
Keyword(s):  
Polynomially Solvable ◽  
Solvable Problems

scholarly journals The subset sum game revisited

2021 ◽  
Author(s):  
Astrid Pieterse ◽  
Gerhard J. Woeginger
Keyword(s):  
Weak Sense ◽  
Constant Factor ◽  
The Other ◽  
Total Weight ◽  
Subset Sum ◽  
Common Resource ◽  
Game Theoretic ◽  
Np Complete ◽  
Polynomially Solvable ◽  
Solvable Problems

AbstractWe discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player. We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS (unless P=NP).

scholarly journals Dispersing Obnoxious Facilities on a Graph

Algorithmica ◽  
2021 ◽  
Author(s):  
Alexander Grigoriev ◽  
Tim A. Hartmann ◽  
Stefan Lendl ◽  
Gerhard J. Woeginger
Keyword(s):  
Facility Location ◽  
Location Problem ◽  
Unit Length ◽  
Facility Location Problem ◽  
Np Hard ◽  
Obnoxious Facilities ◽  
Rational Parameter ◽  
Polynomially Solvable

AbstractWe study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

GENERALISATIONS OF THE GILMORE-GOMORY TRAVELING SALESMAN PROBLEM AND THE GILMORE-GOMORY SCHEME: A SURVEY

2001 ◽  
Vol 03 (02n03) ◽  
pp. 213-235 ◽  
Author(s):  
SANTOSH N. KABADI
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Major Portion ◽  
Polynomially Solvable ◽  
The Traveling Salesman Problem ◽  
Special Case

One of the first and perhaps the most well-known polynomially solvable special case of the traveling salesman problem (TSP) is the Gilmore-Gomory case (G-G TSP). Gilmore and Gomory presented an interesting patching algorithm for this case with a fairly non-trivial proof of its validity. Their work has motivated a great deal of research in the area leading to various generalisations of their results and thereby identification of fairly large polynomially solvable subclasses of the TSP. These results form a major portion of the literature on solvable cases of the TSP. In this paper, we survey the main results on solvable cases of the TSP which are direct generalisations of the G-G TSP and/or the Gilmore-Gomory patching scheme.

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Algebraic Solution ◽  
Explicit Solutions ◽  
Polynomial Map ◽  
Combinatorial Number Theory ◽  
Special Cases ◽  
The Matrix ◽  
Grid Points ◽  
Algebraic Solutions ◽  
Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

A Bicriteria Approach for Single Machine Scheduling with Resource Allocation, Learning Effect and a Deteriorating Maintenance Activity

2017 ◽  
Vol 34 (04) ◽  
pp. 1750011 ◽  
Author(s):  
Zhusong Liu ◽  
Zhenyou Wang ◽  
Yuan-Yuan Lu
Keyword(s):  
Resource Allocation ◽  
Single Machine ◽  
Learning Effect ◽  
Machine Scheduling ◽  
Single Machine Scheduling ◽  
Maintenance Activity ◽  
Optimal Resource ◽  
Deteriorating Maintenance ◽  
Polynomially Solvable ◽  
Completion Times

This paper considers the single machine scheduling with learning effect, resource allocation and deteriorating maintenance activity simultaneously. For the convex resource allocation consumption function, we provide a bicriteria analysis where the first (schedule) criterion is to minimize the total weighted sum of makespan, total completion time and total absolute differences in completion times, and the second (resource) criterion is to minimize the total weighted resource consumption. Our aim is to find the optimal resource allocations and job sequence that minimize the three different models of considering the two criterion. We show that these three models are polynomially solvable respectively.

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