M-Convex Function Minimization Under L1-Distance Constraint and Its Application to Dock Reallocation in Bike-Sharing System

2021 ◽  
Author(s):  
Akiyoshi Shioura
Keyword(s):  
Convex Function ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Steepest Descent ◽  
Nonlinear Integer Programming ◽  
Distance Constraint ◽  
Polynomial Time Algorithms ◽  
Bike Sharing ◽  
Pseudo Polynomial Time

In this paper, we consider a problem of minimizing an M-convex function under an L1-distance constraint (MML1); the constraint is given by an upper bound for L1-distance between a feasible solution and a given “center.” This is motivated by a nonlinear integer programming problem for reallocation of dock capacity in a bike-sharing system discussed by Freund et al. (2017). The main aim of this paper is to better understand the combinatorial structure of the dock reallocation problem through the connection with M-convexity and show its polynomial-time solvability using this connection. For this, we first show that the dock reallocation problem and its generalizations can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on the steepest descent approach. We also propose two polynomial-time algorithms for (MML1) by replacing the L1-distance constraint with a simple linear constraint. Finally, we apply the results for (MML1) to the dock reallocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. For this purpose, we develop a polynomial-time algorithm for a relaxation of the dock reallocation problem by using a proximity-scaling approach, which is of interest in its own right.

scholarly journals Order Statistics for Probabilistic Graphical Models

2017 ◽  
Author(s):  
David Smith ◽  
Sara Rouhani ◽  
Vibhav Gogate
Keyword(s):  
Order Statistics ◽  
Graphical Models ◽  
Polynomial Time ◽  
Graphical Model ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Partition Problem ◽  
Polynomial Time Approximation Algorithm ◽  
Sampling Algorithms ◽  
Pseudo Polynomial Time

We consider the problem of computing r-th order statistics, namely finding an assignment having rank r in a probabilistic graphical model. We show that the problem is NP-hard even when the graphical model has no edges (zero-treewidth models) via a reduction from the partition problem. We use this reduction, specifically a pseudo-polynomial time algorithm for number partitioning to yield a pseudo-polynomial time approximation algorithm for solving the r-th order statistics problem in zero- treewidth models. We then extend this algorithm to arbitrary graphical models by generalizing it to tree decompositions, and demonstrate via experimental evaluation on various datasets that our proposed algorithm is more accurate than sampling algorithms.

scholarly journals Some results on stable sets for k-colorable P₆-free graphs and generalizations

2012 ◽  
Vol Vol. 14 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Stable Set ◽  
Stable Sets ◽  
Polynomial Time Algorithms ◽  
Complexity Status ◽  
International Audience ◽  
Hard Problems ◽  
Structure Properties

Graphs and Algorithms International audience This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).

scholarly journals On Efficiently Explaining Graph-Based Classifiers

2021 ◽  
Author(s):  
Xuanxiang Huang ◽  
Yacine Izza ◽  
Alexey Ignatiev ◽  
Joao Marques-Silva
Keyword(s):  
Decision Trees ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Graph Representation ◽  
Efficient Computation ◽  
Binary Decision ◽  
Polynomial Time Algorithms ◽  
Wide Range ◽  
Novel Algorithms

Recent work has shown that not only decision trees (DTs) may not be interpretable but also proposed a polynomial-time algorithm for computing one PI-explanation of a DT. This paper shows that for a wide range of classifiers, globally referred to as decision graphs, and which include decision trees and binary decision diagrams, but also their multi-valued variants, there exist polynomial-time algorithms for computing one PI-explanation. In addition, the paper also proposes a polynomial-time algorithm for computing one contrastive explanation. These novel algorithms build on explanation graphs (XpG's). XpG's denote a graph representation that enables both theoretical and practically efficient computation of explanations for decision graphs. Furthermore, the paper proposes a practically efficient solution for the enumeration of explanations, and studies the complexity of deciding whether a given feature is included in some explanation. For the concrete case of decision trees, the paper shows that the set of all contrastive explanations can be enumerated in polynomial time. Finally, the experimental results validate the practical applicability of the algorithms proposed in the paper on a wide range of publicly available benchmarks.

MULTI-TERMINAL NETWORK CONNECTEDNESS ON SERIES-PARALLEL NETWORKS

2009 ◽  
Vol 01 (02) ◽  
pp. 253-265 ◽  
Author(s):  
TONI R. FARLEY ◽  
CHARLES J. COLBOURN
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Polynomial Time Algorithms ◽  
Parallel Networks ◽  
Network Connectedness ◽  
Network Operation ◽  
Terminal Reliability ◽  
Special Case ◽  
General Networks

Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.

scholarly journals A Framework for Sampling-Based XML Data Pricing

2017 ◽  
Author(s):  
Ruiming Tang ◽  
Antoine Amarilli ◽  
Pierre Senellart ◽  
Stéphane Bressan
Keyword(s):  
Data Quality ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Uniform Sampling ◽  
First Case ◽  
Sampling Problem ◽  
Uniform Random Sampling ◽  
Data Consumer ◽  
Pseudo Polynomial Time

While price and data quality should define the major tradeoff for consumers in data markets, prices are usually prescribed by vendors and data quality is not negotiable. In this paper we study a model where data quality can be traded for a discount. We focus on the case of XML documents and consider completeness as the quality dimension. In our setting, the data provider offers an XML document, and sets both the price of the document and a weight to each node of the document, depending on its potential worth. The data consumer proposes a price. If the proposed price is lower than that of the entire document, then the data consumer receives a sample, i.e., a random rooted subtree of the document whose selection depends on the discounted price and the weight of nodes. By requesting several samples, the data consumer can iteratively explore the data in the document.We present a pseudo-polynomial time algorithm to select a rooted subtree with prescribed weight uniformly at random, but show that this problem is unfortunately intractable. Yet, we are able to identify several practical cases where our algorithm runs in polynomial time. The first case is uniform random sampling of a rooted subtree with prescribed size rather than weights; the second case restricts to binary weights.As a more challenging scenario for the sampling problem, we also study the uniform sampling of a rooted subtree of prescribed weight and prescribed height. We adapt our pseudo-polynomial time algorithm to this setting and identify tractable cases.

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