Generalized Max-Flows and Min-Cuts in Simplicial Complexes

We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For d-dimensional simplicial complexes embedded into ?^{d+1} we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt

Network flow algorithms and applications

This paper looks at several methods for solving network flow problems. The first chapter gives a brief background for linear programming (LP) problems. It includes basic definitions and theorems. The second chapter gives an overview of graph theory including definitions, theorems, and examples. Chapters 3-5 are the heart of this thesis. Chapter 3 includes algorithms and applications for maximum flow problems. It includes a look at a very important theorem. Maximum Flow/Minimum Cut Theorem. There is also a section on the Augmenting Path Algorithm. Chapter 4 Deals with shortest path problem. It includes Dijsksta\u27s Algorithm and the All-Pairs Labeling Algorithm. Chapter 5 includes information on algorithms and applications for the minimum cost flow(MCF)problem. The algorithms covered include the Cycle Canceling,Successive ShortestPath,and Primal-Dual Algorithms. Each of these chapters 3-5 contain definitions,theorems,and algorithms to solve network flow problems. Throughout the paper the computer program LINDO is used. It serves a couple of functions. First it is a way of checking each solution. The second use is to expose the reader to a very valuable tool in linear programming

New Auction Algorithms for the Assignment Problem and Extensions

We consider the classical linear assignment problem, and we introduce new auction algorithms for its optimal and suboptimal solution. The algorithms are founded on duality theory, and are related to ideas of competitive bidding by persons for objects and the attendant market equilibrium, which underlie real-life auction processes. We distinguish between two fundamentally different types of bidding mechanisms: aggressive and cooperative. Mathematically, aggressive bidding relies on a notion of approximate coordinate descent in dual space, an epsilon-complementary slackness condition to regulate the amount of descent approximation, and the idea of epsilon-scaling to resolve efficiently the price wars that occur naturally as multiple bidders compete for a smaller number of valuable objects. Cooperative bidding avoids price wars through detection and cooperative resolution of any competitive impasse that involves a group of persons. We discuss the relations between the aggressive and the cooperative bidding approaches, we derive new algorithms and variations that combine ideas from both of them, and we also make connections with other primal-dual methods, including the Hungarian method. Furthermore, our discussion points the way to algorithmic extensions that apply more broadly to network optimization, including shortest path, max-flow, transportation, and minimum cost flow problems with both linear and convex cost functions

Efficient algorithms for the minimum cost perfect matching problem on general graphs

Ankara : Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent Univ., 1993.Thesis (Master's) -- Bilkent University, 1993Includes bibliographical refences.The minimum cost perfect matching problem is one of the rare combinatorial optimization problems for which polynomial time algorithms exist. Matching algorithms find applications in Postman Problem, Planar Multicommodity Flow Problem, in heuristics to the well known Traveling Salesman Problem, Vehicle Scheduling Problem, Graph Partitioning Problem, Set Partitioning Problem, in VLSI, et cetera. In this thesis, reviewing the existing primal-dual approaches in the literature, we present two efficient algorithms for the minimum cost perfect matching problem on general graphs. In both of the algorithms, we achieved drastic reductions in the total number of time consuming operations such as scanning, updating dual variables and reduced costs. Detailed computational analysis on randomly generated graphs has shown the proposed algorithms to be several times faster than other algorithms in the literature. Hence, we conjecture that employment of the new algorithms in the solution methods of above stated important problems would speed them up significantly.Atamtürk, AlperM.S

Algoritmi za traženje maksimalnog toka minimalne cijene u mreži

U ovom radu proučavali smo problem maksimalnog toka minimalne cijene u mreži. Radi se o poznatom problemu u području teorije grafova, a kao što smo i pokazali, ima široku primjenu i u svakodnevnom životu. Kako bismo istražili teoretsku pozadinu ovog problema, uveli smo pojmove potencijala i reduciranih cijena. To nam je omogućilo da dokažemo nekoliko rezultata u pogledu optimalnih uvjeta za rješenje problema, a koji su nam poslužili kao osnova za dokazivanje točnosti danih algoritama. Pokazali smo da teorija maksimalnog toka minimalne cijene povezuje neke poznate rezultate iz teorije maksimalnog toka i pronalaska najkraćih puteva u mreži. Najveći dio ovog rada odnosi se na tri algoritma: algoritam s poništavanjem negativnih ciklusa, algoritam s traženjem najkraćih puteva i primal-dual algoritam. Pokazali smo nekoliko svojstava tih algoritama te analizirali njihove vremenske složenosti. Testiranjem na skupu podataka koji je uključivao mreže različitih broja vrhova i bridova potvrdili smo pretpostavku da primal-dual algoritam ima najbolje performanse od sva tri prikazana algoritma.In this work we study the minimum cost flow problem. This problem is well known in graph theory and has wide applications in everyday life. In order to explore the theoretical background of the problem, we introduced the notions of potentials and reduced costs. This enabled us to prove several results regarding optimality conditions for the problem solution, which served as a basis for showing the correctness of algorithms that solve the problem. The theory of the minimum cost flow problem combines several well known facts from the theory of the maximum flow and the shortest path problems. Large part of this work is focused on three algorithms: the cycle-canceling algorithm, the successive shortest path algorithm and the primal-dual algorithm. We presented a number of properties of these algorithms and analyzed their time complexities. Testing on a dataset that included graphs of varying sizes and densities has confirmed our assumption that the primal-dual algorithm has the best performance among the presented algorithms for solving the minimum cost flow problem

Efficient Algorithms for Geometric Partial Matching

Let A and B be two point sets in the plane of sizes r and n respectively (assume r <= n), and let k be a parameter. A matching between A and B is a family of pairs in A x B so that any point of A cup B appears in at most one pair. Given two positive integers p and q, we define the cost of matching M to be c(M) = sum_{(a, b) in M}||a-b||_p^q where ||*||_p is the L_p-norm. The geometric partial matching problem asks to find the minimum-cost size-k matching between A and B. We present efficient algorithms for geometric partial matching problem that work for any powers of L_p-norm matching objective: An exact algorithm that runs in O((n + k^2)polylog n) time, and a (1 + epsilon)-approximation algorithm that runs in O((n + k sqrt{k})polylog n * log epsilon^{-1}) time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in O(min{n^2, rn^{3/2}}polylog n) time

Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page