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16
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension
 JOURNAL OF THE ACM
, 1985
"... It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplextype algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the socalled selfdual method, is analyzed. The algorithm is not started ..."
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It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplextype algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the socalled selfdual method, is analyzed. The algorithm is not started at the traditional point (1,..., but points of the form (1, e, e2,...)T, with t sufficiently small, are used. The result is better, in two respects, than those of the previous analyses. First, it is shown that the expected number of steps is bounded between two quadratic functions cl(min(m, n))' and cz(min(m, n)) ' of the smaller dimension of the problem. This should be compared with the previous two major results in the field. Borgwardt proves an upper bound of 0(n4m1'(n1') under a model that implies that the zero vector satisfies all the constraints, and also the algorithm under his consideration solves only problems from that particular subclass. Smale analyzes the selfdual algorithm starting at (1,..., He shows that for any fixed m there is a constant c(m) such the expected number of steps is less than ~(m)(lnn)&quot;'(&quot;+~); Megiddo has shown that, under Smale's model, an upper bound C(m) exists. Thus, for the first time, a polynomial upper bound with no restrictions (except for nondegeneracy) on the problem is proved, and, for the first time, a nontrivial lower bound of precisely the same order of magnitude is established. Both Borgwardt and Smale require the input vectors to be drawn from
Coalition Formation: From Software Agents to Robots
 J INTELL ROBOT SYST (2007) 50:85–118
, 2007
"... A problem that has recently attracted the attention of the research community is the autonomous formation of robot teams to perform complex multirobot tasks. The corresponding problem for software agents is also known in the multiagent community as the coalition formation problem. Numerous algorith ..."
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A problem that has recently attracted the attention of the research community is the autonomous formation of robot teams to perform complex multirobot tasks. The corresponding problem for software agents is also known in the multiagent community as the coalition formation problem. Numerous algorithms for software agent coalition formation have been provided that allow for efficient cooperation in both competitive and cooperative environments. However, despite the plethora of relevant literature on the software agent coalition formation problem, and the existence of similar problems in theoretical computer science, the multirobot coalition formation problem has not been sufficiently grounded for different tasks and task environments. In this paper, comparisons are drawn to highlight the differences between software agents and robotics, and parallel problems from theoretical computer science are identified. This paper further explores robot coalition formation in different practical robotic environments. A heuristicbased coalition formation algorithm from our previous work was extended to operate in precedence ordered cooperative environments. In order to explore coalition formation in competitive environments, the paper also studies the RACHNA system, a market based coalition formation system. Finally, the paper investigates the notion of task preemption for complex multirobot tasks in random allocation environments.
On Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems
, 2003
"... In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of ..."
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In this paper we study the distribution tails and the moments of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax 0, x 6= 0, with A 2 IR . We consider the case where A is a Gaussian random matrix. For this input model we characterise the exact decay rates of the distribution tails, we improve the existing moment estimates, and we prove various limit theorems for the cases where either n or m and n tend to in nity. Our results are of complexity theoretic interest, because interiorpoint methods and relaxation methods for the solution of Ax 0, x 6= 0 have running times that are bounded in terms of log C (A) and C (A) respectively. AMS Classi cation: primary 90C31,15A52; secondary 90C05,90C60,62H10. Key Words: condition number, random matrices, linear programming, probabilistic analysis, complexity theory.
IMPROVED ASYMPTOTIC ANALYSIS OF THE AVERAGE NUMBER OF STEPS PERFORMED BY THE SELFDUAL SIMPLEX ALGORITHM
, 1986
"... In this paper we analyze the average number of steps performed by the selfdual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem of n variables with m constraints. Smale established that for every numbe ..."
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In this paper we analyze the average number of steps performed by the selfdual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem of n variables with m constraints. Smale established that for every number of constraints m, there is a constant c(m) such that the number of pivot steps of the selfdual algorithm, p(m, n), is less than c(m)(ln n)&quot;&quot;&quot;'+&quot;. We improve upon this estimate by showing that p(m, n) is bounded by a function of m only. The symmetry of the function in m and n implies that p(m, n) is in fact bounded by a function of the smaller of m and n.
Conditioning of random conic systems under a general family of input distributions
 Found. Comput. Math., DOI
, 2009
"... Abstract. We consider the conic feasibility problem associated with the linear homogeneous system Ax ≤ 0, x 6 = 0. The complexity of iterative algorithms for solving this problem depends on a condition number C (A). When studying the typical behaviour of algorithms under stochastic input one is ther ..."
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Abstract. We consider the conic feasibility problem associated with the linear homogeneous system Ax ≤ 0, x 6 = 0. The complexity of iterative algorithms for solving this problem depends on a condition number C (A). When studying the typical behaviour of algorithms under stochastic input one is therefore naturally led to investigate the fatness of the tails of the distribution of C (A). Introducing the very general class of uniformly absolutely continuous probability models for the random matrix A, we show that the distribution tails of C (A) decrease at algebraic rates, both for the GoffinCheungCucker number CG and the Renegar number CR. The exponent that drives the decay arises naturally in the theory of uniform absolute continuity, which we also develop in this paper. In the case of CG we also discuss lower bounds on the tail probabilities and show that there exist absolutely continuous input models for which the tail decay is subalgebraic. AMS subject classifications. Primary 90C31, 15A52; secondary 90C05, 90C60, 62H10. Key words. Condition numbers, random matrices, linear programming, conic feasibility prob
Community Structures of Networks
"... Abstract. We present an approach to studying the community structures of networks by using linear programming (LP). Starting with a network in terms of (a) a collection of nodes and (b) a collection of edges connecting some of these nodes, we use a new LPbased method for simultaneously (i) nding, ..."
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Abstract. We present an approach to studying the community structures of networks by using linear programming (LP). Starting with a network in terms of (a) a collection of nodes and (b) a collection of edges connecting some of these nodes, we use a new LPbased method for simultaneously (i) nding, at minimal cost, a second edge set by deleting existing and inserting additional edges so that the network becomes a disjoint union of cliques and (ii) appropriately calibrating the costs for doing so. We provide examples that suggest that, in practice, this approach provides a surprisingly good strategy for detecting community structures in given networks.
Network Design for MultiHour Traffic Profile
"... The level of traffic and the distribution of traffic tend to vary in a telecommunications network. Obtaining a minimumcost network for multihour traffic profile is a largescale optimization problem. We formulate this as a linear programming problem and use this as a basis for comparing the effica ..."
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The level of traffic and the distribution of traffic tend to vary in a telecommunications network. Obtaining a minimumcost network for multihour traffic profile is a largescale optimization problem. We formulate this as a linear programming problem and use this as a basis for comparing the efficacy of the heuristics we introduce. Our heuristic is to view this as a multiperiod network planning problem where there is growth in aggregate traffic from period to period, as well as redistribution of traffic. Initial numerical experience indicates a difference of less than 10% between network cost obtained by this heuristic and that obtained by the largescale linear programming formulation while the heuristic model is about N&sup2_t times faster to solve where N_t is the number of time periods.
THE EXPECTED NUMBER OF EXTREME POINTS OF A RANDOM LINEAR PROGRAM
, 1986
"... There has been increasing attention recently on average case algorithmic performance measures since worst case measures can be qualitatively quite different. An important characteristic of a linear program, relating to Simplex Method performance, is the number of vertices of the feasible region. We ..."
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There has been increasing attention recently on average case algorithmic performance measures since worst case measures can be qualitatively quite different. An important characteristic of a linear program, relating to Simplex Method performance, is the number of vertices of the feasible region. We show 2 ~ to be an upper bound on the mean number of extreme points of a randomly generated feasible region with arbitrary probability distributions on the constraint matrix and right hand side vector. The only assumption made is that inequality directions are chosen independently in accordance with a series of independent fair coin tosses.
CSC2411 Linear Programming and Combinatorial Optimization ∗ Lecture 6: The Ellipsoid Method: an Introduction
, 2007
"... Summary: This lecture introduces the Ellipsoid Method, the first polynomialtime algorithm to solve LP. We start by discussing the historical significance ..."
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Summary: This lecture introduces the Ellipsoid Method, the first polynomialtime algorithm to solve LP. We start by discussing the historical significance