Small Forbidden Configurations IV: The 3 Rowed Case

The present paper continues the work begun by Anstee, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let F be a k×l (0,1)-matrix (the forbidden configuration). Small refers to the size of k and in this paper k = 3. Assume A is an m×n simple matrix which has no submatrix which is a row and column permutation of F. We define forb(m,F) as the best possible upper bound on n, for such a matrix A, which depends on m and F. We complete the classification for all 3-rowed (0,1)-matrices of forb (m,F) as either Θ(m), Θ(m²) or Θ(m³) (with constants depending on F). * Research is supported in part by NSERC. † Research was done while the second author visited the University of British Columbia supported by NSERC of first author. Research was partially supported by Hungarian National Research Fund (OTKA) numbers T034702 and T037846.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

Lower bound on the profile of degree pairs in cross-intersecting set systems

We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity. Research supported in part by the Hungarian Research Fund under grant OTKA T-032969.     

A new condition number for linear programming

In this paper we define a new condition number ?(A) for the following problem: given a m by n matrix A, find x∈ℝ ⁿ , s.t. Ax<0. We characterize this condition number in terms of distance to ill-posedness and we compare it with existing condition numbers for the same problem. Received: November 5, 1999 / Accepted: November 2000?Published online September 17, 2001     

An Approximation Scheme For Cake Division With A Linear Number Of Cuts

In the cake cutting problem, n≥2 players want to cut a cake into n pieces so that every player gets a ‘fair’ share of the cake by his own measure. We prove the following result: For every ε>0, there exists a cake division scheme for n players that uses at most c_εn cuts, and in which each player can enforce to get a share of at least (1-ε)/n of the cake according to his own private measure. * Partially supported by Institute for Theoretical Computer Science, Prague (project LN00A056 of MŠMT ČR) and grant IAA1019401 of GA AV ČR.     

An O(n 2) active set algorithm for the solution of a parametric quadratic program

In this paper, an O(n ²) active set method is presented for minimizing the parametric quadratic function (1/2)x′Dx-a′x + λmax(c - γ x,0) subject to l ⩽ x ⩽ b, for all nonnegative values of the parameter γ. Here, D is a positive diagonal n x n matrix, a and γ are arbitrary N-vectors, c is an arbitrary scalar, l and b are arbitrary n-vectors, such thatl ⩽ b. An extension of this algorithm is presented for minimizing the parametric function (1/2)x′Dx-a ^′ x + λ |γ′x - c| subject to l ⩽ x ⩽ b. It is also shown that these problems arise naturally in a tax programming problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

The monotone circuit complexity of boolean functions

Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(m ^s /(logm)^2s ) for fixeds, and sizem ^Ω(logm) form/4]. In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)^2/3 requires monotone circuits exp (Ω((m/logm)^1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm) ^s ) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n ^1/4 · (logn)^1/2)), improving a recent result of exp (Ω(n ^1/8-ε)) due to Andreev. First author supported in part by Allon Fellowship, by Bat Sheva de-Rotschild Foundation by the Fund for basic research administered by the Israel Academy of Sciences. Second author supported in part by a National Science Foundation Graduate Fellowship.     

On The Structure Of Triangle-Free Graphs Of Large Minimum Degree

It is shown that for every ε>0 there exists a constant L such that every triangle-free graph on n vertices with minimum degree at least (1/3+ε)n is homomorphic to a triangle-free graph on at most L vertices. * Research partially supported by KBN grant 2 P03A 016 23.     

A new polynomial-time algorithm for linear programming

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n ^3.5 L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n ^2.5). We prove that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time. This is a substantially revised version of the paper presented at the Symposium on Theory of Computing, Washington D. C., April 1984.     

Bounds for arrays of dots with distinct slopes or lengths

Ann×m sonar sequence is a subset of then×m grid with exactly one point in each column, such that the vectors determined by them are all distinct. We show that for fixedn the maximalm for which a sonar sequence exists satisfiesn–Cn ^11/20<m<n+4n ^2/3 for alln andm>n+c logn log logn for infinitely manyn.Another problem concerns the maximal numberD of points that can be selected from then×m grid so that all the vectors have slopes. We proven ^1/2Dn ^4/5 Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901Research conducted by Herbert Taylor was sponsored in part by the Office of Naval Research under ONR Contract No. N00014-90-J-1341.     

Convex Minimization over the Fixed Point Set of Demicontractive Mappings

This paper deals with a viscosity iteration method, in a real Hilbert space , for minimizing a convex function over the fixed point set of , a mapping in the class of demicontractive operators, including the classes of quasi-nonexpansive and strictly pseudocontractive operators. The considered algorithm is written as: x _n+1 := (1 − w) v _n + w T v _n , v _n := x _n − α _n Θ′(x _n ), where w ∈ (0,1) and , Θ′ is the Gateaux derivative of Θ. Under classical conditions on T, Θ, Θ′ and the parameters, we prove that the sequence (x _n ) generated, with an arbitrary , by this scheme converges strongly to some element in Argmin _Fix(T) Θ.      

On The Span Of A Random Channel Assignment Problem

In the radio channel assignment problems considered here, we must assign a ‘channel’ from the set 1,2,... of positive integers to each of n transmitters, and we wish to minimise the span of channels used, subject to the assignment leading to an acceptable level of interference. A standard form of this problem is the ‘constraint matrix’ model. The simplest case of this model (the 0, 1 case) is essentially graph colouring. We consider here a random model for the next simplest case (with lengths 0, 1 or 2), and determine the asymptotic behaviour of the span of channels needed as n→∞. We find that there is a ‘phase change’ in this behaviour, depending on the probabilities for the different lengths.     

Adaptive policies for time-varying stochastic systems under discounted criterion

We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to a discrete-time equation x _{n + 1}=G _n (x _n , a _n , ξ_n), n=0, 1, … , where the ξ_n are i.i.d. ℜ^k-valued random vectors whose common density is unknown, and the G _n are given functions converging, in a restricted way, to some function G _∞ as n→∞. Assuming observability of ξ_n, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system x _n+1=G _∞ (x _n , a _n , ξ_n).     

Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

This paper presents two fast cycle canceling algorithms for the submodular ow problem. The rst uses an assignment problem whose optimal solution identies most negative node-disjoint cycles in an auxiliary network. Canceling these cycles lexicographically makes it possible to obtain an optimal submodular ow in O(n ⁴ h log(nC)) time, which almost matches the current fastest weakly polynomial time for submodular flow (where n is the number of nodes, h is the time for computing an exchange capacity, and C is the maximum absolute value of arc costs). The second algorithm generalizes Goldbergs cycle canceling algorithm for min cost flow to submodular flow to also get a running time of O(n ⁴ h log(nC)).. We show how to modify these algorithms to make them strongly polynomial, with running times of O(n ⁶ h log n), which matches the fastest strongly polynomial time bound for submodular flow. We also show how to extend both algorithms to solve submodular flow with separable convex objectives. * An extended abstract of a preliminary version of part of this paper appeared in [22]. Research supported in part by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan. Research supported by an NSERC Operating Grant. Part of this research was done during a sabbatical leave at Cornell SORIE.§ Research supported in part by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.     

The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization

We show the existence of a sequence (λ _n ) of scalars withλ _n =o(n) such that, for any symmetric compact convex bodyB ⊂R ⁿ , there is an affine transformationT satisfyingQ ⊂T(B) ⊂λ _n Q, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλ _n . We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l ₂ ^m →X andβ:X→l _∞ ^m with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum. Research performed while this author was visiting IHES. Supported in part by the NSF Grant DMS-8702058 and the Sloan Research Fellowship.     

On the number of edge disjoint cliques in graphs of given size

In this paper, we prove that any graph ofn vertices andt _r–1(n)+m edges, wheret _r–1(n) is the Turán number, contains (1–o(1)m edge disjointK _r'sifm=o(n ²). Furthermore, we determine the maximumm such that every graph ofn vertices andt _r–1(n)+m edges containsm edge disjointK _r's ifn is sufficiently large.Research partially supported by Hungarian National Foundation for Scientific Research Grant no. 1812.     

P 3-factorization of complete multipartite graphs

In this note it is shown that a necessary and sufficient condition for the existence of a P3-factorizatlon of complete multipartite graph λK, is (1) m≥3, (2) mn≡0(mod 3) and (3)λ(m-1)n≡0(mod 4).     

On the separation of maximally violated mod-k cuts

Separation is of fundamental importance in cutting-plane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of well-structured cuts. In this paper we address the separation of Chvátal rank-1 inequalities in the context of general ILP’s of the form min{c ^T x:Ax≤b,x integer}, where A is an m×n integer matrix and b an m-dimensional integer vector. In particular, for any given integer k we study mod-k cuts of the form λ ^T Ax≤⌊λ ^T b⌋ for any λ∈{0,1/k,...,(k−1)/k} ^m such that λ ^T A is integer. Following the line of research recently proposed for mod-2 cuts by Applegate, Bixby, Chvátal and Cook [1] and Fleischer and Tardos [19], we restrict to maximally violated cuts, i.e., to inequalities which are violated by (k−1)/k by the given fractional point. We show that, for any given k, such a separation requires O(mn min{m,n}) time. Applications to both the symmetric and asymmetric TSP are discussed. In particular, for any given k, we propose an O(|V|²|E ^*|)-time exact separation algorithm for mod-k cuts which are maximally violated by a given fractional (symmetric or asymmetric) TSP solution with support graph G ^*=(V,E ^*). This implies that we can identify a maximally violated cut for the symmetric TSP whenever a maximally violated (extended) comb inequality exists. Finally, facet-defining mod-k cuts for the symmetric and asymmetric TSP are studied. Received May 29, 1997 / Revised version received May 10, 1999?Published online November 9, 1999     

The structure of area-minimizing double bubbles

We show that the least area required to enclose two volumes in ℝⁿ orS ⁿ forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝⁿ have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝⁿ. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝⁿ, based on an idea due to Brian White. Supported in part by NSF DMS-9409166.     

Families of finite sets with three intersections

LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n ²), and show that in some casesm(n, k, L)=O(n ^3/2), which is quite surprising.