Skew-representable matroids form a fundamental class in matroid theory, bridging combinatorics and linear algebra. They play an important role in areas such as coding theory, optimization, and combinatorial geometry, where linear structure is crucial for both theoretical insights and algorithmic applications. Since deciding skew-representability is computationally intractable, much effort has been focused on identifying necessary or sufficient conditions for a matroid to be skew-representable.

Abstract. Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they found applications in the theory of finitely additive measures, nonlinear integrals, and electric capacities. Recently, a number of connections between these two branches have been established, and the aim of this paper is to generalize further results on submodular set functions on finite sets to the analytic setting. We first extend the notion of duality of matroids to submodular set functions and characterize the uniquely determined decomposition of a submodular set function into the sum of a nonnegative charge and an increasing submodular set function in which the charge is maximal. Then, we describe basic properties of infinite-alternating set functions, a subclass of submodular set functions that serves as an analytic counterpart of coverage functions. By relaxing the monotonicity assumption in the definition, we introduce a new class of submodular functions with distinguished structural properties that includes, among others, weighted cut functions of graphs. We prove that, unlike general submodular set functions over an infinite domain, any infinite-alternating set function can be written as the sum of an increasing and a decreasing submodular function or as the difference of two increasing submodular functions, thus giving an extension of results on monotonic decompositions in the finite case. Finally, motivated by its connections to graph parameters such as the maximum size of a cut and the maximum size of a fractional triangle packing, we study the structure of such decompositions for weighted cut functions of undirected graphs.

The study of matroid products traces back to the 1970s, when Lovász and Mason studied the existence of various types of matroid products with different strengths. Among these, the tensor product is arguably the most important, which can be considered as an extension of the tensor product from linear algebra. However, Las Vergnas showed that the tensor product of two matroids does not always exist. Over the following four decades, matroid products remained surprisingly underexplored, regaining attention only in recent years due to applications in tropical geometry, information theory, and the limit theory of matroids. In this paper, inspired by the concept of coupling in probability theory, we introduce the notion of coupling for matroids – or, more generally, for submodular set functions. This operation can be viewed as a relaxation of the tensor product. Unlike the tensor product, however, we prove that a coupling always exists for any two submodular functions and can be chosen to be increasing if the original functions are increasing. As a corollary, we show that two matroids always admit a matroid coupling, leading to a novel operation on matroids. Our construction is algorithmic, providing an oracle for the coupling matroid through a polynomial number of oracle calls to the original matroids. We apply this construction to derive new necessary conditions for matroid representability and establish connection between tensor products and Ingleton’s inequality. In addition, we verify the existence of set functions that are universal with respect to a given property, meaning any set function over a finite domain with that property can be obtained as a quotient.

Motivated by applications in combinatorial optimization, we study the extent to which the global properties of a metric space, and especially its embeddability into $\ell_1$ with low distortion, are determined by the properties of its small subspaces. We establish both upper and lower bounds on the distortion of embedding locally constrained metrics into various target spaces. Other aspects of locally constrained metrics are studied as well, in particular, how far are those metrics from general metrics.

A flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge. In this article, we study single-commodity confluent flow problems, where we need to route given node demands to a single destination using a confluent flow. Single- and multi-commodity confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are (multi-commodity) confluent flows since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for minimizing congestion in single-commodity confluent flows. The maximum edge congestion of a single-commodity confluent flow occurs at one of the incoming edges of the destination. Therefore, finding a minimum-congestion confluent flow is equivalent to the following problem: given a directed graph G with k sinks and non-negative demands on all the nodes of G , determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. The main result of this article is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln( k ) in G , if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than H k , the k th harmonic number, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the congestion of an optimal confluent flow to within a factor of (log 2 k )/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand. We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph is k -connected. In particular, we prove that k -connected graphs with k sinks admit confluent flows of congestion less than C + d max , where C is the congestion of the best splittable flow, and d max is the maximum demand of any node in G . The proof of this existence theorem is non-constructive and relies on topological techniques introduced by Lovász.

We prove that the hit-and-run random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of Lovasz and Vempala (2004), where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we derive asymptotically faster algorithms in the general oracle model for sampling, rounding, integration and maximization of logconcave functions, improving or generalizing the main results of Lovasz and Vempala (2003), Applegate and Kannan (1990) and Kalai and Vempala respectively. The algorithms for integration and optimization both use sampling and are surprisingly similar

We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. As examples, we provide short proofs of the testability of MaxCut and the recent result of Alon and Shapira about the testability of hereditary graph properties.

We show that the hit-and-run random walk mixes rapidly starting from any interior point of a convex body. This is the first random walk known to have this property. In contrast, the ball walk can take exponentially many steps from some starting points. The proof extends to sampling an exponential density over a convex body.

We propose a scheme for building peer-to-peer overlay networks for broadcasting using network coding. The scheme addresses many practical issues such as scalability, robustness, constraints on bandwidth, and locality of decisions. We analyze the system theoretically and prove near optimal bounds on the parameters defining robustness and scalability. As a result we show that the effects of failures are contained locally, allowing the network to grow exponentially with server load. We also argue that adversarial failures are no more harmful than random failures.

A flow is said to be confluent if at any node all the flow leaves along a single edge. Given a directed graph G with k sinks and non-negative demands on all the nodes of G, we consider the problem of determining a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. Confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are confluent since Internet routing is destination based.We present near-tight approximation algorithms, hardness results, and existence theorems for confluent flows. The main result of this paper is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln(k) in G, if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than Hk, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the congestion of an optimal confluent flow to within a factor of (lg k)/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand.We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph were k connected. In particular, we prove that k-connected graphs with k sinks admit confluent flows of congestion less than C + dmax, where C is the congestion of the best splittable flow, and dmax is the maximum demand of any node in G. The proof of this existence theorem is non-constructive and relies on topological techniques introduced in [16].

We show that the hit-and-run random walk mixes rapidly starting from any interior point of a convex body. This is the first random walk known to have this property. In contrast, the ball walk can take exponentially many steps from some starting points.

We present a new algorithm for computing the volume of a convex body in R/sup n/. The main ingredient of the algorithm is a "morphing" technique that can be viewed as a variant of simulated annealing. Its complexity is O*(n/sup 4/), improving on the previous best algorithm by a factor of n.

The class of logconcave functions in R/sup n/ is a common generalization of Gaussians and of indicator functions of convex sets. Motivated by the problem of sampling from a logconcave density function, we study their geometry and introduce an analysis technique for "smoothing" them out. This leads to efficient sampling algorithms with no assumptions on the local smoothness of the density function. After appropriate preprocessing, both the ball walk (with a Metropolis filter) and a generalization of hit-and-run produce a point from approximately the right distribution in time O*(n/sup 4/), and in amortized time O*(n/sup 3/) if many sample points are needed (where the asterisk indicates that dependence on the error parameter and factors of log n are not shown). The bounds are optimal in terms of a "roundness" parameter and match the best-known bounds for the special case of the uniform density over a convex set.

Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a case-by-case basis. We initiate a more systematic approach. We prove an integrality gap of 2-o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.

We observe a certain random process on a graph "locally", i.e., in the neighborhood of a node, and would like to derive information about "global" properties of the graph. For example, what can we know about a graph based on observing the returns of a random walk to a given node? Our main result concerns a graph embedded in an orientable surface with genus g, and a process, consisting of random excitations of edges and random balancing around nodes and faces. It is shown how to obtain the genus of the surface in polynomial time from local observations of the process restricted to a connected subgraph whose size is (essentially) O(g/sup 2/).

We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds (P. Matthews, 1988) on the cover time, C, and prove that M/2

There are several examples where the mixing time of a Markov chain can be reduced substantially, often to about its square root, by "lifting", i.e., by splitting each state into several states.In several examples of random walks on groups, the lifted chain not only mixes better, but is easier to analyze.We characterize the best mixing time achievable through lifting in terms of multicommodity flows.We show that the reduction to square root is best possible.If the lifted chain is time-reversible, then the gain is smaller, at most a factor of log(l/na), where 110 is the smallest stationary probability of any state.We give an example showing that a gain of a factor of log(l/~o)/log log(l/rro) is possible.

The notion of conductance introduced by Jerrum and Sinclair [JS] has been widely used to prove rapid mixing of Markov chains.Here we introduce a variant of this -instead of measuring the conductance of the worst subset of states, we show that it is enough to bound a certain weighted a~wage conductance (where the average is taken over subsets of states with different sizes.)In the case of convex bodies, we show that this average conductance is better than the known bounds for the worst cage; this helps us save a factor of O(n) which is incurred in all proofs as a 'penalty" for a "bad start" (i.e., because the starting distribution may be arbitrary).We show that in a convex body in !R", with diameter D, random walk with steps in a ball with radius 6 mixes in O'(nD2/$) time (if idle steps at the boundary are not counted).This gives an O'(n3) sampling algorithm after appropriate preprocessing, improving the previous bound of O'(n').

We describe a deterministic polynomial-time algorithm which, for a convex body K in Euclidean n-space, finds upper and lower bounds on K's diameter which differ by a factor of O(/spl radic/n/logn). We show that this is, within a constant factor, the best approximation to the diameter that a polynomial-time algorithm can produce even if randomization is allowed. We also show that the above results hold for other quantities similar to the diameter-namely; inradius, circumradius, width, and maximization of the norm over K. In addition to these results for Euclidean spaces, we give tight results for the error of deterministic polynomial-time approximations of radii and norm-maxima for convex bodies in finite-dimensional l/sub p/ spaces.

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions.

A thrackte is a graph that can be drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs.About thirty years ago, J. H. Conway conjectured that the number of edges of a thrackle cannot exceed the number of its vertices.We show that a thrackle has at most twice as many edges as vert ices.Some related problems and generalizations are also considered.

Let lf be the transition matrix, and a the initial state distribution.for a discrete-time finite-state irreducible Markov chain.A stopping rule for M is an algorithm which observes the progress of the chain and then stops it at some random time r; the distribution of the final state is denoted by ar.We give a useful characterization for stopping rules which are optimal for given target distribution r, in the sense that Some of the work described herein is joint with David Aldous.

The relative power of determinism, randomness, and nondeterminism for search problems in the Boolean decision tree model is studied. It is shown that the gaffs between the nondeterministic, the randomized, and the deterministic complexities can be arbitrarily large for search problems. An interesting connection of this model to the complexity of resolution proofs is also mentioned.

Let A be an $n \times n$ matrix with 0-1 valued entries, and let ${\operatorname{per}}(A)$ be the permanent of A. This paper describes a Monte-Carlo algorithm that produces a “good in the relative sense” estimate of ${\operatorname{per}}(A)$ and has running time ${\operatorname{poly}}(n)2^{{n / 2}} $, where ${\operatorname{poly}}(n)$ denotes a function that grows polynomially with n.

The authors give an O(n/sup 7/log/sup 2/n) randomised algorithm to approximate the volume of a convex body, and an O(n/sup 6/log n) algorithm to sample a point from the uniform distribution over a convex body. For convex polytopes the algorithm runs in O(n/sup 7/log/sup 4/n) steps. Several tools are developed that may be interesting on their own. They extend results of Sinclair-Jerrum (1988) and the authors (1990) on the mixing rate of Markov chains from finite to arbitrary Markov chains. They describe an algorithm to integrate a function with respect to the stationary distribution of a general Markov chain. They also analyze the mixing rate of various random walks on convex bodies, in particular the random walk with steps from the uniform distribution over a unit ball. In several previous positive and negative results, the problem of computing the diameter of a convex body behaved similarly as the volume problem. In contrast to this, they show that there is no polynomial randomized algorithm to compute the diameter within a factor of n/sup 1/4/.>

Article Free Access Share on Linear decision trees: volume estimates and topological bounds Authors: Anders Björner Royal Institute of Technology, Stockholm, Sweden S-100 44 Royal Institute of Technology, Stockholm, Sweden S-100 44View Profile , László Lovász Eötvös Loránd University, Budapest, Hungary H-1088, Princeton University, Princeton, NJ Eötvös Loránd University, Budapest, Hungary H-1088, Princeton University, Princeton, NJView Profile , Andrew C. C. Yao Princeton University, Princeton, NJ Princeton University, Princeton, NJView Profile Authors Info & Claims STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of ComputingJuly 1992 Pages 170–177https://doi.org/10.1145/129712.129730Published:01 July 1992Publication History 41citation642DownloadsMetricsTotal Citations41Total Downloads642Last 12 Months28Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We characterize the power of two-prover one-round (MIP(2,1)) proof systems, showing that MIP(2,1)=NEXPTIME. However, the following intriguing question remains open: Does parallel repetition decrease the error probability of MIP(2,1) proof systems?.

The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P.>

The relative power of determinism, randomness, and nondeterminism for search problems in the Boolean decision tree model is studied. It is shown that the CNF search problem is complete for all the variants of decision trees. It is then shown that the gaps between the nondeterministic, the randomized, and the deterministic complexities can be arbitrarily large for search problems. The special case of nondeterministic complexity is discussed. >

A. Sinclair and M. Jerrum (1988) derived a bound on the mixing rate of time-reversible Markov chains in terms of their conductance. The authors generalize this result by not assuming time reversibility and using a weaker notion of conductance. They prove an isoperimetric inequality for subsets of a convex body. These results are combined to simplify an algorithm of M. Dyer et al. (1989) for approximating the volume of a convex body and to improve running-time bounds.>

Let S ⊂ R3 be an n-set in general position. A plane containing three of the points is called a halving plane if it dissects S into two parts of equal cardinality. It is proved that the number of halving planes is at most Ο(n2.998).

A general framework for the study of a broad class of communication problems is developed. It is based on a recent analysis of the communication complexity of graph connectivity. The approach makes use of combinatorial lattice theory.>

Linial and Saks [2] have shown that $O(\log N)$ evaluations of an order preserving map $f:p \to \mathbb{R}$ are necessary and sufficient to determine whether $\alpha \in f(P)$, where N is the number of ideals of N and $\alpha \in \mathbb{R}$ is a given real number. In this paper, we investigate the problem of how to perform the evaluations so that Linial and Saks’ bound is guaranteed, and solve the problem for the classes of interval and series-parallel orders and hence, in particular, for rooted trees. We observe that the greedy-type binary search algorithm, which is optimal for chains, already need not be optimal for general rooted trees. We furthermore discuss the computational complexity of the general search problem and obtain results indicating that the general problem might be hard.

An ear-decomposition of a digraph is a representation of it as the union of (open or closed) directed paths, each having its endpoints in common with the union of the previous paths but nothing else. We prove that finding an ear-decomposition of a strongly directed graph is in NC, i.e. an eardecomposition can be constructed in parallel in polylog time, using a polynomial number of processors. Using a similar technique, we show that the problem of finding a minimum weight spanning arborescence in an arcweighted rooted digraph is in NC.

It is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences, given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients. All algorithms work in polynomial time

It is proved that the Shannon zero-error capacity of the pentagon is\sqrt{5}. The method is then generalized to obtain upper bounds on the capacity of an arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special graphs; for example, the Petersen graph has capacity four and a self-complementary graph with n points and with a vertex-transitive automorphism group has capacity\sqrt{5}.