arXiv subjects

Leslie Ann Goldberg

Publications and source records attributed to Leslie Ann Goldberg.

At least 91 records · Page 5Linked to original sources

arXiv2012

The Complexity of Approximately Counting Stable Roommate Assignments

We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the $k$-attribute model, in which the preference lists are determined by dot products of "preference vectors" with "attribute vectors" and (ii) the $k$-Euclidean model, in which the preference lists are determined by the closeness of the "positions" of the people to their "preferred positions". Exactly counting the number of assignments is #P-complete, since Irving and Leather demonstrated #P-completeness for the special case of the stable marriage problem. We show that counting the number of stable roommate assignments in the $k$-attribute model ($k \geq 4$) and the 3-Euclidean model($k \geq 3$) is interreducible, in an approximation-preserving sense, with counting independent sets (of all sizes) (#IS) in a graph, or counting the number of satisfying assignments of a Boolean formula (#SAT). This means that there can be no FPRAS for any of these problems unless NP=RP. As a consequence, we infer that there is no FPRAS for counting stable roommate assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we give an approximation-preserving reduction from counting the number of independent sets in a bipartite graph (#BIS) to counting the number of stable roommate assignments both in the 3-attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class $#RHΠ_1$. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an FPRAS for all of $#RHΠ_1$. We also show that the 1-attribute stable roommate problem always has either one or two stable roommate assignments, so the number of assignments can be determined exactly in polynomial time.

cs.CC↗

arXiv2011

The complexity of weighted and unweighted #CSP

We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that a recent dichotomy for unweighted #CSP can be extended to rational-weighted #CSP.

cs.CC↗

arXiv2011

Adaptive Drift Analysis

We show that, for any c>0, the (1+1) evolutionary algorithm using an arbitrary mutation rate p_n = c/n finds the optimum of a linear objective function over bit strings of length n in expected time Theta(n log n). Previously, this was only known for c at most 1. Since previous work also shows that universal drift functions cannot exist for c larger than a certain constant, we instead define drift functions which depend crucially on the relevant objective functions (and also on c itself). Using these carefully-constructed drift functions, we prove that the expected optimisation time is Theta(n log n). By giving an alternative proof of the multiplicative drift theorem, we also show that our optimisation-time bound holds with high probability.

cs.DS↗

arXiv2011

A Counterexample to rapid mixing of the Ge-Stefankovic Process

Ge and Stefankovic have recently introduced a novel two-variable graph polynomial. When specialised to a bipartite graphs G and evaluated at the point (1/2,1) this polynomial gives the number of independent sets in the graph. Inspired by this polynomial, they also introduced a Markov chain which, if rapidly mixing, would provide an efficient sampling procedure for independent sets in G. This sampling procedure in turn would imply the existence of efficient approximation algorithms for a number of significant counting problems whose complexity is so far unresolved. The proposed Markov chain is promising, in the sense that it overcomes the most obvious barrier to mixing. However, we show here, by exhibiting a sequence of counterexamples, that the mixing time of their Markov chain is exponential in the size of the input when the input is chosen from a particular infinite family of bipartite graphs.

math.PR↗

arXiv2011

The Complexity of Approximating Bounded-Degree Boolean \sharp CSP

The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

cs.CC↗

arXiv2011

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G;x,y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x,y) is on the hyperbola H_q given by (x-1)(y-1)=q for q=1 or q=2 or if (x,y) is one of the two special points (x,y)=(-1,-1) or (x,y)=(1,1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G;x,y), in the usual sense of "fully polynomial randomised approximation scheme" or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x,y) plane. In particular, there is no FPRAS if x>1, y<-1 or if y>1, x<-1 or if x<0, y<0 and q>5. Also, there is no FPRAS if x<1, y<1 and q=3. For q>5, our result is intriguing because it shows that there is no FPRAS at (x,y)=(1-q/(1+epsilon),-epsilon) for any positive epsilon but it leaves open the limit point epsilon=0, which corresponds to approximately counting q-colourings of a planar graph.

cs.CC↗

arXiv2010

The Complexity of Approximating Bounded-Degree Boolean #CSP (Extended Abstract)

The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum degree is at least 25 we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

cs.CC↗

arXiv2010

A complexity dichotomy for hypergraph partition functions

We consider the complexity of counting homomorphisms from an $r$-uniform hypergraph $G$ to a symmetric $r$-ary relation $H$. We give a dichotomy theorem for $r>2$, showing for which $H$ this problem is in FP and for which $H$ it is #P-complete. This generalises a theorem of Dyer and Greenhill (2000) for the case $r=2$, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation $H$ is weighted, and the goal is to compute the \emph{partition function}, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of $r$ sites interact symmetrically. In the weighted case, our dichotomy theorem generalises a result of Bulatov and Grohe (2005) for graphs, where $r=2$. When $r=2$, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all $r>2$ the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.

cs.CC↗

arXiv2009

An approximation trichotomy for Boolean #CSP

We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving reductions in the complexity class #RHΠ_1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximation-preserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP.

cs.CC↗

arXiv2009

The Complexity of Weighted Boolean #CSP with Mixed Signs

We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight function as a product of the form (-1)^s g, where the polynomial s determines the sign of the weight and the non-negative function g determines its magnitude. We show that the problem of computing the partition function (the sum of the weights of all possible variable assignments) is in polynomial time if either every weight function can be defined by a "pure affine" magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of "product type" with a linear sign polynomial. In all other cases, computing the partition function is FP^#P-complete.

cs.CC↗

arXiv2009

A complexity dichotomy for partition functions with mixed signs

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition functions of certain "spin glass" models of statistical physics such as the Ising model. Building on earlier work by Dyer, Greenhill and Bulatov, Grohe, we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices -- these turn out to be central in our proofs -- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those "representable" by a quadratic polynomial over the field GF(2).

cs.CC↗

arXiv2009

Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.

math.PR↗

arXiv2008

The Complexity of Weighted Boolean #CSP

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that the partition function, i.e. the sum of the weights of all configurations, can be computed in polynomial time if either (1) every function in F is of ``product type'', or (2) every function in F is ``pure affine''. For every other fixed set F, computing the partition function is FP^{#P}-complete.

cs.CC↗

arXiv2008

The Mixing Time of Glauber Dynamics for Colouring Regular Trees

We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of the $n$-vertex complete $b$-ary tree when $3\leq q\leq b/2\ln(b)$. We give both upper and lower bounds on the mixing time. For fixed $q$ and $b$, our upper bound is $n^{O(b/\log b)}$ and our lower bound is $n^{Ω(b/q \log(b))}$, where the constants implicit in the $O()$ and $Ω()$ notation do not depend upon $n$, $q$ or $b$.

cs.CC↗

arXiv2007

Inapproximability of the Tutte polynomial

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G;x,y). Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola (x-1)(y-1)=1 and at four special points. We are interested in determining for which points (x,y) there is a "fully polynomial randomised approximation scheme" (FPRAS) for T(G;x,y). Under the assumption RP is not equal to NP, we prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes x<-1 or y<-1 (excluding the easy-to-compute cases mentioned above). Two exceptions to this result are the half-line x<-1, y=1 (which is still open) and the portion of the hyperbola (x-1)(y-1)=2 corresponding to y<-1 which we show to be equivalent in difficulty to approximately counting perfect matchings. We give further intractability results for (x,y) in the vicinity of the origin. A corollary of our results is that, under the assumption RP is not equal to NP, there is no FPRAS at the point (x,y)=(0,1--lambda) when λ>2 is a positive integer. Thus there is no FPRAS for counting nowhere-zero λflows for λ>2. This is an interesting consequence of our work since the corresponding decision problem is in P for example for λ=6.

cs.CC↗

arXiv2007

Frugality ratios and improved truthful mechanisms for vertex cover

In {\em set-system auctions}, there are several overlapping teams of agents, and a task that can be completed by any of these teams. The buyer's goal is to hire a team and pay as little as possible. Recently, Karlin, Kempe and Tamir introduced a new definition of {\em frugality ratio} for this setting. Informally, the frugality ratio is the ratio of the total payment of a mechanism to perceived fair cost. In this paper, we study this together with alternative notions of fair cost, and how the resulting frugality ratios relate to each other for various kinds of set systems. We propose a new truthful polynomial-time auction for the vertex cover problem (where the feasible sets correspond to the vertex covers of a given graph), based on the {\em local ratio} algorithm of Bar-Yehuda and Even. The mechanism guarantees to find a winning set whose cost is at most twice the optimal. In this situation, even though it is NP-hard to find a lowest-cost feasible set, we show that {\em local optimality} of a solution can be used to derive frugality bounds that are within a constant factor of best possible. To prove this result, we use our alternative notions of frugality via a bootstrapping technique, which may be of independent interest.

cs.GT↗

arXiv2007

Distributed Selfish Load Balancing

Suppose that a set of $m$ tasks are to be shared as equally as possible amongst a set of $n$ resources. A game-theoretic mechanism to find a suitable allocation is to associate each task with a ``selfish agent'', and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For $m\gg n$, the system becomes approximately balanced (an $ε$-Nash equilibrium) in expected time $O(\log\log m)$. We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time $O(\log\log m+n^4)$. We also give a lower bound of $Ω(\max\{\log\log m,n\})$ for the convergence time.

cs.GT↗

arXiv2007

Computing Good Nash Equilibria in Graphical Games

This paper addresses the problem of fair equilibrium selection in graphical games. Our approach is based on the data structure called the {\em best response policy}, which was proposed by Kearns et al. \cite{kls} as a way to represent all Nash equilibria of a graphical game. In \cite{egg}, it was shown that the best response policy has polynomial size as long as the underlying graph is a path. In this paper, we show that if the underlying graph is a bounded-degree tree and the best response policy has polynomial size then there is an efficient algorithm which constructs a Nash equilibrium that guarantees certain payoffs to all participants. Another attractive solution concept is a Nash equilibrium that maximizes the social welfare. We show that, while exactly computing the latter is infeasible (we prove that solving this problem may involve algebraic numbers of an arbitrarily high degree), there exists an FPTAS for finding such an equilibrium as long as the best response policy has polynomial size. These two algorithms can be combined to produce Nash equilibria that satisfy various fairness criteria.

cs.GT↗