arXiv subjects
Publications and source records attributed to Petteri Kaski.
We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The objective is to maximise $ω$ subject to $Ax \le 1$, $Cx \ge ω1$, and $x \ge 0$ for nonnegative matrices $A$ and $C$. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.
We study the applicability of distributed, local algorithms to 0/1 max-min LPs where the objective is to maximise ${\min_k \sum_v c_{kv} x_v}$ subject to ${\sum_v a_{iv} x_v \le 1}$ for each $i$ and ${x_v \ge 0}$ for each $v$. Here $c_{kv} \in \{0,1\}$, $a_{iv} \in \{0,1\}$, and the support sets ${V_i = \{v : a_{iv} > 0 \}}$ and ${V_k = \{v : c_{kv}>0 \}}$ have bounded size; in particular, we study the case $|V_k| \le 2$. Each agent $v$ is responsible for choosing the value of $x_v$ based on information within its constant-size neighbourhood; the communication network is the hypergraph where the sets $V_k$ and $V_i$ constitute the hyperedges. We present a local approximation algorithm which achieves an approximation ratio arbitrarily close to the theoretical lower bound presented in prior work.
In a bipartite max-min LP, we are given a bipartite graph $\myG = (V \cup I \cup K, E)$, where each agent $v \in V$ is adjacent to exactly one constraint $i \in I$ and exactly one objective $k \in K$. Each agent $v$ controls a variable $x_v$. For each $i \in I$ we have a nonnegative linear constraint on the variables of adjacent agents. For each $k \in K$ we have a nonnegative linear objective function of the variables of adjacent agents. The task is to maximise the minimum of the objective functions. We study local algorithms where each agent $v$ must choose $x_v$ based on input within its constant-radius neighbourhood in $\myG$. We show that for every $ε>0$ there exists a local algorithm achieving the approximation ratio ${Δ_I (1 - 1/Δ_K)} + ε$. We also show that this result is the best possible -- no local algorithm can achieve the approximation ratio ${Δ_I (1 - 1/Δ_K)}$. Here $Δ_I$ is the maximum degree of a vertex $i \in I$, and $Δ_K$ is the maximum degree of a vertex $k \in K$. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.
The deletion--contraction algorithm is perhaps the most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin--Kasteleyn in statistical physics. Prior to this work, deletion--contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph. Here, we give a substantially faster algorithm that computes the Tutte polynomial--and hence, all the aforementioned invariants and more--of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Graham's cover polynomial. An implementation of the algorithm outperforms deletion--contraction also in practice.
We study ways to expedite Yates's algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an $n$-element universe $U$ and a family $\scr F$ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of $U$ with $k$ sets from $\scr F$ in time within a polynomial factor (in $n$) of the number of supersets of the members of $\scr F$. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree $Δ$. In particular, we show how to compute the Domatic Number in time within a polynomial factor of $(2^{Δ+1-2)^{n/(Δ+1)$ and the Chromatic Number in time within a polynomial factor of $(2^{Δ+1-Δ-1)^{n/(Δ+1)$. For any constant $Δ$, these bounds are $O\bigl((2-ε)^n\bigr)$ for $ε>0$ independent of the number of vertices $n$.
We establish by means of a computer search that a complete graph on 14 vertices has 98,758,655,816,833,727,741,338,583,040 distinct and 1,132,835,421,602,062,347 nonisomorphic one-factorizations. The enumeration is constructive for the 10,305,262,573 isomorphism classes that admit a nontrivial automorphism.
Let $Ψ(t,k)$ denote the set of pairs $(v,λ)$ for which there exists a graphical $t$-$(v,k,λ)$ design. Most results on graphical designs have gone to show the finiteness of $Ψ(t,k)$ when $t$ and $k$ satisfy certain conditions. The exact determination of $Ψ(t,k)$ for specified $t$ and $k$ is a hard problem and only $Ψ(2,3)$, $Ψ(2,4)$, $Ψ(3,4)$, $Ψ(4,5)$, and $Ψ(5,6)$ have been determined. In this paper, we determine completely the sets $Ψ(2,5)$ and $Ψ(3,5)$. As a result, we find more than 270000 inequivalent graphical designs, and more than 8000 new parameter sets for which there exists a graphical design. Prior to this, graphical designs are known for only 574 parameter sets.
We study the performance of stochastic local search algorithms for random instances of the $K$-satisfiability ($K$-SAT) problem. We introduce a new stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by {\em never going upwards} in energy. ChainSAT is a \emph{focused} algorithm in the sense that it considers only variables occurring in unsatisfied clauses. We show by extensive numerical investigations that ChainSAT and other focused algorithms solve large $K$-SAT instances almost surely in linear time, up to high clause-to-variable ratios $α$; for example, for K=4 we observe linear-time performance well beyond the recently postulated clustering and condensation transitions in the solution space. The performance of ChainSAT is a surprise given that by design the algorithm gets trapped into the first local energy minimum it encounters, yet no such minima are encountered. We also study the geometry of the solution space as accessed by stochastic local search algorithms.
A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise $\min_k \sum_v c_{kv} x_v$ subject to $\sum_v a_{iv} x_v \le 1$ for each $i$ and $x_v \ge 0$ for each $v$. Here $c_{kv} \ge 0$, $a_{iv} \ge 0$, and the support sets $V_i = \{v : a_{iv} > 0 \}$, $V_k = \{v : c_{kv}>0 \}$, $I_v = \{i : a_{iv} > 0 \}$ and $K_v = \{k : c_{kv} > 0 \}$ have bounded size. In the distributed setting, each agent $v$ is responsible for choosing the value of $x_v$, and the communication network is a hypergraph $\mathcal{H}$ where the sets $V_k$ and $V_i$ constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if $|V_i|$ and $|V_k|$ are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in $\mathcal{H}$.
We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N, compute their subset convolution f*g, defined for all S\subseteq N by (f * g)(S) = \sum_{T \subseteq S}f(T) g(S\setminus T), where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n^2 2^n) additions and multiplications, substantially improving upon the straightforward O(3^n) algorithm. Specifically, if the input functions have an integer range {-M,-M+1,...,M}, their subset convolution over the ordinary sum-product ring can be computed in O^*(2^n log M) time; the notation O^* suppresses polylogarithmic factors. Furthermore, using a standard embedding technique we can compute the subset convolution over the max-sum or min-sum semiring in O^*(2^n M) time. To demonstrate the applicability of fast subset convolution, we present the first O^*(2^k n^2 + n m) algorithm for the minimum Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O^*(3^k n + 2^k n^2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O^*(2^n)-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).
A local search algorithm operating on an instance of a Boolean constraint satisfaction problem (in particular, k-SAT) can be viewed as a stochastic process traversing successive adjacent states in an ``energy landscape'' defined by the problem instance on the n-dimensional Boolean hypercube. We investigate analytically the worst-case topography of such landscapes in the context of satisfiable k-SAT via a random ensemble of satisfiable ``k-regular'' linear equations modulo 2. We show that for each fixed k=3,4,..., the typical k-SAT energy landscape induced by an instance drawn from the ensemble has a set of 2^{Ω(n)} local energy minima, each separated by an unconditional Ω(n) energy barrier from each of the O(1) ground states, that is, solution states with zero energy. The main technical aspect of the analysis is that a random k-regular 0/1 matrix constitutes a strong boundary expander with almost full GF(2)-linear rank, a property which also enables us to prove a 2^{Ω(n)} lower bound for the expected number of steps required by the focused random walk heuristic to solve typical instances drawn from the ensemble. These results paint a grim picture of the worst-case topography of k-SAT for local search, and constitute apparently the first rigorous analysis of the growth of energy barriers in a random ensemble of k-SAT landscapes as the number of variables n is increased.