Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cs/0005011

An Average Analysis of Backtracking on Random Constraint Satisfaction Problems

Ke Xu·Wei Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we propose a random CSP model, called Model GB, which is a natural generalization of standard Model B. It is proved that Model GB in which each constraint is easy to satisfy exhibits non-trivial behaviour (not trivially satisfiable or unsatisfiable) as the number of variables approaches infinity. A detailed analysis to obtain an asymptotic estimate (good to 1+o(1)) of the average number of nodes in a search tree used by the backtracking algorithm on Model GB is also presented. It is shown that the average number of nodes required for finding all solutions or proving that no solution exists grows exponentially with the number of variables. So this model might be an interesting distribution for studying the nature of hard instances and evaluating the performance of CSP algorithms. In addition, we further investigate the behaviour of the average number of nodes as r (the ratio of constraints to variables) varies. The results indicate that as r increases, random CSP instances get easier and easier to solve, and the base for the average number of nodes that is exponential in r tends to 1 as r approaches infinity. Therefore, although the average number of nodes used by the backtracking algorithm on random CSP is exponential, many CSP instances will be very easy to solve when r is sufficiently large.

Explore related subjects

cs.CCcs.AI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ke Xu, Wei Li. 2000-05-09. An Average Analysis of Backtracking on Random Constraint Satisfaction Problems. https://arxiv.org/abs/cs/0005011

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Improved Algorithms for the Remote Point Problem

The Remote Point Problem (RPP) is an algorithmic problem that asks, given a linear subspace $L \subseteq \mathbb{F}^n$ of dimension $k$, to deterministically find a vector $v \in \mathbb{F}^n$ far in Hamming distance from $L$. This problem was introduced by Alon, Panigrahy and Yekhanin [APY09], motivated in part by the matrix rigidity approach for proving circuit lower bounds. An algorithm is said to achieve remoteness $d$ if it finds a vector $v$ whose Hamming distance from $L$ is at least $d$. We observe that over the rational numbers, the problem admits a deterministic polynomial-time algorithm that achieves optimal remoteness $n-k$. Over finite fields, we obtain a (modest) improvement of a result of Alon, Panigrahy and Yekhanin [APY09], and give an algorithm that achieves remoteness $Ω\left(\frac{n}{\max\{k, \log n\}} \log n\right)$.

cs.CC↗
arXiv2026

Bit-counting complexity classes

We define bit-counting complexity classes whose membership depends on the binary profile of the number of accepting paths of non-deterministic polynomial time Turing machines. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the complexity class ${\bf PP}$ is contained in our comparison based bit-counting complexity classes ${\bf B_{|0|=|1|}P}$, ${\bf B_{|0|<|1|}P}$ and ${\bf B_{|0|>|1|}P}$. We then show that the comparison based bit-counting complexity classes and the complexity class ${\bf PP}$ are Turing equivalent, that is ${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$. We then prove that the complexity classes ${\bf NP}$ and ${\bf CoNP}$ are contained in both of our parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We also show that the Turing closures of the parity based bit-counting complexity classes coincide, that is ${\bf P}^{{\bf B_{|0|\oplus}P}}={\bf P}^{{\bf B_{|1|\oplus}P}}$. We do this by proving that when either parity based bit-counting complexity class is provided as an oracle for a polynomial time Turing machine, then it can simulate the other one, that is ${\bf B_{|1| \oplus}P}\subseteq {\bf P}^{{\bf B_{|0| \oplus}P}}$ and ${\bf B_{|0| \oplus}P}\subseteq {\bf P}^{{\bf B_{|1| \oplus}P}}$.

cs.CC↗
arXiv2026

Geometric Complexity Theory and Graph Isomorphism

We investigate ideas from the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001) in the setting of graph isomorphism. This provides us a playground of finite combinatorial objects on which to explore these techniques. We seek to separate non-isomorphic pairs of graphs using vector spaces of polynomials that are set-wise invariant under permutations (so-called separating modules). We characterize the power of this method for distinguishing graphs under several different complexity measures: - We show that separating modules of "support-degree" $k$ are equivalent in power to the counts of $O(k)$-vertex subgraphs. - We show that separating modules of symmetric algebraic circuit size $n^{Θ(k)}$ are equivalent to $Θ(k)$-dimensional Weisfeiler-Leman. This generalizes and strengthens the result of Dawar & Wilsenach (CSL '18; ICALP '20; ACM Trans. Comput. Log., 2022; Theory Comput., 2025). - When considering only the representation-theoretic multiplicities of separating modules, we show that two graphs are separated by multiplicities if and only if their automorphism groups have different multiplicity of cycle types (cycle index). The latter result is notable in the analogy with GCT, as it is the only result we are aware of in which the multiplicity approach to separating isomorphism types of objects has been given an "intrinsic" characterization in terms of the objects themselves. We show that for graphs, multiplicity obstructions are stronger than occurrence obstructions. We also connect support size (from the study of WL) to complexity measures on $S_n$ (Dafni, Filmus, Lifshitz, Lindzey, & Vinyals, ITCS '21); as well as connections between invariant polynomials, the Graph Reconstruction Conjectures, and Forman's "invariants of finite type" (Adv. Math., 2004).

cs.CC↗