Abstract

We introduce a new algebraic sieving technique to detect constrained multilinear monomials in multivariate polynomial generating functions given by an evaluation oracle. The polynomials are assumed to have coefficients from a field of characteristic two. As applications of the technique, we show an $$O^*(2^k)$$O?(2k)-time polynomial space algorithm for the $$k$$k-sized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, Min-Substitute Graph Motif, and Min-Add Graph Motif. Finally, we provide a piece of evidence that our result might be essentially tight: the existence of an $$O^*((2-\epsilon )^k)$$O?((2-∈)k)-time algorithm for the Graph Motif problem implies an $$O((2-\epsilon ')^n)$$O((2-∈?)n)-time algorithm for Set Cover.

Highlights

  • Many hard combinatorial problems can be reduced to the framework of detecting whether a multivariate polynomial P(x) = P(x1, x2, . . . , xn) has a monomial with specific properties of interest

  • Our objectives in this paper are to (1) present an improved algebraic technique for constrained multilinear detection in polynomials over fields of characteristic 2, (2) generalize the technique to allow for approximate matching at cost, and (3) derive improved algorithms for graph motif problems, together with evidence that our algorithms may be optimal in the exponential part of their running time

  • We introduce a new common generalization—the closest graph motif problem—that tracks the weighted edit distance between the target motif and each candidate pattern; this in particular generalizes both the minimum substitution and minimum addition variants of the graph motif problem introduced by Dondi et al [10]

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Summary

Introduction

Xn) has a monomial with specific properties of interest In such a setup, P(x) is not available in explicit symbolic form but is implicitly defined by the problem instance at hand, and our access to P(x) is restricted to having an efficient algorithm for computing values of P(x) at points of our choosing. Our objectives in this paper are to (1) present an improved algebraic technique for constrained multilinear detection in polynomials over fields of characteristic 2, (2) generalize the technique to allow for approximate matching at cost, and (3) derive improved algorithms for graph motif problems, together with evidence that our algorithms may be optimal in the exponential part of their running time. All the algebraic contributions rely essentially on what can be called the “substitution-sieving” method in characteristic 2 [2,4]

Multilinearity
Constrained Multilinearity
Cost-Constrained Multilinearity
Graph Motif Problems
Background
Weighted Edit Distance and the Closest Motif Problem
A Lower Bound
Algebraic Tools
Proof of Lemma 3
Remarks
An Algorithm for the Maximum Graph Motif Problem
Branching Walks
A Generating Polynomial for Branching Walks
Evaluating the Generating Polynomial
The Algorithm
Running Time
Correctness
Minor Variants and Extensions
An Algorithm for the Closest Graph Motif Problem
Optimum Edit Sequences
A Lower Bound Reduction from Set Cover
Proof of Theorem 6
Full Text
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