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Computers And Intractability A Guide To The Theory Of Np Completeness

Membership Inference Attack (MIA) determines the presence of a record in a machine learning model's training data by querying the model Prior work has shown that the attack is feasible when the model is overfitted to its training data or when the adversary controls the training algorithm However, when the model is not overfitted and the adversary does not control the training algorithm, the threat is not well understood In this paper, we report a study that discovers overfitting to be a sufficient but not a necessary condition for an MIA to succeed More specifically, we demonstrate that even a well-generalized model contains vulnerable instances subject to a new generalized MIA (GMIA) In GMIA, we use novel techniques for selecting vulnerable instances and detecting their subtle influences ignored by overfitting metrics Specifically, we successfully identify individual records with high precision in real-world datasets by querying black-box machine learning models Further we show that a vulnerable record can even be indirectly attacked by querying other related records and existing generalization techniques are found to be less effective in protecting the vulnerable instances Our findings sharpen the understanding of the fundamental cause of the problem: the unique influences the training instance may have on the model

Understanding Membership Inferences on Well-Generalized Learning Models

In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. 
We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy. 
On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.

Fixed-parameter tractability, definability, and model checking

Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given ...

New Limits to Classical and Quantum Instance Compression.

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_t$-free unit $d$-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions, witness that the twin-width is at most $d$. We show that FO model checking, that is deciding if a given first-order formula $\phi$ evaluates to true for a given binary structure $G$ on a domain $D$, is FPT in $|\phi|$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$-contraction sequence for $G$, our algorithm runs in time $f(d,|\phi|) \cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS '15].

Twin-width I: tractable FO model checking

The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most $d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in $G$ We show that if a graph admits a $d$-contraction sequence, then it also has a linear-arity tree of $f(d)$-contractions, for some function $f$ First this permits to show that every bounded twin-width class is small, ie, has at most $n!c^n$ graphs labeled by $[n]$, for some constant $c$ This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al, JCTB '06] The second consequence is an $O(\log n)$-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture We then explore the "small conjecture" that, conversely, every small hereditary class has bounded twin-width Inspired by sorting networks of logarithmic depth, we show that $\log_{\Theta(\log \log d)}n$-subdivisions of $K_n$ (a small class when $d$ is constant) have twin-width at most $d$ We obtain a rather sharp converse with a surprisingly direct proof: the $\log_{d+1}n$-subdivision of $K_n$ has twin-width at least $d$ Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width Thirdly we show that cubic expanders obtained by iterated random 2-lifts from $K_4$~[Bilu and Linial, Combinatorica '06] have bounded twin-width, too We suggest a promising connection between the small conjecture and group theory Finally we define a robust notion of sparse twin-width and discuss how it compares with other sparse classes

Twin-width II: small classes

The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most $d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in $G$. We show that if a graph admits a $d$-contraction sequence, then it also has a linear-arity tree of $f(d)$-contractions, for some function $f$. First this permits to show that every bounded twin-width class is small, i.e., has at most $n!c^n$ graphs labeled by $[n]$, for some constant $c$. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. The second consequence is an $O(\log n)$-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the "small conjecture" that, conversely, every small hereditary class has bounded twin-width. Inspired by sorting networks of logarithmic depth, we show that $\log_{\Theta(\log \log d)}n$-subdivisions of $K_n$ (a small class when $d$ is constant) have twin-width at most $d$. We obtain a rather sharp converse with a surprisingly direct proof: the $\log_{d+1}n$-subdivision of $K_n$ has twin-width at least $d$. Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from $K_4$~[Bilu and Linial, Combinatorica '06] have bounded twin-width, too. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width and discuss how it compares with other sparse classes.

Resiliency is a relatively new topic in the context of access control. Informally, it refers to the extent to which a multi-user computer system, subject to an authorization policy, is able to continue functioning if a number of authorized users are unavailable. Several interesting problems connected to resiliency were introduced by Li, Wang and Tripunitara [13], many of which were found to be intractable. In this paper, we show that these resiliency problems have unexpected connections with the workflow satisfiability problem (WSP). In particular, we show that an instance of the resiliency checking problem (RCP) may be reduced to an instance of WSP. We then demonstrate that recent advances in our understanding of WSP enable us to develop fixed-parameter tractable algorithms for RCP. Moreover, these algorithms are likely to be useful in practice, given recent experimental work demonstrating the advantages of bespoke algorithms to solve WSP. We also generalize RCP in several different ways, showing in each case how to adapt the reduction to WSP. Li et al also showed that the coexistence of resiliency policies and static separation-of-duty policies gives rise to further interesting questions. We show how our reduction of RCP to WSP may be extended to solve these problems as well and establish that they are also fixed-parameter tractable.

https://dl.acm.org/doi/pdf/10.1145/2914642.2914650

Resiliency Policies in Access Control Revisited

We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Clique, and $k$-Dominating Set when an $O(1)$-sequence is provided. We further show how to solve weighted $k$-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in time $2^{O(k \log k)}n$. These algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example of this reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, establishing that bounded twin-width classes are $\chi$-bounded. This significantly extends the $\chi$-boundedness of bounded rank-width classes, and does so with a very concise proof. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in sublinear time $O(n \log n)$ and time $O(n^2 \log n)$, respectively.

Twin-width III: Max Independent Set and Coloring

In this paper,
we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases We first show that MIS remains W[1]-hard in graphs forbidding simultaneously 
$$K_{1, 4}$$

, any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices In particular, this answers an open question of Dabrowski et al concerning 
$$C_4$$

-free graphs Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey’s theorem Iterative expansion is a maximization version of the so-called iterative compression We believe that our framework can be of independent interest for solving other similar graph problems Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H

https://hal.archives-ouvertes.fr/hal-03015353/document

Rémi Watrigant

Papers

Twin-width II: small classes

Twin-width II: small classes

Resiliency Policies in Access Control Revisited

Twin-width III: Max Independent Set and Coloring

Parameterized Complexity of Independent Set in H-Free Graphs