107 research outputs found
List Locally Surjective Homomorphisms in Hereditary Graph Classes
A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time.
We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family ? consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ? {P?,C?} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ? ? (unless the ETH fails)
Complexity of the List Homomorphism Problem in Hereditary Graph Classes
A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). For a fixed graph H, in the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is equipped with a list L(v) ? V(H). We ask if there exists a homomorphism f from G to H, in which f(v) ? L(v) for every v ? V(G). Feder, Hell, and Huang [JGT 2003] proved that LHom(H) is polynomial time-solvable if H is a so-called bi-arc-graph, and NP-complete otherwise.
We are interested in the complexity of the LHom(H) problem in F-free graphs, i.e., graphs excluding a copy of some fixed graph F as an induced subgraph. It is known that if F is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom(H) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs F.
If F is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if H is not predacious, then for every fixed t the LHom(H) problem can be solved in quasi-polynomial time in P_t-free graphs. On the other hand, if H is predacious, then there exists t, such that the existence of a subexponential-time algorithm for LHom(H) in P_t-free graphs would violate the ETH.
If F is a subdivided claw, we show a full dichotomy in two important cases: for H being irreflexive (i.e., with no loops), and for H being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive H the LHom(H) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if H is non-predacious and triangle-free. On the other hand, if H is reflexive, then LHom(H) cannot be solved in subexponential time whenever H is not a bi-arc graph
Finding Large H-Colorable Subgraphs in Hereditary Graph Classes
We study the \textsc{Max Partial -Coloring} problem: given a graph ,
find the largest induced subgraph of that admits a homomorphism into ,
where is a fixed pattern graph without loops. Note that when is a
complete graph on vertices, the problem reduces to finding the largest
induced -colorable subgraph, which for is equivalent (by
complementation) to \textsc{Odd Cycle Transversal}.
We prove that for every fixed pattern graph without loops, \textsc{Max
Partial -Coloring} can be solved:
in -free graphs in polynomial time, whenever is a
threshold graph;
in -free graphs in polynomial time;
in -free graphs in time ;
in -free graphs in time
.
Here, is the number of vertices of the input graph and is
the maximum size of a clique in~. Furthermore, combining the mentioned
algorithms for -free and for -free
graphs with a simple branching procedure, we obtain subexponential-time
algorithms for \textsc{Max Partial -Coloring} in these classes of graphs.
Finally, we show that even a restricted variant of \textsc{Max Partial
-Coloring} is -hard in the considered subclasses of -free
graphs, if we allow loops on
2-Server PIR with sub-polynomial communication
A 2-server Private Information Retrieval (PIR) scheme allows a user to
retrieve the th bit of an -bit database replicated among two servers
(which do not communicate) while not revealing any information about to
either server. In this work we construct a 1-round 2-server PIR with total
communication cost . This improves over the
currently known 2-server protocols which require communication and
matches the communication cost of known 3-server PIR schemes. Our improvement
comes from reducing the number of servers in existing protocols, based on
Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing
these protocols in an algebraic way (using polynomial interpolation) and
extending them using partial derivatives
On space efficiency of algorithms working on structural decompositions of graphs
Dynamic programming on path and tree decompositions of graphs is a technique
that is ubiquitous in the field of parameterized and exponential-time
algorithms. However, one of its drawbacks is that the space usage is
exponential in the decomposition's width. Following the work of Allender et al.
[Theory of Computing, '14], we investigate whether this space complexity
explosion is unavoidable. Using the idea of reparameterization of Cai and
Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely
related to a conjecture that the Longest Common Subsequence problem
parameterized by the number of input strings does not admit an algorithm that
simultaneously uses XP time and FPT space. Moreover, we complete the complexity
landscape sketched for pathwidth and treewidth by Allender et al. by
considering the parameter tree-depth. We prove that computations on tree-depth
decompositions correspond to a model of non-deterministic machines that work in
polynomial time and logarithmic space, with access to an auxiliary stack of
maximum height equal to the decomposition's depth. Together with the results of
Allender et al., this describes a hierarchy of complexity classes for
polynomial-time non-deterministic machines with different restrictions on the
access to working space, which mirrors the classic relations between treewidth,
pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new
version is augmented with a space-efficient algorithm for Dominating Set
using the Chinese remainder theore
Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs
A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ? V(G) it holds that h(v) ? L(v).
The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. The authors showed that the problem is polynomial-time solvable if H belongs to the class called bi-arc graphs, and for all other graphs H it is NP-complete.
We are interested in the complexity of the LHom(H) problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rz??ewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H, i.e., if every vertex has a loop.
In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHom(H) problem is NP-hard. For every such H we find a constant k = k(H), such that the LHom(H) problem on instances G with n vertices and treewidth t
- can be solved in time k^t ? n^?(1), provided that G is given along with a tree decomposition of width t,
- cannot be solved in time (k-?)^t ? n^?(1), for any ? > 0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|.
Obtaining matching upper and lower bounds shows that the set of algorithmic tools that we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of the LHom(H) problem, e.g. with different parameterizations
SL2 homomorphic hash functions: Worst case to average case reduction and short collision search
We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the
field with elements.
Modulo a well supported number theoretic hypothesis, which holds in particular for concrete
homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions:
upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism.
For a family of homomorphisms containing several concrete proposals in the literature,
we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)).
For general homomorphisms we offer an algorithm that, heuristically and according to experiments,
in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q.
While exponetial time, our algorithms are faster in practice than all earlier generic algorithms,
and produce much shorter collisions
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