Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

Jansen, Bmp Bart; Pieterse, Astrid

doi:10.1007/s00453-016-0189-9

Cited by 14 publications

(9 citation statements)

References 20 publications

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“…A key result in this area is Schaefer's dichotomy theorem [20], which classifies each CSP over the Boolean domain as polynomial-time solvable or NP-complete. Continuing a recent line of investigation [12,14,17], we aim to understand for which NP-complete CSPs an instance can be sparsified in polynomial time, without changing the answer. In particular, we investigate the following questions.…”

Section: Introduction Backgroundmentioning

confidence: 99%

“…This pessimistic state of affairs concerning nontrivial sparsification algorithms changed several years ago, when a subset of the authors [14] showed that the d-Not-All-Equal SAT problem does have a nontrivial sparsification. In this problem, clauses have size at most d and are satisfied if the literals do not all evaluate to the same value.…”

Section: Introduction Backgroundmentioning

confidence: 99%

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Best-Case and Worst-Case Sparsifiability of Boolean CSPs

2020

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We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation (the constraint language). Two algorithmic results are presented. The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR (up to negations). Our second result concerns linear sparsification, that is, a reduction to an equivalent instance with O(n) constraints. Using linear algebra over rings of integers modulo prime powers, we give an elegant necessary and sufficient condition for a constraint type to be captured by a degree-1 polynomial over such a ring, which yields linear sparsifications. The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs. For NP-complete Boolean CSPs whose constraints are symmetric (the satisfaction depends only on the number of 1 values in the assignment, not on their positions), we give a complete characterization of which constraint languages allow for a linear sparsification. For Boolean CSPs in which every constraint has arity at most three, we characterize the optimal size of sparsifications in terms of the largest OR that can be expressed by the constraint language. ACM Subject ClassificationComputing methodologies → Symbolic and algebraic algorithms, Theory of computation → Problems, reductions and completeness, Theory of computation → Parameterized complexity and exact algorithms

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Section: Introduction Backgroundmentioning

confidence: 99%

Section: Introduction Backgroundmentioning

confidence: 99%

Best-Case and Worst-Case Sparsifiability of Boolean CSPs

2020

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“…The problem Dominating Set takes as an input a graph G and an integer k, and the goal is to decide whether there exists X ⊆ V (G) of size at most k, such that for each v ∈ V (G), X ∩ N [v] = ∅. Jansen and Pieterse proved that Dominating Set does not admit a compression of bit size O(n 2− ), for any > 0 unless NP ⊆ coNP/poly, where n is the number of vertices in the input graph [15]. This result directly implies the following (see, for instance [1], for a formal statement).…”

Section: Lower Bounds On the Size Of Kernelsmentioning

confidence: 99%

Parameterized Complexity of Maximum Edge Colorable Subgraph

Agrawal¹,

Kundu²,

Sahu³

et al. 2020

Preprint

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A graph H is p-edge colorable if there is a coloring ψ : E(H) → {1, 2, . . . , p}, such that for distinct uv, vw ∈ E(H), we have ψ(uv) = ψ(vw). The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p, and the objective is to find a subgraph H of G and a p-edge-coloring of H, such that |E(H)| ≥ l. We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G, by using Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to Rainbow Matching, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p + k, where k is one of the following: (1) the solution size, l, (2) the vertex cover number of G, and (3) l − mm(G), where mm(G) is the size of a maximum matching in G; we show that the (decision version of the) problem admits a kernel with O(k • p) vertices. Furthermore, we show that there is no kernel of size O(k 1− • f (p)), for any > 0 and computable function f , unless NP ⊆ coNP/poly.

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“…Having presented the gadget we use in our construction, we define the source problem for the cross-composition. This problem was also used as the starting problem for a crosscomposition in our earlier sparsification lower bound for 4-Coloring [15]. [15] Input: A graph G with a partition of its vertex set into U ∪ V such that G[U ] is an edgeless graph and G[V ] is a disjoint union of triangles.…”

Section: Sparsification Lower Bound For 3-coloringmentioning

confidence: 99%

“…Our second main result concerns the parameterization by the number of vertices n. The current authors showed in earlier work [15] that for a number of graph problems it is impossible to give a kernel of size O(n 2−ε ), unless NP ⊆ coNP/poly. This implies that the number of edges cannot efficiently be reduced to a subquadratic amount without changing the answer, a task that is also known as sparsification.…”

Section: Introductionmentioning

confidence: 98%

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

Jansen

Pieterse

2019

Algorithmica

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The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring.First, we obtain a kernel of bitsize O(k q−1 log k) for q-Coloring parameterized by Vertex Cover for any q ≥ 3. This size bound is optimal up to k o(1) factors assuming NP ⊆ coNP/poly, and improves on the previous-best kernel of size O(k q ). We generalize this result for deciding q-colorability of a graph G, to deciding the existence of a homomorphism from G to an arbitrary fixed graph H. Furthermore, we can replace the parameter vertex cover by the less restrictive parameter twin-cover. We prove that H-Coloring parameterized by Twin-Cover has a kernel of size O(k ∆(H) log k).Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP ⊆ coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n 2−ε ) for any ε > 0. Previously, such a lower bound was only known for coloring with q ≥ 4 colors. We show how to extend this coloring to G. Since we assumed that |P | ≤ ω(H), H has a clique X of size |P |. Use the colors of X to assign a different color to each vertex in P . Since N G (P ) = ∅, this gives a proper H-coloring of G.Lemma 20. Applying Reduction rule 2 or 3 to a graph G does not increase the size of a minimum twin-cover of G.Proof. Let P ⊆ V (G). It is easy to see that if S is a twin-cover of G, then it is also a twin-cover of G − P . Thereby, the statement holds for Reduction rule 3.Let Π be the twin decomposition of G and let P = P ∈ Π. Let F := E G (P , P ) be the set of edges between P and P . Let S be a twin-cover of G. We show that S is a twin-cover of G \ F . Clearly, any edges that were previously covered, are still covered. We show that all vertices that were twins in G, are also twins in G \ F to conclude the proof. Let u, v be twins in G, and let P ∈ Π such that u, v ∈ P . If P = P and P = P , it is obvious that u

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Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

Cited by 14 publications

References 20 publications

Best-Case and Worst-Case Sparsifiability of Boolean CSPs

Best-Case and Worst-Case Sparsifiability of Boolean CSPs

Parameterized Complexity of Maximum Edge Colorable Subgraph

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

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