Approximation Algorithms and Hardness for Strong Unique Games

Ghoshal, Suprovat; Louis, Anand

doi:10.1137/1.9781611976465.26

Cited by 4 publications

(6 citation statements)

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“…The above theorem illustrates the tractability of the STRONGUNIQUEGAMES problem in the setting where just the instance induced on the satisfiable set has low threshold rank. To put the above result in perspective, [GL21] showed that given a STRONGUNIQUEGAMES instance with value (1 − δ), it is Unique Games hard to output a subset of relative size (1 − Ω( δ log d log k)), where d is the maximum degree of the graph and k is the label set size. We remark that the exponent in the fraction of vertices deleted (i.e., δ 1/12 ) might be improvable and we have not made further attempts towards optimizing it.…”

Section: Our Resultsmentioning

confidence: 99%

“…For both ODDCYCLETRANSVERSAL as well as BALANCEDVERTEXSEPARATOR, the best known approximation algorithm for general instances have an approximation guarantee of O( log |V|) [FHL08,ACMM05]. Furthermore, [GL21] showed that given a (1 − δ)satisfiable instance of ODDCYCLETRANSVERSAL, assuming UGC, it is NP-Hard to find set of size (1 − Ω( δ log d)) which induces a bipartite graph. It is important to note that our results hold for more restrictive setting where we assume the low threshold rank guarantee on the good set.…”

Section: Our Resultsmentioning

confidence: 99%

“…Strong Unique Games Ghoshal and Louis [GL21] gave an algorithm that takes as input an instance of STRONGUNIQUEGAMES having a set of size (1 − ε)n such that the instance induced on that set has value 1, and outputs a set of size at least 1 − Õ k 2 ε log n n such that instance induced on the set has value 1. They gave another algorithm that produced a set of size 1 − Õ k 2 ε log d n such that the instance instance on that set has value 1, where d is the largest vertex degree of the instance.…”

Section: Related Workmentioning

confidence: 99%

“…ODDCYCLETRANSVERSAL is a well studied problem. In general, it is known be constant factor inapproximable [BK09] (assuming the Unique Games Conjecture), and the best known upper bounds (in terms of fraction of vertices deleted) are O(δ log |V|) [ACMM05] and O( δ log d) [GL21] -where δ is the optimal fraction of vertices to be deleted and d is the maximum degree of the graph -the latter bound is also tight upto constant factors assuming the Unique Games Conjecture [GL21]. Given these worst case bounds, one might ask if there are natural classes of instances under which ODDCYCLETRANSVERSAL admits better approximation?…”

Section: Introductionmentioning

confidence: 99%

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Approximating CSPs with Outliers

Ghoshal¹,

Louis²

2022

Preprint

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Constraint satisfaction problems (CSPs) are ubiquitous in theoretical computer science. We study the problem of STRONG-CSPs, i.e. instances where a large induced sub-instance has a satisfying assignment. More formally, given a CSP instance G(V, E, [k], {Π ij } (i,j)∈E ) consisting of a set of vertices V, a set of edges E, alphabet [k], a constraint Π ij ⊂ [k] × [k] for each (i, j) ∈ E, the goal of this problem is to compute the largest subset S ⊆ V such that the instance induced on S has an assignment that satisfies all the constraints.In this paper, we study approximation algorithms for UNIQUEGAMES and related problems under the STRONG-CSP framework when the underlying constraint graph satisfies mild expansion properties. In particular, we show that given a STRONGUNIQUEGAMES instance whose optimal solution S * is supported on a regular low threshold rank graph, there exists an algorithm that runs in time exponential in the threshold rank, and recovers a large satisfiable sub-instance whose size is independent on the label set size and maximum degree of the graph. Our algorithm combines the techniques of , with several new ideas and runs in time exponential in the threshold rank of the optimal set. A key component of our algorithm is a new threshold rank based spectral decomposition, which is used to compute a "large" induced subgraph of "small" threshold rank; our techniques build on the work of Oveis Gharan and Rezaei (SODA'17) and could be of independent interest. there are spectral characterizations under which ODDCYCLETRANSVERSAL (and more generally, STRONG-CSP's) admit improved approximation. In particular, we study instances which are expanding, or more generally, have low threshold rank 2 . Formally, the threshold rank of a graph is defined as follows.Definition 1.2 (Threshold rank) Given an undirected graph G = (V, E), let A denote its weighted adjacency matrix G and let D denote the diagonal matrix where D(i, i) is the weighted degree of vertex i. The (1 − ε) threshold rank of G, denoted by rank ≥1−ε (G) is defined as the number of eigenvalues of D − 1 2 AD − 1 2 that are greater than or equal to 1 − ε.In the setting of CSPs, low threshold rank instances have been studied extensively -the study of such instances was instrumental in the development of sub-exponential time algorithms for UNIQUEGAMES and SMALLSETEDGEEXPANSION [Kol10, ABS15, BRS11]. In particular, for the edge deletion analogue of ODDCYCLETRANSVERSAL i.e., MAX-CUT, [BRS11] gave a (1/λ t )approximation algorithm running in time n poly (t) , where λ t is the t th smallest eigenvalue of the normalized Laplacian. Surprisingly, to the best of our knowledge, no such analogous results are known for ODDCYCLETRANSVERSAL. Furthermore, random instances of CSPs are expanding, and naturally have low threshold rank. This motivates us to explore the approximability of ODDCYCLETRANSVERSAL and other STRONG-CSP's in low threshold instances. In fact, we study them under the more stringent setting where only the graph induced on good vertices (constituting t...

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximating CSPs with Outliers

Ghoshal¹,

Louis²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the worst-case instance of the problem, the work [ACMM05] gives an algorithm that computes a set with at least 1 − O ε log n fraction of vertices which induces a bipartite graph, when it is promised that the graph contains an induced bipartite graph having (1 − ε) n fraction of the vertices. The work [GL21] gives an efficient randomized algorithm that computes an induced bipartite subgraph having 1 − O ε log d fraction of the vertices where d is the bound on the maximum degree of the graph. They also give a matching (up to constant factors) Unique Games hardness for certain regimes of parameters.…”

Section: Introductionmentioning

confidence: 99%