LP-Based Parameterized Algorithms for Separation Problems

D. Lokshtanov, N.S. NarayanaswamyV. Raman, M.S. Ramanujan S. Saurabh

Message of this talk

It was open for quite a while whether Odd Cycle Transversal and Almost 2-Sat are FPT.

A simple branching algorithm for Vertex Cover known since the mid 90’s solves both problems in time .

Some more work gives .

Results

(Above LP) Multiway Cut

(Above LP) Vertex Cover

Almost 2-SAT

Odd Cycle Transversal

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2.32𝑘−𝐿𝑃

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How does one get a 4k-LP algorithm?

Branching: on both sides k-LP decreases by at least ½.

How to improve? Decrease k-LP more.

Multiway Cut

In: Graph G, set T of vertices, integer k.Question: such that no component of G\S has at least two vertices of T?

FPT by Marx, 04Faster FPT by Chen et al, 07Fastest FPT and FPT/k-LP by Cygan et al, 11

Vertex Cover

Long story...Here: .

In: G, tQuestion: such that every edge in G has an endpoint in S?

Almost 2-SAT

FPT by Razgon and O’Sullivan, 08Here: .

In: 2-SAT formula, integer kQuestion: Can we remove k variables from and make it satisfiable?

Odd Cycle Transversal

FPT: by Reed et al.Here: .

In: G, kQuestion: such that G\S is bipartite?

Vertex CoverIn: G, tQuestion: such that every edge in G has an endpoint in S?

Minimize ∀𝑢𝑣∈𝐸 (𝐺 ) :𝑥𝑢+𝑥𝑣≥1𝑥𝑖≥0

Z

Vertex Cover Above LP

In: G, tQuestion: such that every edge in G has an endpoint in S?Running Time: , where LP is the value of the optimum LP solution.

𝜇=𝑡− 𝐿𝑃

Odd Cycle Transversal Almost 2-Sat

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Almost 2-SAT Vertex Cover/t-LP

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Nemhauser Trotter Theorem

(a) There is always an optimal solution to Vertex Cover LP that sets variables to .

(b) For any –solution there is a matching from the 1-vertices to the 0-vertices, saturating the 1-vertices.

Nemhauser Trotter Proof

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Reduction Rule

If exists optimal LP solution that sets xv to 1, then exists optimal vertex cover that selects v.

Remove v from G and decrease t by 1.

Correctness follows from Nemhauser TrotterPolynomial time by LP solving.

Branching

Pick an edge uv. Solve (G\u, t-1) and (G\v, t-1).

since otherwise there is an optimal LP solution for G that sets u to 1.

Then

Branching - Analysis

LP – t drops by ½ ... in both branches!

Total time:

Caveat: The reduction does not increase the measure!

Moral

Nemhauser Trotter reduction + classic «branch on an edge» gives time algorithm for Vertex Cover and time algorithm for Odd Cycle Transversal and Almost 2-Sat.

Can we do better?

Surplus

The surplus of a set I is |N(I)| – |I|. The surplus can be negative!

In any -LP solution, the total weight is n/2 + surplus(V0)/2.

Solving the Vertex Cover LP is equivalent to finding an independent set I of minimum surplus.

Surplus and Reductions

If «all ½» is the unique LP optimum then surplus(I) > 0 for all independent sets.

Can we say anything meaningful for independent sets of surplus 1? 2? k?

Surplus Branching Lemma

Let I be an independent set in G with minimum surplus. There exists an optimal vertex cover C that either contains I or avoids I.

Surplus Branching Lemma Proof

I N(I) R

Branching Rule

Find an independent set I of minimum surplus. Solve (G\I, t-|I|) and (G\N(I), t-|N(I)|).

LP(G\I) > LP(G) - |I|, since otherwise LP(G) has an optimal solution that sets I to 1.

So

t-LP drops by at least ½.

Branching Rule Analysis Cont’d

Analyzing the (G\N(I), t-N(I)) side:

So

t-LP drops by at least

Branching Summary

The measure k-LP drops by (½, surplus(I)/2).

Will see that independent sets of surplus 1 can be reduced in polynomial time!

Measure drops by (½,1) giving a time algorithm for Vertex Cover

Reducing Surplus 1 sets.Lemma: If surplus(I) = 1, I has minimum surplus and N(I) is not independent then there exists an optimum vertex cover containing N(I).

I N(I)R

Reducing Surplus 1 sets.Reduction Rule: If surplus(I) = 1, I has minimum surplus and N(I) is independent then solve (G’,t-|I|) where G’ is G with N[I] contracted to a single vertex v.

I N(I)R

Summary

Nemhauser Trotter lets us assume surplus > 0

More rules let us assume surplus > 1 ()*

If surplus then branching yields time for Vertex Cover

The correctness of these rules were also proved by NT!

Can we do better?

Can get down to by more clever branching rules. Yields for Almost 2-SAT and Odd Cycle Transversal.

Should not be the end of the story.

Better OCT?

Can we get down to for Odd Cycle Transversal?

LP Branching in other cases

I believe many more problems should have FPT algorithms by LP-guided branching.

What about ... (Directed) Feedback Vertex Set, parameterized by solution size k?

Thank You!