Hardness and approximation of submodular minimum linear ordering problems

Abstract

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost

$$f(\cdot )$$

f ( · )

due to an ordering

$$\sigma $$

σ

of the items (say [n]), i.e.,

$$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$

min σ

∑

i ∈ [ n ]

f

(

E

i , σ

)

, where

$$E_{i,\sigma }$$

E

i , σ

is the set of items mapped by

$$\sigma $$

σ

to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a

$$(2-\frac{1+\ell _{f}}{1+|E|})$$

( 2 -

1 +

ℓ f

1 + | E |

)

-approximation for monotone submodular MLOP where

$$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$

ℓ f

=

f ( E )

max

x ∈ E

f

(

{ x }

)

satisfies

$$1 \le \ell _f \le |E|$$

1 ≤

ℓ f

≤

| E |

. Our theory provides new approximation bounds for special cases of the problem, in particular a

$$(2-\frac{1+r(E)}{1+|E|})$$

( 2 -

1 + r ( E )

1 + | E |

)

-approximation for the matroid MLOP, where

$$f = r$$

f = r

is the rank function of a matroid. We further show that minimum latency vertex cover is

$$\frac{4}{3}$$

4 3

-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.